Special Session 149: Recent developments in Free Boundary Problems and Nonlinear PDEs

Multiscale Schnauzer Estimates

Hector A Chang Lara Universidad de Coimbra Portugal Co-Author(s):

Abstract: We present a flexible framework for proving regularity estimates for solutions of partial differential equations that does not rely on scaling invariance. This makes it particularly suited to problems where classical blow-up methods are unavailable. The approach has been used in the study of Holder regularity for parabolic equations driven by integro-differential operators, as well as in the analysis of free boundary regularity for the Hele-Shaw problem. It also applies to establish $C^{1,\alpha}$ estimates for solutions of nonlocal equations with rough kernels. As a new application, we consider the logarithmic Laplacian, a zero-order operator for which rescaling produces a non-integrable tail, creating a serious obstruction to standard compactness and blow-up arguments. In this setting, we obtain optimal Schauder-type logarithmic Holder estimates.

Constructing nontrivial global periodic solutions to the Bernoulli free boundary problem in 3D

Nikola Kamburov Pontificia Universidad Catolica de Chile Chile Co-Author(s):    Jose Basulto

Abstract: I this talk I will present the construction of a new family of Delaunay-type solutions to the Bernoulli one-phase free boundary problem in dimension 3, starting from the double hairpin solution of Hauwirth-H\`elein-Pacard in $\mathbb{R}^2$. This is joint work with J. Basulto (UC).

Regularity of thin constraint maps

Sunghan Kim Uppsala University Sweden Co-Author(s):    Marvin Weidner, Hui Yu

Abstract: I would like to discuss recent progress in the regularity of thin constraint maps. These are energy-minimizing maps subject to a thin constraint, which is a natural vectorial extension of the thin obstacle problem. In this sense, the problem is closely related to nonlocal obstacle problems, but it is also related to free boundary problems for harmonic maps and minimal surfaces. Analogous to thick constraint maps, thin constraint maps also develop free boundaries, due to the presence of the thin obstacle, while they also develop discontinuities, often for topological reasons. While the partial regularity theory of these maps was established in the nineties, little has been known concerning their free boundaries. In this talk, I will introduce several new results for thin constraint maps including (i) a partial regularity result that is optimal in both regularity and dimension, and (ii) the full regularity in a uniform neighborhood of their non-coincidence set even for slightly nonconvex thin obstacles. The latter result is surprisingly stronger than the thick obstacle case, where mere convexity is insufficient. My talk will be based on a joint work with Marvin Weidner (University of Bonn) and Hui Yu (National University of Singapore).

A Free Boundary Problem with Nonlocal Obstacle

Hayk Mikayelyan University of Nottingham Ningbo China Peoples Rep of China Co-Author(s):    M. Chipot, Zh. Lin

Abstract: We consider a functional with with the nonlocal obstacle acting on the function $V(x')=\int_0^1 U(x', t) dt $ $$ \int_\Omega \frac{1}{2}|\nabla U(x)|^2dx +\int_D V(x')^+\,dx'. $$ The minimizer solves the equation $$ \Delta U(x',x_n) = \chi_{\{V>0\}}(x') + \chi_{\{V=0\}}(x') [\partial_\nu U (x',0) + \partial_\nu U (x',1)], $$ where $\partial_\nu U$ is the exterior normal derivative of $U$. Several regularity results are proven. It is shown that the comparison principle does not hold for minimizers, which makes numerical approximation we present somehow challenging.

A mathematical approach to epidermal wound healing

Katerina Nik KAUST Saudi Arabia Co-Author(s):

Abstract: A mathematical model for epidermal wound closure is considered. The model is based on a moving boundary problem in which the wound edge moves once a generic epidermal growth factor exceeds a given threshold. Solutions are constructed explicitly and the existence of a waiting time before boundary motion sets in is demonstrated. The natural emergence of delay in the solution is then discussed for cases where diffusion of the growth factor concentration fails to balance the rapid motion of the wound edge.

