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Special Session 82: Recent Advances in Nonlinear PDEs and Free Boundary Problems
Nonlinear gradient estimates for degenerate elliptic equations with nonstandard growth
Sun-Sig Byun
Seoul National University
Korea
Co-Author(s):    Sun-Sig Byun
Abstract:
This talk is primarily concerned with providing an optimal regularity theory for nonlinear elliptic equations with matrix weights
Regularity Results for Stationary Mean-Field Games with Logarithmic Couplings
Diogo Gomes
KAUST
Saudi Arabia
Co-Author(s):    Tigran Bakaryan and Giuseppe Di Fazio
Abstract:
In this joint work with Tigran Bakaryan and Giuseppe Di Fazio, we present recent findings on the regularity properties of stationary mean-field games (MFGs) on the torus, focusing on systems with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The goal is to bridge the gap between known low-regularity results for bounded diffusions and the smooth solutions typically associated with the Laplacian. By employing the Hopf-Cole transformation, we reformulate the system into a scalar elliptic equation, enabling us to establish the existence of C1,α solutions. These results have significant implications for understanding the fine structure of equilibria in MFG models, especially in applications with non-linear and non-smooth dynamics.
Existence of weak solutions of nonlinear drift-diffusion equations
Sukjung Hwang
Chungbuk National University
Korea
Co-Author(s):    
Abstract:
In this presentation, we discuss recent existence results for porous medium and fast diffusion equations with a divergence-type drift term, which are widely applicable to various reaction-diffusion equations, including Keller-Segel models. Our focus is on identifying suitable functional spaces for the drift, primarily determined by the nonlinear diffusion and the initial data. By adapting techniques from Wasserstein spaces, we construct weak solutions and establish some regularity properties of the solutions.
One bubble dynamics for the Sobolev critical fast diffusion equation in bounded domains
Tianling Jin
The Hong Kong University of Science and Technology
Hong Kong
Co-Author(s):    Jingang Xiong
Abstract:
We will discuss the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold assumption on the initial data, we will show the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble.
The Well-posedness of Cylindrical Jets with Surface Tension
Aram Karakhanyan
The University of Edinburgh
Scotland
Co-Author(s):    Dr Yukon Huang
Abstract:
In 1879 Rayleigh studied the stability of infinite cylindrical jets, inspired by the experiments of Plateau. The principal question that Rayleigh asked is: under what circumstances the jet is stable, for small displacements. In this talk I will discuss the short time stability for the initial condition belonging to some Sobolev space, and the initial jet boundary being uniformly bounded away from the axis of symmetry. This is proved by the method of paradifferential calculus and paralinearization. The salient feature of these results is that no smallness assumption is imposed on the initial condition. The results are taken from a joint paper with Dr Yucong Huang.
Sharp regularity for singular/degenerate fully nonlinear free boundary problems with singular absorption terms
Seunghyun Kim
Seoul National University
Korea
Co-Author(s):    Jose Miguel Urbano
Abstract:
We investigate the sharp local regularity of viscosity solutions to singular or degenerate fully nonlinear elliptic free boundary problems with singular absorption terms. By employing suitable substitutions and the Ishii-Lions method, we establish regularity estimates for the associated auxiliary problems. Additionally, we explore the parabolic case.
A Free Boundary Problem with Nonlocal Obstacle
Hayk Mikayelyan
University of Nottingham Ningbo China
Peoples Rep of China
Co-Author(s):    Michel Chipot, Zhilin Li
Abstract:
Consider the following optimal minimization problem in the cylindrical domain Ω=D×(0,1): min where is the unique solution of , and . We show the existence of the unique minimizer. Moreover, we show that for a particular the function minimizes the functional with nonlocal obstacle acting on function and solves the equation where is the exterior normal derivative of . Several further regularity results are proven. It is shown that the comparison principle does not hold for minimizers, which makes numerical approximation we developed in [LM] somewhat challenging. Keywords: rearrangement problems, free boundary, nonlocal obstacle MSC Classification: 35R11, 35J60, 35R35, 35B51, 49J20, 65N06 [CM] Chipot, Michel; Mikayelyan, Hayk, , Nonlinear Anal. 217 (2022), Paper No. 112754, 17 pp. [LM] Li, Zhilin; Mikayelyan, Hayk, , Appl. Math. Lett. 135 (2023), Paper No. 108414, 6 pp. [M] Mikayelyan, Hayk, , 2018, ESAIM - Control, Optimisation and Calculus of Variations. 24, 2, p. 859-872 14 p.
Free boundaries and the minimal surface system
Connor Mooney

USA
Co-Author(s):    
Abstract:
We will discuss two free boundary problems that arise in the study of minimal surfaces of high codimension. The first one shows that viscosity solutions to the special Lagrangian equation need not be , or have minimal gradient graph. The second one shows that large interior singularities can arise when minimizing area among graphs in codimension two or larger. This is joint work with O. Savin.
Variational principles in mean-field games and related problems
Levon Nurbekyan
Emory University
USA
Co-Author(s):    
Abstract:
Mean-field game (MFG) systems are challenging nonlinear PDEs, often requiring a non-standard analysis approach. In this talk, I`ll discuss several variational principles for MFG that substantially aid the analyses by importing tools from the calculus of variations. I`ll also discuss ensuing future research directions.
Regularity results for n-Laplace systems with antisymmetric potential
Armin Schikorra
University of Pittsburgh
USA
Co-Author(s):    
Abstract:
n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat. I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione. Joint work with Dorian Martino
Elliptic regularity estimates with optimized constants and applications
Boyan Sirakov
PUC-Rio
Brazil
Co-Author(s):    Philippe Souplet
Abstract:
We revisit the classical theory of linear second-order uniformly elliptic equations in divergence form whose solutions have H\older continuous gradients, and prove versions of the generalized maximum principle, the -estimate, the Hopf-Oleinik lemma, the boundary weak Harnack inequality and the differential Harnack inequality, in which the constant is optimized with respect to the norms of the coefficients of the operator and the size of the domain. Our estimates are complemented by counterexamples which show their optimality. We also give applications to the Landis conjecture and spectral estimates.