Jointly organized by Juncheng Wei at Chinese University of Hong Kong and
Jingang Xiong at Beijing Normal University.
Suspended for Fall 2027
Time
Speaker
Title
Zoom
27 Feb 2026, 9:30am Beijing time.
Rafe Mazzeo (Stanford University)
Minimal submanifolds with stratified branching sets
Abstract: Multi-valued harmonic functions arise naturally in several different contexts, including in the description of higher codimension minimal submanifolds. Using some insights that emerged originally in gauge theory I will describe some new ways to construct minimal submanifolds with intricate branching sets. This is joint work with Federico Francheschini and Paul Minter. More broadly, I will also discuss some of the emerging theory of multivalued harmonic functions, as developed by a number of researchers.
ID: 975 8674 7895
Passcode: 331444
6 Mar 2026, 9:30am Beijing time.
Zhiyuan Geng (Purdue University)
Asymptotics for 2D vector-valued Allen-Cahn minimizers
Abstract: For the scalar two-phase Allen-Cahn equation, there is a rich literature on the celebrated De Giorgi conjecture, which reveals deep connections between diffuse interfaces and minimal surfaces. On the other hand, for three or more equally preferred phases, a vector-valued order parameter is required, and the resulting diffuse interfaces are expected to resemble weighted minimal partitions.
In this talk, I will present recent results on minimizers of a 2D Allen-Cahn system with a multi-well potential. We describe the asymptotic behavior near the junction of three phases by analyzing the blow-up limit, which is a global minimizing solution converging at infinity to a Y-shaped minimal cone. We derive sharp energy upper and lower bounds via a novel slicing argument, which allows us to localize the diffuse interface within a small neighborhood of the sharp interface. In addition, we obtain a classification of global 2D minimizers in terms of their blow-down limits at infinity. This is joint work with Nicholas Alikakos.
ID: 985 9910 8529
Passcode: 534602
13 Mar 2026, 9:30am Beijing time.
Mingyang Li (Stony Brook University)
Gravitational instantons and harmonic maps
Abstract: Axisymmetric stationary black holes in general relativity admit a classical reduction to axisymmetric harmonic maps into the hyperbolic plane H^2. In a Riemannian analogue, four-dimensional Ricci-flat metrics with torus symmetry can likewise be described locally by harmonic maps to H^2, now subject to an additional tameness condition that encodes regularity of the metric. We study such harmonic maps and application includes a construction of infinitely many new complete, asymptotically flat, Ricci-flat 4-manifolds with arbitrarily large second Betti number b_2.
If time permits, I will also briefly describe how the same framework can be run in the opposite direction: in low degree one can classify such tame axisymmetric harmonic maps based on our study of gravitational instantons. Joint work with Song Sun.
ID: 932 5027 7484
Passcode: 399689
27 Mar 2026, 9:30am Beijing time.
Robert McCann (University of Toronto)
A geometric approach to apriori estimates for optimal transport maps
Abstract: A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma-Trudinger-Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior modulus of the differential estimate for smooth optimal maps. We describe a new derivation of this estimate with Brendle, Leger and Rankin which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.
ID: 932 8426 6467
Passcode: 417967
16 Apr 2026, 4pm Beijing time.
Serena Dipierro (The University of Western Australia)
Regularity of the trace of nonlocal minimal graphs
Abstract: Nonlocal minimal surfaces are the fractional counterpart of the classical minimizers of the perimeter functional. A special subclass is given by nonlocal minimal graphs, namely nonlocal minimal surfaces that possess a graphical structure.
Because of the presence of long-range interactions, nonlocal minimal graphs behave quite differently from their classical analogues, most notably through the typical appearance of boundary discontinuities. Nevertheless, nonlocal minimal graphs remain continuous from the interior up to the boundary of the reference domain, and therefore admit a boundary trace.
In this talk, we discuss regularity properties of the trace of nonlocal minimal graphs at points of stickiness.
ID: 935 6949 5878
Passcode: 801558
7 May 2026, 4pm Beijing time.
Enrico Valdinoci (The University of Western Australia)
A strict maximum principle for nonlocal minimal surfaces
Abstract: Suppose that two nonlocal minimal surfaces are included one into the other and touch at a point. Then, they must coincide. But this is perhaps less obvious than what it seems at first glance.
ID: 936 4924 5131
Passcode: 289085
Time
Speaker
Title
Zoom
4 Sept 2025, 4pm Beijing time.
Marvin Weidner (University of Bonn)
Optimal regularity for kinetic equations in domains
Abstract: The Boltzmann equation is one of the central equations in statistical mechanics and models the evolution of a gas through particle interactions. In recent years, groundbreaking work by Imbert and Silvestre has led to a conditional regularity theory for periodic solutions of the Boltzmann equation. They established that any possible singularity of a periodic solution to the Boltzmann equation must be visible macroscopically. A major open challenge is whether such a theory can be extended to bounded domains with physically relevant boundary conditions.
As a first step toward understanding the boundary case, in this talk I will discuss the smoothness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection condition. While the interior regularity of such equations is well understood, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov's equation. Finally, I will report on recent joint work with Xavier Ros-Oton, in which we establish sharp boundary regularity results for this class of equations.
ID: 992 0251 2373
Passcode: 129157
12 Sept 2025, 9:30am Beijing time.
In-Jee Jeong (Seoul National University)
Long time dynamics and stability of multi-vortex solution
Abstract: Classical variational approach of maximizing the kinetic energy with various constraints provides vortex stability in several special cases, but in general this approach fails when the vorticity is concentrated at several points in the fluid domain. This is simply because such configurations are not local kinetic energy maximizers, even when we restrict the admissible class using all the other coercive conserved quantities of fluid motion. In this talk, we present several results on the stability of multi-vortex solutions, obtained by combining classical variational approaches with dynamical bootstrapping schemes. We focus on the case of multiple Lamb dipoles weakly interacting with each other. This is based on joint works with Ken Abe, Kyudong Choi, and Yao Yao.
ID: 966 2308 3524
Passcode: 054468
18 Sept 2025, 4pm Beijing time.
Paul Laurain (Universite Gustave Eiffel)
Multi-bubble analysis for the Brezis-Nirenberg problem
Abstract: In a celebrated paper, Brezis and Nirenberg showed that the behavior of the Yamabe-type equation
\[
\Delta u + a u = u^{2^*-1},
\]
where \(2^*\) is the critical Sobolev exponent, can vary dramatically depending on the data of the problem. In particular, there is a notion of a critical potential \(a\), below which no positive solution exists. We are interested in the behavior of solutions as the potential approaches this critical value. We will examine how the location and rate of blow-up of these solutions are constrained by the Robin function of the operator \(\Delta + a\). This is joint work with Tobias König.
ID: 934 2237 4331
Passcode: 644054
26 Sept 2025, 9:30am Beijing time.
Inwon Kim (UCLA)
A model on tumor growth with nutrients
Abstract: We discuss a model that describes tumor growth with nutrient, where the tumor density satisfies an "incompressibility" condition. The model, which can be formulated as a Hele-Shaw type free boundary problem, generates evolution of tumor patches, where its density is uniform. We will discuss interesting features on the dynamics of tumor patch evolution, some established results, and some open questions.
