Special Session 15: Qualitative properties for solutions to nonlinear elliptic and parabolic equations

Regularity for Local-Nonlocal Elliptic Equations with $(p,q)$-Growth and Matrix Weights

Sun-Sig Byun Seoul National University Korea Co-Author(s):

Abstract: This talk studies mixed local and nonlocal elliptic equations with degenerate and singular matrix weights. Using a frozen-weight comparison method, we obtain higher integrability and gradient estimates while addressing intrinsic geometry and nonlocal tail effects.

Stability of Sobolev inequality and Sobolev equation in the endpoint case.

Lu Chen Beijing Institute of Technology Chile Co-Author(s):

Abstract: In this talk, I will first revisit some new progress on the optimal stability of geometric inequality including Sobolev inequality, HLS inequality and fractional Sobolev inequality etc. Then I will present our recent results on the stability of Sobolev inequality and Sobolev equation in the end-point case. Finally, I will explain the relationship between these quantitative stability inequality and Aubin` conjecture, Chang-Yang conjecture, Keller-Segel equation

Monotonicity for solutions to a fractional nonlinear problem in a half space

Wenxiong Chen Yeshiva University USA Co-Author(s):    Yahong Guo, Leyun Wu

Abstract: We study nonlinear fractional equations $$(-\Delta)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$ to be globally bounded in $\mathbb{R}^n$. We substantially weaken this condition by assuming only that $u$ be bounded in each slab. As a crucial ingredient, we obtained a boundary H\{older} regularity estimate that requires only the boundedness of $u$ near the boundary. This represents a significant improvement over existing results, which often assumed global boundedness of $u$ throughout $\mathbb{R}^n$. To derive the monotonicity, we employ the method of moving planes. We first obtain a narrow region principle in unbounded domains, then we apply averaging effects four times to ensure that the planes can be moved continuously all the way to $x_n = \infty$. Compared with the traditional approaches, methods based on this new technique-the averaging effect require substantially weaker regularity assumptions and can even accommodate unbounded solutions.

Non-radial solutions for the critical quasi-linear Henon equation

Wei Dai Beihang University (BUAA) Peoples Rep of China Co-Author(s):    Lixiu Duan, Changfeng Gui, Yuan Li

Abstract: In this talk, we show the existence of non-radial solutions for the critical quasi-linear Henon equation involving $p$-Laplace operator ($1

Global wellposedness for the Muskat problem

Hongjie Dong Brown University USA Co-Author(s):

Abstract: We study the free boundary problem for 2D and 3D incompressible flow in porous media, known as the one-phase Muskat problem. In the absence of surface tension, we prove that when the initial interface is given by a Lipschitz graph, there exists a unique global Lipschitz strong solution. When surface tension is included, we establish small-data global well-posedness and time-decay results in both the whole-space and periodic settings. This talk is based on joint work with Francisco Gancedo (Universidad de Sevilla), Huy Q. Nguyen (University of Maryland), and Hyunwoo (Will) Kwon (Brown University).

Strichartz estimates and well-posedness for a Schr\odinger equation with a nonlinear Neumann condition

Nicola Garofalo Arizona State University USA Co-Author(s):    Gigliola Staffilani

Abstract: I present the well-posedness in a Cauchy problem for a Schr\odinger equation with a nonlinear Neumann condition. This represents the appropriate Caffarelli-Silvestre extension problem for a nonlocal nonlinear Schr\odinger equation with memory.

Boundary regularity and a priori estimates for fractional equations on unbounded domains

Yahong Guo Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Congming Li, Yugao Ouyang

Abstract: In this talk, we first give a a local version of boundary H\older regularity for nonnegative solutions to the fractional Dirichlet problem in unbounded domains with C^{1,1} boundary. Then as an important application, we establish a priori estimates for solutions to a family of nonlinear fractional equations on unbounded domains with boundaries.

CLASSIFICATION OF SOLUTIONS TO A SINGULAR LANE-EMDEN-FOWLER EQUATION IN THE HALF SPACE

Yahong Guo Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: We focus on the classification of positive solutions to $(-\Delta)^s u=\frac{1}{u^\gamma}$ in the half space with $\gamma>0$, subject to the Dirichlet condition. We show that when $\gamma>1$, all positive solutions exhibit one-dimensional symmetry and are monotone increasing in $x_n$. Moreover, we provide a complete classification of all such one-dimensional solutions via their ``asymptotic $s$-order slope.

