Related papers: Existence and regularity of monotone solutions to …
In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle…
In this paper we study the Castelnuovo-Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the path ideals of gap free and claw…
We consider viscosity solutions to a one-phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We apply the tools developed in \cite{D} to prove that flat free boundaries are $C^{1,\alpha}$. Moreover, we obtain…
Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…
Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a…
We study a general class of elliptic free boundary problems equipped with a Dirichlet boundary condition. Our primary result establishes an optimal $C^{1,1}$-regularity estimate for $L^p$-strong solutions at points where the free and fixed…
We present in this paper a result about existence and convexity of solutions to a free boundary problem of Bernoulli type, with non constant gradient boundary constraint depending on the outer unit normal. In particular we prove that, in…
We study the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in \cite{MonsaingonNovikovRoquejoffre}. We set up a finite difference scheme…
We propose a numerical method to approximate viscosity solutions of fully nonlinear free transmission problems. The method discretises a two-layer regularisation of a PDE, involving a functional and a vanishing parameter. The former is…
Exact solutions of a classical problem of a plane unsteady potential flow of an ideal incompressible fluid with a free boundary are presented. The fluid occupies a semi-infinite strip bounded by the free surface (from above) and (from the…
We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved…
Energy-minimizing constraint maps are a natural extension of the obstacle problem within a vectorial framework. Due to inherent topological constraints, these maps manifest a diverse structure that includes singularities similar to harmonic…
We continue our study of the free boundary regularity in the thin one-phase problem and show that $C^{2,\alpha}$ free boundaries are smooth.
The interior free boundary theory for linear elliptic operators in higher dimensions was developed by Caffarelli in the low regularity context. In these notes, the up-to-the boundary free boundary regularity is discussed for nonlinear…
We present results of existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under…
In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish…
The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving…
We consider the one-phase Stefan problem describing the evolution of melting ice. On the one hand, we focus on understanding the evolution of the free boundary near isolated singular points, and we establish for the first time upper and…
In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_{\Omega}\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad…
In this paper, we study the regularity of the "regular" part of the free boundary for almost minimizers in the parabolic Signorini problem with zero thin obstacle. This work is a continuation of our earlier research on the regularity of…