ID: 912 2829 7810
Passcode: 274784
16 Oct 2025, 4pm Beijing time.
Francois Hamel (Aix-Marseille Universite)
Symmetry of Euler flows in cylinders and annuli
Abstract: In this talk, I will discuss some rigidity results for steady incompressible flows away from stagnation in infinite cylinders with tangential boundary conditions. In two-dimensional strips, Euler flows turn out to be parallel flows. In any dimension, Euler or Navier-Stokes potential flows turn out to be constant. I will also discuss various counter-examples without the main assumptions. The case of flows in two-dimensional annuli will also be discussed. The proofs rely on the study of the geometric properties of the streamlines and a combination of ODE and PDE arguments. The talk is based on some joint works with A. Karakhanyan and N. Nadirashvili.
ID: 929 2091 1866
Passcode: 610142
23 Oct 2025, 4pm Beijing time.
José A. Carrillo (University of Oxford)
Stein-Log-Sobolev inequalities for the continuous Stein variational gradient descent method
Abstract: The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between \(H^{-1}(\mathbb{R}^n)\) and \(H^{1}(\mathbb{R}^n)\), which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.
ID: 954 7274 3525
Passcode: 174291
6 Nov 2025, 4pm Beijing time.
Joaquim Serra (ETH)
Recent progress on stable solutions of the Allen-Cahn equation
Abstract: I will present recent results and open problems concerning stable solutions of the Allen-Cahn equation and its free boundary version. In particular, I will discuss the long-standing problem of classifying stable solutions to the Allen-Cahn equation, both with and without area bounds, in low dimensions, and the consequences of these classifications.
I will outline the classical results and highlight more recent developments, emphasizing the main difficulties in the problem and some of the key ideas underlying the proofs of our recent results.
The talk is based on two papers: one joint with Chan, Figalli, and Fernández-Real, and another joint with Florit and Simon.
ID: 990 3365 0021
Passcode: 075159
13 Nov 2025, 4pm Beijing time.
Benjamin Gess (Technische Universität Berlin & Max Planck Institute)
From large deviations around porous media, to PDEs with irregular coefficients, to gradient flow structures
Abstract: We consider the large deviations of the rescaled zero-range process about its hydrodynamic limit, the porous medium equation. This leads to the analysis of the skeleton equation, an energy-critical, degenerate parabolic-hyperbolic PDE with irregular drift. In this talk, we present a robust well-posedness theory for such PDEs based on concepts of renormalized solutions, the equation's kinetic form, and commutator estimates. The relationship of these large deviations principles to a formal gradient flow interpretation of the porous medium equation will be demonstrated by deducing an entropy dissipation equality from the large deviations and reversibility.
ID: 995 8284 2944
Passcode: 046828
21 Nov 2025, 9:30am Beijing time. (This is the last seminar of this semester.)
Tristan C. Collins (University of Toronto)
Boundary regularity of optimal transport maps on convex domains
Abstract: I will discuss a new approach to boundary regularity of optimal transport maps (for the quadratic cost) on convex domains based on ideas from geometric regularity theory and a new monotonicity formula. As an application I will highlight some applications of these ideas to global regularity for non-degenerate densities. This is joint
work with Freid Tong.
.
ID: 916 8504 2077
Passcode: 318482
Time
Speaker
Title
Zoom
27 Feb 2025, 4pm Beijing time.
Laurent Betermin (Institut Camille Jordan, Université Claude Bernard Lyon 1)
On crystallization in the plane for pair potentials with an arbitrary norm
Abstract: Imagine N points on the plane interacting through a pair potential depending on distances, i.e. on the chosen norm. Imagine now that you want to minimize the potential energy of such system and that you wish optimal point configurations to be on a periodic structure, as atoms on a crystal. This is called a crystallization problem. What kind of potential should you choose to get such phenomenon? How the minimizing structure depends on the chosen norm? This is obviously a deep and complicated question that I will discuss in this talk, based on a joint work with Camille Furlanetto (Université Lyon 1).
ID: 916 9421 3932
Passcode: 821196
6 Mar 2025, 4pm Beijing time.
Frédéric Robert (Institut Élie Cartan,
Université de Lorraine)
Localization of bubbling for high order nonlinear equations
Abstract: We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation
\[
P_\alpha{u_\alpha}=\Delta_g^k{u_\alpha} +\text{l.o.t.}=|{u_\alpha}|^{2^\star-2-{\epsilon_\alpha}}{u_\alpha}\text{ in }M
\]
that behave like
\[
{u_\alpha}=u_0+B_\alpha+o(1)\text{ in }H_k^2(M)
\]
where \(B=(B_\alpha)_\alpha\) is a Bubble, also called a Peak. We give obstructions for such a concentration to occur: depending on the dimension, they involve the mass of the associated Green's function or the difference between \(P_\alpha\) and the conformally invariant GJMS operator. The bulk of this analysis is the proof of the pointwise control
\[
|{u_\alpha}(x)|\leq C\|u_0\|_\infty^{(2^\star-1)^2}+C\left(\frac{{\mu_\alpha}^{2}}{{\mu_\alpha}^{2}+d_g(x,{x_\alpha})^{2}}\right)^{\frac{n-2k}{2}}\text{ for all }x\in M\text{ and }\alpha>0,
\]
where \(|{u_\alpha}({x_\alpha})|=\max_M|u_\alpha|\to +\infty\) and \({\mu_\alpha}:=|{u_\alpha}({x_\alpha})|^{-\frac{2}{n-2k}}\). The key to obtain this estimate is a sharp control of the Green's function for elliptic operators involving a Hardy potential.
ID: 965 6529 2221
Passcode: 269806
14 Mar 2025, 9:30am Beijing time.
Yuming Paul Zhang (Auburn University)
A Hele-Shaw type flow with interior and free boundary oscillation
Abstract: We investigate a Hele-Shaw type free boundary problem in one spatial dimension, where heterogeneities appear both on the free boundary and within the interior of the positivity set. Our contributions are twofold. First, we establish well-posedness and a comparison principle for the problem by introducing a novel notion of viscosity flows. Second, assuming the coefficients are stationary ergodic, we prove a stochastic homogenization result. To derive the effective free boundary velocity, we use a new approximation that accounts for interior homogenization as well as for the free boundary propagation. This is joint work with Olga Turanova.
ID: 919 4349 4293
Passcode: 354618
21 Mar 2025, 9:30am Beijing time.
Gong Chen (Georgia Institute of Technology)
Asymptotic stability of the sine-Gordon kink outside symmetry
Abstract: We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \(\varphi 4\) model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes.
We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model.
This is a joint work with Jonas Lührmann (Texas A&M).
ID: 963 5371 1289
Passcode: 246196
28 Mar 2025, 9:30am Beijing time.
Antoine Mellet (University of Maryland)
Phase separation and free boundary problems for aggregation-diffusion phenomena
Abstract: We investigate phase separation phenomena that arise as a result of the competition between nonlocal attraction and local repulsion in classical aggregation-diffusion models (we will mainly discuss the Patlak-Keller-Segel model). We will present some recent results showing that when a large population of organisms is observed at some appropriate scales (in time and space), a sharp interface appears, separating regions of high and low density. The derivation of free boundary problems to describe the motion of this interface shows that the collective dynamics is driven, at macroscopic scales, by surface tension and contact angle phenomena as in classical models of fluid dynamics..
ID: 991 1148 4217
Passcode: 600657
11 Apr 2025, 9:30am Beijing time.
Christopher Henderson (University of Arizona)
Nash inequalities and boundary behavior of kinetic equations
Abstract: Kinetic equations model systems, such as a gas, where particles move through space according to a velocity that is diffusing (due to, say, collisions with other particles). The presence of spatial boundaries in these models causes technical issues because they are first order in the spatial variable and therefore cannot be defined everywhere on the boundary. In this talk, I will present \(L^1-L^\infty\) estimates that yield sharp bounds on the behavior at the spatial boundary. The main estimate is a kinetic version of the Nash inequality. This is a joint work with Giacomo Lucertini and Weinan Wang.