The exterior Bernoulli problem for the half Laplacian

Sven Jarohs Goethe University Frankfurt Germany Co-Author(s):    Tadeusz Kulczycki and Paolo Salani

Abstract: Given a smooth bounded domain $K$ in $\mathbb{R}^N$ and a parameter $\lambda>0$, the exterior Bernoulli problem (EBP) for the half Laplacian is to find a function $u:\mathbb{R}^N\to\mathbb R$ and a smooth open subset $\Omega$ of $\mathbb{R}^N$ such that $\overline{K}\subset \Omega$ and $u$ is a solution to the problem $$ (-\Delta)^{1/2}u=0\quad\text{in $\Omega\setminus \overline{K}$,}\quad u=0\quad\text{in $\mathbb{R}^N\setminus \Omega$,}\quad u=1\quad\text{in $\overline{K}$,} $$ with $$ D_{\Omega}^{1/2}u(\theta)=\lim_{t\to 0^+}\frac{u(\theta+t\nu(\theta)}{t^{1/2}}=\lambda \quad\text{for all $\theta\in \partial\Omega$,} $$ where $\nu(\theta)$ denotes the interior unit normal at $\theta\in \partial\Omega$. In this talk, the existence of a solution to the EBP with its geometric properties and resulting regularity is discussed. Furthermore qualitative properties related to the asymptotic behavior of the free boundary of solutions when the Bernoulli`s gradient parameter $\lambda$ tends to $0^+$ or to $+\infty$ are presented. The talk is based on two joint works with Tadeusz Kulczycki and Paolo Salani.

Some recent work on qualitative analysis of nonlocal PDEs

Congming Li Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: We present some recent work on the analysis of nonlocal problems, focusing on maximum principles, Liouville type theorems, and the classification of solutions. Related results on a priori estimates and regularity are also discussed. Key topics include Hardy-Littlewood-Sobolev type systems and curvature-driven geometric equations.

Global existence and zero relaxation limit for a hyperbolic system arising in traffic flow with large data

Tong Li University of Iowa USA Co-Author(s):    Jose David Beltran

Abstract: This talk is concerned with the existence of global weak solutions and their zero relaxation limit for a non-strictly hyperbolic system arising in traffic flow with large initial data. We use the vanishing viscosity method and the compensated compactness framework to prove the existence of admissible weak solutions and to study their behavior as the relaxation parameter tends to zero. By constructing convex dissipative entropies that display special compatibility conditions with the local equilibrium equation of the system, we establish the existence and uniqueness of the zero relaxation limit solution for large initial data.

Uniqueness of entire solutions to quasilinear equations with a sub-natural growth term and measure data

Phuc C Nguyen Louisiana State University USA Co-Author(s):    Igor E. Verbitsky

Abstract: We present the uniqueness of nontrivial solutions to the problem \begin{equation*} \left\{ \begin{array}{ll} - \Delta_p u = \sigma u^q + \mu, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} in the sub-natural growth case $0\lt q\lt p-1$, where $\mu, \sigma$ are nonnegative locally finite measures in $\mathbb{R}^n$ absolutely continuous with respect to the $p$-capacity. Here $\Delta_p u:={\rm div}(|\nabla u|^{p-2}\nabla u)$, $1\lt p\lt \infty$, is the $p$-Laplace operator. The uniqueness is obtained in the class of \emph{reachable} solutions, and moreover, if the condition $\displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0$ is replaced by the stronger condition $\displaystyle{\lim_{|x|\rightarrow \infty}}\, u = 0$, then such uniqueness is obtained in the larger class of $p$-superharmonic solutions. This talk is based on joint work with Igor E. Verbitsky.

Uniform Calder\`{o}n-Zygmund estimates in multiscale elliptic homogenization

Weisheng Niu Anhui University Peoples Rep of China Co-Author(s):    Jinping Zhuge

Abstract: We report some recent results on the uniform Calderon-Zygmund estimate in the homogenization of elliptic equations with multiscales. Our result includes the uniform Calderon-Zygmund estimate in quasiperiodic elliptic homogenization (even without the Diophantine condition), which was previously unknown. The proof combines the Dirichlet`s theorem on the simultaneous Diophantine approximation from number theory, a technique of reperiodization, reiterated periodic homogenization and a large-scale real-variable argument. This a joint work with Jinping Zhuge.

