相关论文: Weak Solutions and Regularity of the Interface in …

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…

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 second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions. We prove existence and uniqueness of weak solutions and their continuity up to the boundary of…

We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable…

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-\Delta)_p^s u= \lambda f(u),\; u> 0 ~\text{in}~\Omega;\; u=0\;\text{in}~ \mathbb{R}^N\setminus\Omega$, where $p>1$,…

We explore regularity properties of solutions to a two-phase elliptic free boundary problem near a Neumann fixed boundary in two dimensions. Consider a function u, which is harmonic where it is not zero and satisfies a gradient jump…

We prove partial regularity of weakly stationary harmonic maps with (partially) free boundary data on manifolds where the domain metric may degenerate or become singular along the free boundary at the rate $d^\alpha$ for the distance…

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the…

This paper studies the regularity of the free boundary for viscosity solutions to a parabolic Bernoulli-type free boundary problem with variable coefficients. The main result is that Lipschitz free boundaries are $C^1$ with a normal vector…

We investigate the regularity in $L^p$ ($p>2$) of the gradient of any weak solution of a Cauchy problem with mixed Neumann-power type boundary conditions. Under suitable assumptions we prove the existence of weak solutions that satisfy…

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the…

We consider the equation $-\Delta_p u=f(u)$ in a smooth bounded domain of $\mathbb{R}^n $, where $\Delta_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\geq p+4p/(p-1)$. Instead, when…

In this article we prove the global existence of weak solutions to an initial boundary value problem with an exponential and p-Laplacian nonlinearity. The equation is a continuum limit of a family of kinetic Monte Carlo models of crystal…

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.

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…

We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,\alpha}$…

For any $\Omega\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $\Omega$ in a neat interval depending only by the best constant of the Sobolev…

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $\Omega\subset\mathbb{R}^n$ is…

In this paper, we characterize the geometry of solutions to one-phase inhomogeneous fully nonlinear Stefan problem with flat free boundaries under a new nondegeneracy assumption. This continues the study of regularity of flat free…

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 $u(x)\ge 0$…