ID: 960 3968 5429
Passcode: 096885
24 Apr 2025, 4pm Beijing time.
Tristan Rivière (ETH Zurich)
Area Variations with Differential Constraints
Abstract: The study of the variation of the area of sub-manifolds under pointwise constraints on the tangent plane has numerous motivation from various branches of mathematics such as control theory, elasticity, symplectic and calibrated geometry or minimal surface theory. We will explain the difficulties for developing the analysis of the PDEs such variational problems are generating and expose various regularity results obtained in this direction so far. We will then come to a recent work in collaboration with Alessandro Pigati in which we are proving that surfaces which are critical points of the area under Legendrian constraint, that is surfaces which are
tangent to a non integrable plane distributions and critical point of the area under this sub-Riemannian constraint, are smooth away from isolated conical singularities. We will then come to the still mysterious problem relative to the location of these conical singularities and explain why it is relevant to fundamental questions in differential geometry which have to do with the realization or not of special homology classes by calibrated surfaces.
ID: 992 3077 4811
Passcode: 896212
8 May 2025, 4pm Beijing time. (This is the last seminar of this semester.)
Junichi Harada (Akita University)
Finite energy solutions to the 6D Fujita equation in the Sobolev critical case
Abstract: We discuss the dynamics near the ground states for the 6D Fujita equation. This result presents a 6D version of the classification in a higher dimensional setting, obtained by Charles Collot - Frank Merle - Pierre Raphael (2017). In contrast to their results (they assume only that \(u_0\in \dot H^1(\mathbb{R}^n)\)), our result requires the additional integrability conditions on the initial data \(u_0\in L^2(\mathbb{R}^n)\). We also point out that the assumption \(u_0\in \dot H^1(\mathbb{R}^n)\) alone is not sufficient for the stabilization of the solution in the case \(n=6\).
ID: 976 9249 9717
Passcode: 816427
Time
Speaker
Title
Zoom
6 Sept 2024, 9:30am Beijing time.
Van Tien Nguyen (National Taiwan University)
Blowup classification for the three-dimensional energy-critical nonlinear heat equation
Abstract: Advanced mathematical techniques have been developed in the last two decades for rigorous constructions of blowup solutions within the class of finite energy. However, the question of a complete classification of blowup dynamics remains open. In this talk, we consider the energy-critical nonlinear heat equation in \(\mathbb{R}^3\) as a model example to illustrate our technique to archive such a blowup classification. It consists of two parts: 1) a review of different techniques for constructing finite-energy blowup solutions, and 2) a rigidity result of the finite-energy blowup.
ID: 937 7346 9392
Passcode: 977902
12 Sept 2024, 4pm Beijing time.
Azahara DeLaTorre (Sapienza Universita di Roma)
Uniqueness of least-energy solutions to the fractional Lane-Emden equation in the ball
Abstract: In this talk we will show the uniqueness of least-energy solutions for the fractional Lane-Emden equation posed in the ball under homogeneous Dirichlet exterior conditions. This is a non-local semilinear equation with a superlinear and subcritical nonlinearity. Existence of positive solutions follows easily from variational methods, but uniqueness is quite complicated. In the local case, the uniqueness of positive solutions follows from the result of Gidas, Ni and Nirenberg. Indeed, by using the moving plane method, they proved radial symmetry of the solutions which allows the application of ODE techniques. In the non-local case, these arguments don't seem to work. Our proof makes use of Morse theory, and it is inspired by some results obtained by C. S. Lin in the '90s. The talk is based on a joint work with Enea Parini.
ID: 996 1061 1987
Passcode: 671240
19 Sept 2024, 4pm Beijing time.
Hardy Chan (Universität Basel)
Local and nonlocal ODEs in the singular fractional Yamabe problem
Abstract: In conformal geometry, the Yamabe problem asks for Yamabe metrics, or conformal metrics of constant scalar curvature. In search of singular Yamabe metrics, one is led to the study of the Lane-Emden equation with a Sobolev-subcritical exponent that depends on the dimension of the singularity. The radial profile, which solves a classical ODE, is well-understood.
One could pose the same problem concerning the fractional curvature, a general notion that includes the scalar curvature, the curvatures associated to Paneitz and GJMS operators, as well as those with non-integer order. For the investigation of the corresponding radial profile, we discuss the development of the nonlocal ODE theory. Apart from the localizing Caffarelli-Silvestre extension, we show that nonlocal ODE can also be understood as a coupled infinite system of second order ODEs. Finally, we also mention a simple while surprising transformation that reduces the nonlocal ODE into almost a scalar first order ODE.
This is a joint work with Azahara DelaTorre.
ID: 999 4939 5972
Passcode: 045553
26 Sept 2024, 4pm Beijing time.
Riccardo Tione (Max Planck Institute)
Non-classical solutions to the \(p\)-Laplace equation
Abstract: In this talk we will consider the \(p\)-Laplace equation, \(\mbox{div}(|Du|^{p-2}Du) = 0\). In particular, we will focus on the very weak solutions, i.e. solutions \(u\) to the \(p\)-Laplace equation with \(u \in W^{1,q}\), where \(\max\{1,p-1\} < q < p\). In 1994, T. Iwaniec and C. Sbordone showed that if \(q\) is sufficiently close to \(p\), then very weak solutions belong to \(W^{1,p}\), and thus are classical solutions. They conjectured the same to happen for any \(\max\{1,p-1\} < q\). In this talk, I will present a positive result which shows that Iwaniec-Sbordone's conjecture is true if the gradient of \(u\) belongs to suitable cones, and next I will sketch the construction of a counterexample for this conjecture if this additional condition is not fulfilled. This is based on a joint work with Maria Colombo.
ID: 958 8198 9311
Passcode: 485387
17 Oct 2024, 4pm Beijing time.
Emmanuel Hebey (CY Cergy Paris Université)
Schrödinger-Proca type systems in the closed setting
Abstract: We discuss our recent work on Schrödinger-Poisson-Proca systems and Bopp-Podolsky-Schrödinger-Proca systems in the setting of 3-dimensional closed manifolds. We plan to explain where the equations come from and then address (in their electro-magneto-static versions) questions related to existence, stability, compactness and strong convergence of the Bopp-Podolsky-Schrödinger-Proca systems to the Schrödinger-Poisson-Proca systems.
ID: 949 8965 6031
Passcode: 625180
24 Oct 2024, 4pm Beijing time.
Jesse Ratzkin (Universität Würzburg)
Quantitative stability of the total Q-curvature functional
Abstract: I will discuss recent work with João Henrique Andrade, Tobias König and Juncheng Wei exploring the stability of the minimizing set of the total Q-curvature functional. The Q-curvature of order k of a Riemannian metric is an analog of the scalar curvature, except that it transforms according to a PDE of order 2k under a conformal change. Thus, just as in the scalar curvature setting, one can minimize the volume-normalized total Q-curvature to produce conformal invariants. We show that the distance of a metric to the minimizing set is controlled by a power of the Q-curvature deficit. Generically this exponent is two, but we also produce interesting examples such that the exponent is strictly larger than two.
