Related papers: Free boundary regularity for almost every solution…

We study the parabolic free boundary problem of obstacle type $$ \lap u-\frac{\partial u}{\partial t}= f\chi_{{u\ne 0}}. $$ Under the condition that $f=Hv$ for some function $v$ with bounded second order spatial derivatives and bounded…

In this paper a strongly degenerate parabolic equation derived from a density dependent particle flow model is studied. Furthermore, a free boundary problem and its connection to the strongly degenerate parabolic equation is investigated.…

We study the obstacle problem for parabolic operators of the type $\partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-\Delta)^s$, in the supercritical regime $s \in (0,{1/2})$. The best result…

In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is $C^{1,\alpha}$ near regular…

In this article we use flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set…

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3)$ minimizes $$ J(\u)=\int_{B_1^+}|\nabla \u+\nabla^\bot \u|^2+\lambda\div(\u)^2 $$ in the…

We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the…

We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators; more specifically, we consider viscosity solutions of \[ |D u|^\gamma F(x, D^2u) = f(x)\chi_{\{u>\phi\}}…

As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in…

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional…

We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for…

The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. After a general overview, we present some recent results on the structure of singular free boundary…

This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $u_{t}=\Delta{}u^{m}$. Our main objective is to improve the H$\ddot{o}$lder estimate obtained by pioneers…

The thin obstacle problem or $n$-dimensional Signorini problem is a classical variational problem arising in several applications, starting with its first introduction in elasticity theory. The vast literature concerns mostly quadratic…

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of…

Let $u$ be a solution to the normalized p-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2<p<\infty$, $u\ge 0$ and $$ \d\left( |\nabla u|^{p-2}\nabla u\right)=\chi_{\{u>0\}}\textrm{ in }B_1(0) $$ where $u(x)\ge 0$…

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle…

We study solutions to a variational equation that models heat control on the boundary. This problem can be thought of as the two phase parabolic Signorini problem. We show that when the solution has a sign on the boundary, the study of the…

This is the sequel of the recent work (Du, Huang, Pu, Commun. Math. Phys, 2023, doi: 10.1007/s00220-023-04651-7) on axially symmetric gravity water waves with general vorticities, which has investigated the singular wave profile of the free…

We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the…