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.

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.

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.

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.

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.

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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.

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).

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).

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.

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.

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.

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.

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.

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.

Abstract: This talk is concerned with general elliptic and parabolic equations with matrix weights and measurable nonlinearities leading to Calderon-Zygmund type estimates

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).

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.

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).

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.

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).

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.

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.

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.

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).

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.)

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.

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.

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).

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.

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.

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.

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.