ID: 947 2279 1613
Passcode: 552593
31 Oct 2024, 4pm Beijing time.
Tobias Weth (Goethe-Universität Frankfurt)
The Schiffer problem on the cylinder and on the 2-sphere
Abstract: I will discuss a new result on the existence of a family of compact subdomains of the flat cylinder for which the Neumann eigenvalue problem for the Laplacian admits eigenfunctions with constant Dirichlet values on the boundary. These domains have the property that their boundaries have nonconstant principal curvatures. In the context of ambient Riemannian manifolds, our construction provides the first examples of such domains whose boundaries are neither homogeneous nor isoparametric hypersurfaces. The underlying functional analytic approach we have developed overcomes an inherent loss of regularity of the problem in standard function spaces. With the help of this approach, we also construct a related family of subdomains of the 2-sphere. By this we disprove a conjecture of Souam from 2005. This is joint work with M.M. Fall and I.A. Minlend.
ID: 925 0347 5020
Passcode: 723004
7 Nov 2024, 4pm Beijing time.
Miles H. Wheeler (University of Bath)
Overhanging solitary water waves
Abstract: We construct gravity water waves with constant vorticity having the approximate form of a disk joined to a strip by a thin neck. This is the first rigorous existence result for such waves, which have been seen in numerics since the 80s and 90s. Our method is related to the construction of constant mean curvature surfaces through gluing, and involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, Helein, and Pacard. This is joint work with Juan Davila, Manuel del Pino, and Monica Musso.
ID: 937 8510 8632
Passcode: 840183
14 Nov 2024, 4pm Beijing time. (This is the last seminar of this semester.)
Frank Merle (IHES and CY Cergy Paris Université)
On degenerate blow-up profiles for the semilinear heat equation
Abstract: In this talk, we consider together with Hatem Zaag the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped and natural extension of this result.
ID: 932 8354 9312
Passcode: 860254
Time
Speaker
Title
Zoom
23 Feb 2024, 9:30am Beijing time.
Luis Silvestre (University of Chicago)
The Landau equation does not blow up
Abstract: The Landau equation is one of the main equations in kinetic theory. It models the evolution of the density of particles when they are assumed to repel each other by Coulomb potentials. It is a limit case of the Boltzmann equation with very soft potentials. In the space-homogeneous case, we show that the Fisher information is monotone decreasing in time. As a consequence, we deduce that for any initial data the solutions stay smooth and never blow up, closing a well-known open problem in the area.
ID: 967 1736 3318
Passcode: 679489
29 Feb 2024, 4pm Beijing time.
Antonio Fernandez (Universidad Autónoma de Madrid)
A Schiffer-type problem with applications to stationary Euler flows
Abstract: If on a smooth bounded domain \(\Omega\subset\mathbb{R}^2\) there is a nonconstant Neumann eigenfunction \(u\) that is locally constant on the boundary, must \(\Omega\) be a disk or an annulus? This question can be understood as a weaker analog of the well-known Schiffer conjecture, in that the function \(u\) is here allowed to take a different constant value on each connected component of \(\partial \Omega\) yet many of the known rigidity properties of the original problem are essentially preserved. In this talk we provide a negative answer by constructing a family of nontrivial doubly connected domains \(\Omega\) with the above property. Then, we will show how our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial. The talk is based on a joint work with Alberto Enciso, David Ruiz and Pieralberto Sicbaldi.
ID: 975 0876 4296
Passcode: 583722
7 March 2024, 4pm Beijing time.
Christian Seis (Universität Münster, Germany)
Steady vortex rings with surface tension
Abstract: The existence of steady vortex rings for the two-phase Euler equations with surface tension is studied, describing the evolution of a perfect bubble air ring in water. Such objects are created in nature by cetaceans such as dolphins or beluga whales, and they appear to be surprisingly stable configurations. The mathematical model features a vortex sheet on the surface of the air bubble. We construct such vortex rings with small cross sections with the help of an implicit function theorem and derive the asymptotics of various quantities for small cross sections. Joint work with David Meyer (Münster).
ID: 989 5275 7926
Passcode: 340496
14 March 2024, 4pm Beijing time.
Simon Noah Nowak (Universitat Bielefeld, Germany)
Nonlinear nonlocal potential theory at the gradient level
Abstract: We present pointwise gradient potential estimates for a class of nonlinear nonlocal equations related to strongly convex nonlocal energy functionals. Our pointwise estimates imply that the first-order regularity properties of such general nonlinear nonlocal equations coincide with the sharp ones of the fractional Laplacian. The talk is based on joint work with Lars Diening, Kyeongbae Kim and Ho-Sik Lee.
ID: 996 9739 6030
Passcode: 231607
11 April 2024, 4pm Beijing time.
Martin Mayer (Scuola Superiore Meridionale, Italia)
Classical arguments and their limitations in conformal geometry
Abstract: We consider the classical conformally prescribed scalar curvature problem on a closed Riemannian manifold. In the most generic and simple situations we review the relevant formulations, strategies and arguments to show solvability, in particular exhibiting some recent existence results. Complementarily we will see, that the theory as of today is far from being complete, as in most situations and for generic curvature candidate functions the underlying equation has trivial degree, but is solvable, while only a few obstructions are explicitly known.
ID: 942 9965 6235
Passcode: 154986
18 April 2024, 4pm Beijing time.
Xavier Fernandez-Real (EPFL)
On the one-phase problem
Abstract: In this talk we present an introduction to elliptic free boundary problems, with a particular emphasis on the one-phase problem. We will explain the basic regularity theory for it, and compare it with other classical free boundary problems. At the end of the talk, we will introduce some recent results in collaboration with Max Engelstein and Hui Yu on both the Bernstein problem for one-phase free boundaries, and generic uniqueness and regularity for both the one-phase and Alt- Phillips problem.
ID: 927 1816 1052
Passcode: 657193
25 April 2024, 4pm Beijing time. (This is the last seminar of this semester.)
Xavier Lamy (Universite Paul Sabatier, Toulouse)
Generation of vortices in the Ginzburg Landau heat flow
Abstract:
In the two-dimensional Ginzburg-Landau heat flow, a configuration with a
logarithmic energy bound has well-formed vortices and their motion is
well-understood. For an initial condition with a finite number of
nondegenerate zeros, but possibly very high energy, we show that the
initial zeros are conserved and the flow rapidly enters the logarithmic
energy regime. This is joint work with M.Kowalczyk.
ID: 980 2447 4585
Passcode: 367891
Time
Speaker
Title
Zoom
8 Sept 2023, 9:30am Beijing time.
Matthew Gursky (University of Notre Dame)
Deformation of ASD four-manifolds and a nonlinear PDE from spectral theory
Abstract: In this talk I will give a brief overview of the deformation theory for anti-self-dual 4-manifolds, and introduce a geometric differential operator that plays a key role. I will then report on some recent progress toward understanding when the cokernel of this operator vanishes, and the connection to a nonlinear PDE from spectral geometry. This is joint work with Rod Gover.
