相关论文: Improved regularity for the $p$-Poisson equation

We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity…

In this paper we study optimal control problems with either fractional or regional fractional $p$-Laplace equation, of order $s$ and $p\in [2,\infty)$, as constraints over a bounded open set with Lipschitz continuous boundary. The control,…

We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently…

In this paper, we study the regularity of the solutions of Maxwell's equations in a bounded domain. We consider several different types of low regularity assumptions to the coefficients which are all less than Lipschitz. We first develop a…

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where…

We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To…

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that…

The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type \begin{align*}…

We propose a characterization of a $p$-Laplace higher eigenvalue based on the inverse iteration method with balancing the Rayleigh quotients of the positive and negative parts of solutions to consecutive $p$-Poisson equations. The approach…

In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In…

We examine the interior regularity of solutions to a degenerate normalized $p$-Laplace equation, where the degeneracy is governed by a modulus of continuity whose inverse satisfies a Dini continuity condition. We prove that under very…

We establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal…

In this paper, we study the regularity of solutions to the $p$-Poisson equation for all $1<p<\infty$. In particular, we are interested in smoothness estimates in the adaptivity scale $ B^\sigma_{\tau}(L_{\tau}(\Omega))$, $1/\tau =…

We propose a general framework of iteratively reweighted l1 methods for solving lp regularization problems. We prove that after some iteration k, the iterates generated by the proposed methods have the same support and sign as the limit…

We present and analyze two stabilized finite element methods for solving numerically the Poisson--Nernst--Planck equations. The stabilization we consider is carried out by using a shock detector and a discrete graph Laplacian operator for…

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and H\"{o}lder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable…

A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the…

Higher Sobolev and H\"older regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or}…

We study regularity properties of solutions to nonlinear and nonlocal evolution problems driven by the so-called \emph{$0$-order fractional $p-$Laplacian} type operators: $$ \partial_t u(x,t)=\mathcal{J}_p u(x,t):=\int_{\mathbb{R}^n}…

We present recent advances in the analysis of constrained optimization problems with constraints given by singular mappings obtained within the framework of the $p$-regularity theory developed over the last twenty years. In particular, we…