Second-order regularity properties from one-side geometric control

Edgard Pimentel University of Coimbra Portugal Co-Author(s):

Abstract: We establish second-order regularity properties for viscosity solutions of fully nonlinear PDE, in the presence of geometric control from below. As an application, we examine the regularity of the value functions to impulse optimal control problems.

Spectral partition problems for the divergence operator with volume and inclusion constraints

Makson Santos University of Lisbon Portugal Co-Author(s):    P\^{e}dra Andrade, Dario Mazzoleni, Ederson Moreira dos Santos, Hugo tavares

Abstract: We study the existence and regularity of optimal partitions for a problem with volume and inclusion constraints driven by the divergence operator. In particular, we prove that an optimal partition is connected and the eigenfunction associated with each set is locally Lipschitz continuous, which implies that the optimal sets are at least open sets. We show that there is a variational formulation to our problem that does not involve subsets, only functions, and we prove the desired properties for the minimizer.

Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory

Mariana Smit Vega Garcia Western Washington University USA Co-Author(s):    Blair Davey

Abstract: Since elliptic PDE theory can be seen as a steady-state version of parabolic PDE theory, if a parabolic estimate holds, then by eliminating the time parameter, one can obtain an underlying elliptic statement. Producing a parabolic statement from an elliptic statement, on the other hand, is not as straightforward. In this talk, we will discuss how a high-dimensional limiting technique can be used to prove theorems about solutions to the fractional heat equation from their elliptic counterparts.

On fractional $k$-Hessian operators

Maria Soria-Carro Universidad Autonoma de Madrid Spain Co-Author(s):    Mar Gonz\\`alez and Fernando Charro

Abstract: In this talk, I will present a fractional analogue of the classical $k$-Hessian operator, defined as an infimum of anisotropic fractional Laplacians over a suitable class of matrices. I will focus on the case $k=2$ and show that, under natural assumptions, the fractional 2-Hessian operator is locally uniformly elliptic. This property places the operator within the framework of fully nonlinear integro-differential equations and makes available the regularity theory developed for nonlocal elliptic operators. The key issue is to understand the possible degeneracies in the admissible class of matrices and to prove that these degenerate directions do not contribute to the infimum. These results extend the theory known for the fractional Monge-Amp\`ere operator ($k=n$) developed by Caffarelli et al. in 2015. This is ongoing joint work with Mar Gonz\`alez (UAM) and Fernando Charro (Wayne State).

Improved regularity for a nonlocal dead-core problem

Rafayel Teymurazyan KAUST Saudi Arabia Co-Author(s):    D. dos Prazeres, J.M. Urbano

Abstract: We obtain improved regularity results for solutions to a nonlocal dead-core problem at branching points. Our approach, which does not rely on the maximum principle, introduces a new strategy for analyzing two-phase problems within the local framework, an area that remains largely unexplored.

A sharp differentiability threshold for minimizers of singular energies

Jose Miguel Urbano KAUST Saudi Arabia Co-Author(s):    Damiao Araujo (UFPB), Aelson Sobral (KAUST), and Eduardo Teixeira (OSU)

Abstract: We address the borderline regularity of local minimizers of singular energy functionals. For bounded and measurable potentials, we show that sign-changing minimizers are Log-Lipschitz continuous, which is optimal in this generality. In the one-phase case, however, we derive gradient bounds along the free boundary, uncovering a structural gain in regularity. Our first main result establishes sharp Lipschitz regularity for a merely bounded potential. Most notably, we prove that if the potential is further assumed to be a modulus of continuity, then minimizers become continuously differentiable. We thus identify a sharp threshold for differentiability in terms of the potential`s regularity.

Generic regularity of the free boundary in the Alt-Caffarelli-Phillips problem

Hui Yu National University of Singapore Singapore Co-Author(s):    Xavier Fernandez-Real

Abstract: For generic boundary data, we discuss the improvement on the estimate of the singular dimension in the Alt-Caffarelli-Phillips problem.