ID: 952 9345 5527
Passcode: 002473
14 Sept 2023, 4pm Beijing time.
Sunghan Kim (Uppsala University, Sweden)
Constraint maps with free boundaries
Abstract: In this talk, we shall consider maps that minimize the Dirichlet energy subject to constraints on their image. We shall these map (minimizing) constraint maps. In the manifold level, these maps are precisely harmonic maps into manifolds-with-boundary. On the other hand, the minimization problem can also be considered as canonical extension of the classical obstacle problem to the vectorial setting. The constraint maps were considered several decades ago, mainly by F. Duzaar and M. Fuchs, who established the optimal partial regularity theory. The observations were aligned with the development of the theory for harmonic maps. What differentiates the constraint maps from the (usual) harmonic maps (into manifolds without boundary) is the presence of free boundaries. Although they were studied many years ago, only basic properties of free boundaries were studied for the constraint maps. Recently, together with A. Figalli and H. Shahgholian, I took a closer look, from the perspective of free boundary problems, at the behavior of the mappings in the vicinity of their free boundaries. Our result shows some interesting (vectorial) features, which do not (and cannot) arise in the scalar obstacle problems. In this talk, I will give a brief overview on the development and the characters of the constraint maps, and present the recent result, and if time allows, some interesting, new problems in this direction. The talk will be based on the joint works by A. Figalli, A. Guerra and H. Shahgholian.
ID: 920 4113 2301
Passcode: 393276
21 Sept 2023, 4pm Beijing time.
Angela Pistoia (Sapienza Universita di Roma)
Conformal metrics with prescribed curvatures
Abstract: I will present some recent results concerning the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a n-dimensional compact Riemannian manifold.
ID: 972 7880 3385
Passcode: 524995
28 Sept 2023, 4pm Beijing time.
Joaquim Serra (ETH)
Nonlocal approximation of minimal surfaces: optimal estimates from stability
Abstract: Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture.
In two recent papers --- with Chan, Dipierro and Valdinoci, and with Caselli and Florit--- we set the ground for a new approximation based on nonlocal minimal surfaces.
In the first paper, we prove that stable s-minimal surfaces in the unit ball of \(\mathbb{R}^3\) satisfy curvature estimates that are robust as s approaches 1 (i.e. as the energy approaches that of classical minimal surfaces). Moreover, we obtain optimal sheet separation estimates and show that critical interactions are encoded by nontrivial solutions to a (local) "Toda type" system. As a nontrivial application, we establish that hyperplanes are the only stable s-minimal hypersurfaces in \(\mathbb{R}^4\), for s sufficiently close to 1.
In the second paper, we establish the existence of infinitely many nonlocal minimal surfaces in every closed manifold (i.e., a version of Yau's conjecture).
ID: 942 8487 7910
Passcode: 675302
20 Oct 2023, 9:30am Beijing time.
Ao Sun (Lehigh University)
Genus one singularities in mean curvature flow
Abstract: We show that for certain one-parameter families of initial conditions in \(\mathbb{R}^3\), when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. Among applications, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut, and we proved that the fourth lowest entropy self-shrinker in \(\mathbb{R}^3\) can not be rotationally symmetric. Based on joint work with Adrian Chu.
ID: 968 7788 7622
Passcode: 107767
26 Oct 2023, 4pm Beijing time.
Jose M. Espinar (Universidad de Granada, Spain)
Frankel property and Maximum Principle at Infinity for complete minimal hypersurfaces
Abstract: We extend Mazet's Maximum Principle at infinity for parabolic, two-sided, properly embedded minimal hypersurfaces, up to ambient dimension seven. Parabolicity is a necessary condition in dimension \(n \geq 4\), even in Euclidean space, as the example of the higher-dimensional catenoid shows. Next, inspired by the Tubular Neighborhood Theorem of Meeks-Rosenberg in Euclidean three-space we focus on the existence of an embedded \(\epsilon -\)tube when the ambient manifold \(\mathcal{M}\) has non-negative Ricci curvature. These results will allow us to establish Frankel-type properties and to extend the Anderson-Rodriguez Splitting Theorem under the existence of an area-minimizing \({\rm mod}(2)\) hypersurface \(\Sigma\) in these manifolds \(\mathcal{M}\) (up to dimension seven), the universal covering space of \(\mathcal{M}\) is isometric to \(\Sigma \times \mathbb{R}\) with the product metric.
ID: 941 3577 6085
Passcode: 794539
3 Nov 2023, 9:30am Beijing time.
Yilun Wu (University of Oklahoma)
Existence of rotating stars with variable Entropy
Abstract: Rotating stars can be modeled by steady solutions to the Euler-Poisson equations. An extensive literature has established the existence of rotating stars for given differentially rotating angular velocity profiles. However, all of the existing results require the angular velocity to depend on the distance to the rotation axis, but not on the distance to the equatorial plane. Incidentally, all of these solutions have constant entropy within the star. In this talk, I will present a recent result which is the first that allows a general rotation profile, without restrictions. It is also the first result that allows genuinely changing entropy within the star. The variation of entropy causes the previous methods used to construct steady solutions inapplicable. We discover a div-curl reformulation of the problem and perform analysis on the resulting elliptic-hyperbolic system. This is joint work with Juhi Jang and Walter Strauss.
ID: 985 3186 6133
Passcode: 045613
10 Nov 2023, 9:30am Beijing time.
Jeaheang Bang (The University of Texas at San Antonio)
Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications
Abstract:
Solutions with scaling-invariant bounds, such as self-similar solutions, play an important role in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations. We recently proved that any steady solution satisfying \(|u(x)|\leq C/|x|\) for any constant C in \(\mathbb{R}^n\setminus \{0\}\) with \(n\geq 4\), must be zero without imposing a smallness or self-similarity assumption. Our main idea is to analyze the velocity field and the total head pressure via weighted energy estimates with suitable multipliers, and our proof is pretty elementary and short. These results not only give the Liouville-type theorem for steady solutions in higher dimensions but also help to remove a class of singularities of solutions and give the optimal asymptotic behaviors of solutions at infinity in the exterior domains. This is a joint work with Changfeng Gui, Hao Liu, Yun Wang and Chunjing Xie.
ID: 952 0206 9493
Passcode: 305143
23 Nov 2023, 4pm Beijing time. (This is the last seminar of this semester.)
Christophe Prange (Cergy Paris University)
Some recent trends for the regularity of the 3D Navier-Stokes equations
Abstract: In this talk, I will review some recent developments in collaboration with Dallas Albritton (UW Madison), Tobias Barker (U Bath) and Jin Tan (Cergy Paris U) on the regularity for solutions of the three-dimensional Navier-Stokes equations.
ID: 967 2621 5463
Passcode: 887262
Time
Speaker
Title
Zoom
3 Mar 2023, 9am Beijing time.
Sigurd Angenent (University of Wisconsin at Madison)
Ancient solutions to space curve shortening and where they appear
Abstract: Ancient solutions appear as blow-up limits in the study of singularities of geometric flows. In the case of planar and space curve shortening I will give an overview of various ancient solutions that have been constructed, and discuss ways in which they can appear as blow-ups.
ID: 968 2493 3414
Passcode: 542678
10 Mar 2023, 9am Beijing time.
Connor Mooney (University of California at Irvine)
The anisotropic Bernstein problem
Abstract: The Bernstein problem asks whether entire minimal graphs in
R^{n+1} are necessarily hyperplanes. It is known through spectacular
work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De
Giorgi-Giusti that the answer is positive if and only if n < 8. The
anisotropic Bernstein problem asks the same question about minimizers of
parametric elliptic functionals, which are natural generalizations of
the area functional that both arise in many applications and offer
important technical challenges. We will discuss the recent solution of
this problem (the answer is positive if and only if n < 4). This is
joint work with Y. Yang.
