Related papers: Regularity Theory for Elliptic PDE
Schauder theory is a basic tool in the study of elliptic and parabolic PDEs, asserting that solutions inherit the regularity of the coefficients. It plays a central role in establishing higher regularity for solutions to a broad class of…
These are the notes of a part of the PhD course Regularity for free boundary problems and for elliptic PDEs, held in Pavia in the spring of 2025. The aim is to provide a comprehensive and self-contained treatment of classical interior and…
This paper deals with regularity theory for two-phase free boundary variational problems ruled by degenerate elliptic operators.
PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning…
Using a mixture of classical and probabilistic techniques we investigate the convexity of solutions to the elliptic pde associated with a certain generalized Ornstein-Uhlenbeck process.
This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this…
The paper contains a survey of the results obtained during the last ten years in the theory of elliptic boundary problems in H\"ormander function spaces, developed by the authors, and other related results of modern analysis. The basics of…
This book introduces to the theory of probabilities from the beginning. Assuming that the reader possesses the normal mathematical level acquired at the end of the secondary school, we aim to equip him with a solid basis in probability…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence…
Regularity theorems \`a la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general…
Regularizing neural networks is important for anticipating model behavior in regions of the data space that are not well represented. In this work, we propose a regularization technique for enforcing a level of smoothness in the mapping…
We prove existence of strong solutions to a family of some semilinear parabolic free boundary problems by means of elliptic regularization. Existence of solutions is obtained in two steps: we first show some uniform energy estimates and…
In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.
We give an essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular stochastic PDEs.
In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one…
The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its…
We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and…
In the first part of this doctoral thesis we develop a regularity theory for a polyconvex functional in compressible elasticity. In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity.…
We survey some recent regularity results for fractional p-Laplacian elliptic equations, especially focusing on pure and weighted boundary H\"older continuity of the solutions of related Dirichlet problems. Then, we present some applications…