On Hopf`s Lemma for sign-changing supersolutions to fractional Laplacian equations

Enea Parini Aix Marseille Universite France Co-Author(s):    Azahara DelaTorre

Abstract: Unlike in the local case, Hopf's Lemma does not hold true, in general, for sign-changing supersolutions to equations driven by the fractional Laplacian, as proven in a recent paper by Dipierro, Soave and Valdinoci. We investigate the validity of Hopf's Lemma for a (possibly sign-changing) function $u \in H^s_0(\Omega)$ satisfying \[ (-\Delta)^s u(x) \geq c(x)u(x) \quad \text{in }\Omega,\] where $\Omega \subset \mathbb{R}^N$ is an open set, $c \in L^\infty(\Omega)$ and $(-\Delta)^s u$ is the fractional Laplacian of $u$. We show that, under suitable assumptions, the validity of Hopf`s Lemma for $u$ at a point $x_0 \in \partial \Omega$ is essentially equivalent to the validity of Hopf`s Lemma for the Caffarelli-Silvestre extension of $u$ at the point $(x_0,0) \in \mathbb{R}^N \times \mathbb{R}^+$. The results have been obtained in collaboration with Azahara DelaTorre.

Existence and local uniqueness of solutions to fractional equations with application to fluid dynamics

Guolin Qin Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Daomin Cao, Shanfa Lai, Weicheng Zhan, Changjun Zou

Abstract: This talk is concerned with the study of fractional elliptic equations arising from fluid dynamics. Taking the generalized surface quasi-geostrophic (gSQG) equation as a prototype, we investigate the existence, asymptotic behavior, local uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds.

Asymptotic behavior for anisotropic fractional energies

Ariel Salort CEU San Pablo, Madrid Spain Co-Author(s):    Julian Fernandez Bonder

Abstract: We investigate the asymptotic behavior of anisotropic fractional energies as the fractional parameter $s \in (0, 1)$ approaches the critical limits $s \uparrow 1$ and $s \downarrow 0$, in the spirit of the seminal works by Bourgain-Brezis-Mironescu and Maz`ya-Shaposhnikova. Focusing on the limit $s \uparrow 1$, we analyze the stability and convergence of solutions to the corresponding minimization problems. Finally, we examine the interplay between homogenization effects and the localization phenomena that arise as the operator recovers its local structure in the limit $s \uparrow 1$.

Volume growth and Liouville theorems

Yuhua Sun Nankai University Peoples Rep of China Co-Author(s):

Abstract: Though volume is a very simple geometric quantity, but it plays a very important role when one studies the geometric and probabilistic properties. We will present how to use volume growth to obtain Liouville type results for elliptic and parabolic equations on manifolds and weighted graphs.

Sobolev and Hardy-Sobolev inequalities and supercritical problems

John Villavert University of Texas-Rio Grande Valley USA Co-Author(s):

Abstract: In this talk, we shall describe some supercritical variants of first and second-order Sobolev and Hardy--Sobolev-type embeddings and related elliptic problems that incorporate power nonlinearities with strongly supercritical variable exponents. In particular, we shall describe a variational approach to derive such inequalities. Then, we will discuss how to obtain some existence results for the Dirichlet problem to a class of supercritical elliptic problems with variable exponents in the unit ball, which contrasts with non-existence results for the classical constant exponent case. We will also discuss complimentary non-existence results in both ball domains and the whole space.

Geometric Approach to DeGorgi Nash regularity theorem

Lihe Wang University of Iowa USA Co-Author(s):    Lihe Wang

Abstract: We will present a new and more geometric transparent approach to the local boundedness theorem of DeGiorgi and Nash.

Regularity of Elliptic Double Phase Problems with Measure Data: Exploring the Exponent Ranges

Yeonghun Youn Incheon National University Korea Co-Author(s):    Kyeong Song

Abstract: This talk addresses the regularity theory for elliptic double phase problems with measure data. We begin by providing a brief overview of the historical development of existence and regularity results for elliptic problems with measure data. The main focus of this presentation will then shift to the precise conditions on the range of exponents that are crucial for establishing regularity in double phase problems with measure data. We will heuristically demonstrate the optimality of this exponent range through a scaling argument. This work represents a significant step towards understanding the regularity of solutions to this class of challenging problems, and is a joint work with Dr. Song at KIAS.

Optimal convergence rates and uniform regularity for multiscale elliptic homogenization

Jinping Zhuge Morningside Center of Mathematics, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Zhongwei Shen, Yao Xu, Weisheng Niu

Abstract: I will talk about recent progress in homogenization theory for the linear elliptic equations with coefficients periodically oscillating at multiple different microscopic scales. The optimal convergence rates and uniform regularity in various settings will be discussed.