ID: 953 3934 3887
Passcode: 232558
17 March 2023, 9am Beijing time.
Robin Neumayer (Carnegie Mellon University)
Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit
Abstract: The Alt-Caffarelli-Friedman (ACF) monotonicity formula is an important tool in the study of free boundary problems. More generally, given any pair of nonnegative subharmonic functions with disjoint positivity sets, the ACF formula provides information about the interface between the supports. In this talk we'll show that on the portion of the interface where the ACF formula is asymptotically positive forms an H^{n-1}-rectifiable set, and that the two functions have unique blowups at H^{n-1} almost every such point. This talk is based on joint work with Mark Allen and Dennis Kriventsov.
ID: 979 3708 6485
Passcode: 336205
24 March 2023, 5pm Beijing time.
María del Mar González (Universidad Autónoma de Madrid)
Spectral properties of Levy Fokker-Planck equations
Abstract: We study the spectrum of a fractional Laplacian equation with drift in suitable weighted spaces. This operator arises when studying the fractional heat equation in self-similar variables. We show, in the radially symmetric case, compactness, and then calculate the eigenfunctions in terms of Laguerre polynomials. The proofs involve Mellin transform and complex analysis methods. This is joint work with H. Chan, M. Fontelos and J. Wei.
ID: 934 7331 0348
Passcode: 692453
31 March 2023, 9am Beijing time.
Zhuolun Yang (Brown University)
Optimal estimates for the conductivity problems with closely spaced inclusions of high contrast
Abstract: In this talk, I will describe an elliptic PDE that models electric conduction, and the electric field concentration phenomenon between closely spaced inclusions of high contrast. In the first part, I will present some results on the insulated conductivity problem (jointly with Hongjie Dong and Yanyan Li). We obtained an optimal gradient estimate in terms of the distance between inclusions. This solved one of the major open problems in this area. In the second part, I will present a recent work regarding optimal estimates for higher derivatives of the conductivity problem with circular inclusions in 2D, when the relative conductivities of inclusions have different signs (Jointly with Hongjie Dong). This improves a recent result of Ji and Kang.
ID: 984 5920 0811
Passcode: 834124
14 April 2023, 9am Beijing time.
Tai-Peng Tsai (University of British Columbia)
Growth rates for axisymmetric Euler flows
Abstract: We consider axisymmetric, swirl-free solutions of the Euler equations in three and higher dimensions, of generalized anti-parallel-vortex-tube-pair-type: the initial scalar vorticity has a sign in the half-space, is odd under reflection across the plane, is bounded and decays sufficiently rapidly at the axis and at spatial infinity. We prove lower bounds on the growth of such solutions in all dimensions, improving a lower bound proved by Choi and Jeong in three dimensions. This is based on joint work with Stephen Gustafson and Evan Miller, arXiv:2303.12043.
ID: 910 4275 9581
Passcode: 154270
21 April 2023, 9am Beijing time.
Yifu Zhou (Johns Hopkins University)
Singularity formation for the Landau-Lifshitz-Gilbert equation in dimension two
Abstract: Landau-Lifshitz-Gilbert equation (LLG), which models the evolution of spin fields in continuum ferromagnetism, can be viewed as a coupling between the harmonic map heat flow and the Schrodinger map flow. In this talk, we shall report some recent gluing construction of finite-time singularities for LLG in dimension two. To overcome the difficulties caused by the dispersion, technical ingredients such as distorted Fourier transform and sub-Gaussian estimates are employed. This is based on a joint work with J. Wei and Q. Zhang.
ID: 968 5118 7301
Passcode: 166862
28 April 2023, 9am Beijing time.
Antonio De Rosa (University of Maryland)
Min-max construction of anisotropic CMC surfaces
Abstract: We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension 3. Joint work with G. De Philippis.
ID: 974 9316 6138
Passcode: 421146
5 May 2023, 4pm Beijing time. (This is the last seminar of this semester.)
Monica Musso (University of Bath)
Leapfrogging for Euler equations
Abstract: We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings, and of a helical filament, associated to a translating-rotating helix. In this talk I will consider the case of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).
ID: 917 3528 8155
Passcode: 158116
Time
Speaker
Title
Zoom
16 Sept 2022, 4pm Beijing time.
Matteo Bonforte (Universidad Autónoma de Madrid)
Stability in Gagliardo-Nirenberg-Sobolev inequalities: nonlinear flows, regularity and the entropy method
Abstract: We discuss stability results in Gagliardo-Nirenberg-Sobolev inequalities, a joint project with J. Dolbeault, B. Nazaret and N. Simonov.
We have developped a new quantitative and costructive "flow method", based on entropy methods and sharp regularity estimates for solutions to the fast diffusion equation (FDE). This allows to study refined versions of the Gagliardo-Nirenberg-Sobolev inequalities that are nothing but explicit stability estimates. Using the quantitative regularity estimates, we go beyond the variational results and provide fully constructive estimates, to the price of a small restriction of the functional space which is inherent to the method.
ID: 943 4890 8003
Passcode: 257490
23 Sept 2022, 4pm Beijing time.
Tobias König (Goethe University Frankfurt)
Reverse conformally invariant Sobolev inequalities on the sphere
Abstract: In this talk I will present a result about the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the n-dimensional sphere S^n involving an operator of order 2s > n. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of s. In particular, we obtain the sharp threshold value of s for the validity of a corresponding Sobolev inequality in all dimensions n >= 2. This is joint work with Rupert L. Frank (LMU Munich and Caltech) and Hanli Tang (Beijing Normal University).
ID: 948 0616 1456
Passcode: 826283
30 Sept 2022, 4pm Beijing time.
Juan Dávila (University of Bath, UK)
Infinite time blow-up for the Keller-Segel system in the plane
Abstract: We study the Keller-Segel system in the plane with an initial
condition with sufficient decay and critical mass 8 pi.
We find a function u0 with mass 8 pi such that
for any initial condition sufficiently close to u0 and mass 8 pi,
the solution is globally defined and blows up in infinite time.
This proves the non-radial stability of the infinite-time blow up
for some initial conditions, answering a question by Ghoul and Masmoudi (2018).
This is joint work with Manuel del Pino (U. of Bath), Jean Dolbeault (U. Paris Dauphine), Monica Musso (U. of Bath) and Juncheng Wei (UBC).
ID: 993 1965 1034
Passcode: 711831
7 Oct 2022, 4pm Beijing time.
Xavier Ros-Oton (Universitat de Barcelona)
The singular set in the Stefan problem
Abstract: The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points. The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.
ID: 996 5119 3566
Passcode: 615823
14 Oct 2022, 9am Beijing time.
Beomjun Choi (POSTECH, South Korea)
Higher order asymptotics for fast diffusions on bounded domains
Abstract: Sobolev-subcritical fast diffusion with vanishing boundary condition leads to finite-time extinction, with a vanishing profile selected by the initial datum. In a joint work with R. McCann and C. Seis, we quantify the rate of convergence to this profile for general smooth bounded domains. In rescaled time variable, the solution either converges exponentially fast or algebraically slow. In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes, giving a higher order asymptotics. We also improve on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the presence of zero modes.
ID: 952 3554 4779
Passcode: 991961
21 Oct 2022, 9am Beijing time.
Xiangwen Zhang (University of California, Irvine)
Geometric flows in symplectic geometry
Abstract: Geometric flows have been proven to be powerful tools in the study of many important problems arising from both geometry and theoretical physics. Aiming to study the equations from the flux compactifications of Type IIA superstrings, we introduce the so-called Type IIA flow, which is a flow of closed and primitive 3-forms on a symplectic Calabi-Yau 6-manifold. Remarkably, the Type IIA flow can also be viewed as a flow as a coupling of the Ricci flow with a scalar field. In this talk, we will discuss the recent progress on this flow. This is based on a joint project with Fei, Phong and Picard.
