In this work, we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent αF such that u is C1,αF on either side of the obstacle. In order to do that, we prove ...
We prove new boundary regularity results for minimizers to the one-phase Alt-Caffarelli functional (also known as Bernoulli free boundary problem) in the case of continuous and Hölder-continuous boundary data. As an application, we use them to extend recen ...
We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in L∞ in the case of hard potentials. As a consequence, we derive $C^{\i ...
This monograph offers a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. This class of equations finds relevance in fields such as analysis, probability theory, mathemat ...
This paper studies the infinite-width limit of deep linear neural networks (NNs) initialized with random parameters. We obtain that, when the number of parameters diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a ...
The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one -phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions n >= 3 is completely open. In this context, axial ...
We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient di ...
In this note, we prove that if a subharmonic function Delta u >= 0 has pure second derivatives partial derivative(ii)u that are signed measures, then their negative part (partial derivative(ii)u)- belongs to L-1 (in particular, it is not singular). We then ...
The free boundary for the Signorini problem in Rn+1 is smooth outside of a degenerate set, which can have the same dimension (n - 1) as the free boundary itself. In [15] it was shown that generically, the set where the free boundary is not smooth is at mos ...