ID: 913 6112 7888
Passcode: 877592
28 Oct 2022, 9am Beijing time.
Changyou Wang (Purdue University)
Analysis on isotropic-nematic phase transition and liquid crystal droplet
Abstract: In this talk, I will discuss the phase transition phenomena between the isotropic and nematic states within the framework of Ericksen theory of liquid crystals with variable degrees of orientations. Treating it as the singular perturbation problems within the Gamma convergence theory, we will show that the sharp interface formed between isotropic and nematic states is an area minimizing surface. Under suitable assumptions either on the strong anchoring boundary values on the boundary of a bounded domain or the volume constraint of nematic regions in the entire space, we also show that the limiting nematic liquid configuration in the nematic region is a minimizer of the corresponding Oseen-Frank energy with either homeotropic or planar anchoring on the free sharp interface pending on the relative sizes of leading Frank elasticity coefficients. This is a joint work with Fanghua Lin.
ID: 956 3307 8331
Passcode: 648553
4 Nov 2022, 9am Beijing time.
Moon-Jin Kang (KAIST)
Stability of Riemann solution containing a shock under physically admissible perturbations
Abstract: I will present the so-called "a-contraction with shifts” method, which is basically energy based.
This method is quite useful in studying the stability of Riemann solution containing a shock under physically admissible perturbations.
First, based on the method, we prove the long-time behavior of compressible Navier-Stokes flows perturbed from Riemann solution composed of a shock and other waves of different kinds. This resolves the long-standing problem since the 1980s.
On the other hand, since the method can handle large perturbations of a shock, we can also prove the uniform stability of a shock w.r.t. the strength of viscosity, which provides the result on uniqueness of Riemann solution composed of a shock in the class of inviscid limits of solutions to the compressible Navier-Stokes system.
ID: 952 4020 0735
Passcode: 611512
11 Nov 2022, 4pm Beijing time. (This is joint with Jingshi Distinguished Lecture at Beijing Normal University. This is the last seminar of this semester.)
Juan Luis Vázquez (Universidad Autónoma de Madrid)
Nonlinear Diffusion Equations driven by p-Laplacian operators of nonlocal type
Abstract:
We will first review some basic models of nonlinear diffusion. The usual p-Laplacian is a model of strongly nonlinear operator with a well-developed elliptic and parabolic theory. In recent years, interest has focused on versions of this model that include nonlocal effects of fractional type. We describe recent results obtained for the nonlocal parabolic theory for p finite. Finally, we discuss a nonlocal infinity Laplacian model motivated by a tug-of-war process introduced by Bjorland, Caffarelli and Figalli.
ID: 836 7562 6006
Passcode: 221111
Time
Speaker
Title
Zoom
25 Feb 2022, 9am Beijing time.
Sun-Sig Byun (Seoul National University)
Elliptic and parabolic equations with matrix weights and measurable nonlinearities
Abstract: This talk is concerned with general elliptic and parabolic equations with matrix
weights and measurable nonlinearities leading to Calderon-Zygmund type estimates
ID: 921 7807 0034
Passcode: 990108
4 Mar 2022, 9am Beijing time.
Hui Yu (National University of Singapore)
Rate of blow up in the thin obstacle problem
Abstract: The thin obstacle problem is a classical free boundary problem arising from the study of an elastic membrane resting on a lower-dimensional obstacle. Concerning the behavior of the solution near a contact point between the membrane and the obstacle, many important questions remain open. In this talk, we discuss a unified method that leads to a rate of convergence to `tangent cones' at contact points with integer frequencies.
This talk is based on a recent joint work with Ovidiu Savin (Columbia).
ID: 980 1551 7417
Passcode: 446612
11 Mar 2022, 9am Beijing time.
Marcello Lucia (City University of New York, CSI and Graduate Center)
A variational approach to describe the moduli space of minimal
immersions in hyperbolic manifolds
Abstract:
In a seminal paper, Uhlenbeck studied the set of first and second
fundamental form that arise from minimal immersions of a given surface
into a three dimensional hyperbolic manifold. However, as will be explained in
this talk, such data are not suitable for describing the moduli space
of minimal immersions. Following the ideas of Goncalves-Uhlenbeck, one
must instead consider the “dual data” given by the
cohomology class of (0,1) forms. To justify that approach, one is
led to justify the existence and uniqueness of critical points of a
functional whose lack of regularity and compactness do not allow to
apply classical variational results. This is a joint work with Z. Huang and G.Tarantello.
ID: 968 8518 2169
Passcode: 184804
25 Mar 2022, 9am Beijing time.
Xiaodan Zhou (Okinawa Institute of Science and Technology Graduate University)
Horizontally quasiconvex envelope in the Heisenberg group
Abstract: Various notions of convexity of sets and functions in the Heisenberg group have been studied in the past two decades. In this talk, we focus on the horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group. Inspired by the first-order characterization and construction of quasiconvex envelope by Barron, Goebel and Jensen in the Euclidean space, we obtain a PDE approach to construct the h-quasiconvex envelope for a given function f in the Heisenberg group. In particular, we show the uniqueness and existence of viscosity solutions to a non-local Hamilton-Jacobi equation and iterate the equation to obtain the h-quasiconvex envelope. Relations between h-convex hull of a set and the h-quasiconvex envelopes are also investigated. This is joint work with Antoni Kijowski (OIST) and Qing Liu (Fukuoka University/OIST).
ID: 980 4965 4261
Passcode: 495913
1 Apr 2022, 9am Beijing time.
Kyeongsu Choi (Korea Institute for Advanced Study
)
Translating solitons to the power-of-Gauss curvature flow
Abstract : The evolution of convex surfaces by powers of the Gauss curvature is a fully nonlinear parabolic equation. In particular, its translating solitons are complete convex graphs of solutions to a Monge-Ampere type equation. Hence, the classification of translators is a Liouville type theory for a Monge-Ampere type equation. In this talk, we address the existence and classification of translating surfaces by sub-affine-critical powers of the Gauss curvature.
ID: 992 7314 9445
Passcode: 309990
8 Apr 2022, 4pm Beijing time.
Daniele Bartolucci (University of Rome "Tor Vergata", Italy)
On the uniqueness and monotonicity of solutions of Grad-Shafranov
type equations
Abstract: We report about a series of results concerning a Grad-Shafranov type equation, which in dimension 2 describes the equilibrium configurations of a plasma in a Tokamak. In a neat interval depending only by the Sobolev constant of the domain we deduce uniqueness and monotonicity of the boundary density and of a suitably defined energy. Interestingly enough, in dimension 2 we derive the sharp values of the positivity threshold and of the energy upper bound. We also answer open questions about the lack of free boundary and generic properties of the global bifurcation diagram. This is part of a joint research project with A. Jevnikar (Udine), Y. Hu (Changsha), W. Yang (Wuhan).
ID: 910 4994 6922
Passcode: 399051
22 Apr 2022, 4pm Beijing time.
Jose Miguel Urbano (University of Coimbra, Portugal)
Regularity for singular and degenerate PDEs: qualitative vs. sharp estimates
Abstract: Singular and degenerate partial differential equations are unavoidable in the modelling of several phenomena, like phase transitions and chemotaxis, and are also used in machine learning in the context of semi-supervised learning. They encompass a crucial issue in the analysis of PDEs, namely wether we can still derive analytical estimates when the crucial algebraic assumption of ellipticity collapses. We provide a broad overview of qualitative versus quantitative regularity estimates for solutions of these equations, introducing the method of intrinsic scaling and deriving sharp estimates by means of geometric tangential analysis. We discuss, in particular, recent results concerning the Stefan problem and the parabolic p-Poisson equation.
ID: 931 7473 3026
Passcode: 766088
29 Apr 2022, 9am Beijing time.
Xinliang An (National University of Singapore)
Anisotropic Dynamical Horizons Arising in Gravitational Collapse
Abstract: Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou's monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price's law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.
ID: 975 4285 8380
Passcode: 897811
6 May 2022, 4pm Beijing time. (This is the last seminar of this semester.)
Manuel del Pino (University of Bath, UK)
Dynamics of concentrated vorticities in 2D and 3D Euler flows
Abstract: A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions' existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices. We
rigorously establish the law of motion of "leapfrogging vortex rings", initially conjectured by Helmholtz in 1858.
ID: 967 6138 4440
Passcode: 085839
Time
Speaker
Title
Zoom
17 Sept 2021, 9am Beijing time.
Seunghyeok Kim (Hanyang University)
Infinite-time blowing-up solutions to small perturbations of the Yamabe flow
Abstract: In this talk, we will examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold $M$ exists for all time $t$ and uniformly converges to a solution to the Yamabe problem on $M$ as $t \to \infty$. We will observe that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on $M$ in the infinite time. We also concern the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points. This is joint work with Monica Musso (University of Bath, UK).
ID: 974 4590 7096
Passcode: 875622
24 Sept 2021, 9am Beijing time.
Yao Yao (National University of Singapore)
Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states
Abstract:
The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modeled by nonlinear diffusion, and long-range attraction modeled by nonlocal interaction. In this talk, I will discuss several qualitative properties of its steady states and dynamical solutions.
Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint work with Carrillo, Hittmeir and Volzone). In a recent work, we further investigate whether they are unique within the radial class, and show that for a given mass, uniqueness/non-uniqueness of steady states are determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint work with Delgadino and Yan.)
ID: 919 4845 1738
Passcode: 774559
8 Oct 2021, 9am Beijing time.
Yannick Sire (Johns Hopkins University)
Some results on harmonic maps with free boundary and beyond
Abstract: The theory of harmonic maps with free boundary is an old topic in geometric analysis. I will report on recent results on their Ginzburg-Landau approximation, regularity theory, and their heat flow. I will also describe several models in the theory of liquid crystals where the heat flow of those maps appears, emphasizing on some well-posedness issues and some hints on the construction of blow-up solutions. Several important results in geometric analysis such as extremal metrics for the Steklov eigenvalues for instance make a crucial use of such maps. I’ll give some open problems and will try to explain how to attack few open questions in the field using tools recently developed.
ID: 939 4965 3207
Passcode: 405755
15 Oct 2021, 9am Beijing time.
Goro Akagi (Tohoku University)
Energy method for quantitative analysis of rates of convergence to
asymptotic profiles for fast diffusion
Abstract: In this talk, we shall discuss quantitative analysis of asymptotic
behaviors of (possibly sign-changing)
solutions to the Cauchy-Dirichlet problem for the fast diffusion
equation posed on bounded domains
with Sobolev subcritical exponents. More precisely, rates of
convergence to non-degenerate asymptotic
profiles will be revealed via an energy method. The sharp rate of
convergence to positive asymptotic
profiles was recently discussed by Bonforte and Figalli (2021, CPAM)
based on an entropy method.
An alternative proof for their result will also be provided.
ID: 924 9007 6141
Passcode: 639787
22 Oct 2021, 9am Beijing time.
Hiroyoshi Mitake (University of Tokyo)
On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation
Abstract: In this talk, we consider an initial-boundary value problem for the time-fractional diffusion equation.
We show the equivalence of two notions of weak solutions, viscosity solutions and distributional solutions.
It is worth emphasizing that in general the notion of viscosity solutions is based on the comparison principle, while the notion of distributional solutions is based on the variational principle.
Since two notions of weak solutions are introduced in totally different manners, it is highly nontrivial whether two notions are same or not.
In our approach, we use the discrete scheme for time-fractional diffusion equations which was introduced by Giga-Liu-Mitake (Asymptot. Anal. 2020).
A main difficulty is in proving that the error term which comes from the approximated
solution and the distributional solution converges to zero in a suitable weak sense.
The idea to overcome this difficulty is to introduce an approximation of kernel in consideration
of the discrete scheme. Due to the discrete scheme and kernel approximation, we can get the precise error estimate which enables us to get our main theorem.
This is a joint work with Y. Giga (U. Tokyo) and S. Sato (U. Tokyo).
ID: 936 8185 8116
Passcode: 302432
29 Oct 2021, 9am Beijing time.
Dennis Kriventsov (Rutgers University)
Stability for Faber-Krahn inequalities and the ACF formula
Abstract: The Faber-Krahn inequality states that the first Dirichlet
eigenvalue of the Laplacian on a domain is greater than or equal to that
of a ball of the same volume (and if equality holds, then the domain is
a translate of a ball). Similar inequalities are available on other
manifolds where balls minimize perimeter over sets of a given volume. I
will present a new sharp stability theorem for such inequalities: if the
eigenvalue of a set is close to a ball, then the first eigenfunction of
that set must be close to the first eigenfunction of a ball, with the
closeness quantified in an optimal way. I will also explain an
application of this to the behavior of the Alt-Caffarelli-Friedman
monotonicity formula, which has implications for free boundary problems
with multiple phases. This is based on recent joint work with Mark Allen
and Robin Neumayer.
ID: 937 3422 9393
Passcode: 936334
5 Nov 2021, 3pm Beijing time.
Naian Liao (Universität Salzburg)
Continuity of solutions to doubly nonlinear parabolic equations
Abstract: I will report some recent advances on the local continuity of weak solutions to some doubly nonlinear parabolic equations, which include the parabolic p-Laplacian, the porous medium equation and the Stefan problem as particular cases. Various moduli of continuity are obtained depending upon the type of nonlinearity.
ID: 965 2222 8911
Passcode: 630202
12 Nov 2021, 9am Beijing time.
Qing Liu (Fukuoka University)
Principal eigenvalue problem for infinity Laplacian in metric spaces
Abstract: In this talk, we discuss the Dirichlet eigenvalue problem associated to the infinity Laplacian in metric spaces. We provide a direct PDE approach to find the principal eigenvalue and eigenfunctions for a bounded domain in a proper geodesic space with no measure structure. We give an appropriate notion of solutions to the infinity eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process, introduced by Juutinen-Lindqvist-Manfredi (ARMA,1999), via the variational eigenvalue formulation for $p$-Laplacian in the Euclidean space. Several further results and concrete examples will be given in the case of finite metric graphs. This talk is based on joint work with Ayato Mitsuishi at Fukuoka University.
ID: 986 8580 0264
Passcode: 440023
19 Nov 2021, 9am Beijing time. (This is the last seminar of this semester.)
Juncheng Wei (University of British Columbia)
Title: Stability of Sobolev Inequalities and Related Topics
Abstract: In this talk I will discuss the stability of Sobolev inequalities from critical point theory, thereby solving a conjecture of Figalli-Glaudo. I will also report related surprising results on harmonic map inequalities and half-harmonic map inequalities.