Related papers: $L^p$ regularity theory for even order elliptic sy…

One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations, is to render them from their hyperbolic character to elliptic form. This paper is devoted to the study of the stabilized linear…

This paper deals with the parabolic $(1,\,p)$-Laplace system, a parabolic system that involves the one-Laplace and $p$-Laplace operators with $p\in(1,\,\infty)$. We aim to prove that a spatial gradient is continuous in space and time. An…

The paper recalls two of the regularity results for Burgers' equation, and discusses what happens in the case of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The first regularity result which is considered is…

Suppose that $1<p\leq\infty$ and $\varphi\in L^{p}(\mathbb{B}^{n},\mathbb{R}^{n}).$ In this note, we use H\"{o}lder inequality and some basic properties of hypergeometric functions to establish the sharp constant $C_{p}$ and function…

Regularity theory for diffusive operators is among the finest treasures of the modern mathematical sciences. It appears in several different fields, such as, differential geometry, topology, numerical analysis, dynamical systems,…

In this paper we study solutions, possibly unbounded and sign-changing, of the following problem: -\D_{\lambda} u=|x|_{\lambda}^a |u|^{p-1}u, in R^n,\;n\geq 1,\; p>1, and a \geq 0, where \D_{\lambda} is a strongly degenerate elliptic…

Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic…

We consider elliptic operators in divergence form with lower order terms of the form $Lu=-$div$\nabla u+bu)-c\nabla u-du$, in an open set $\Omega\subset \mathbb{R}^n$, $n\geq 3$, with possibly infinite Lebesgue measure. We assume that the…

We consider finite-entropy solutions of scalar conservation laws $u_t +a(u)_x =0$, that is, bounded weak solutions whose entropy productions are locally finite Radon measures. Under the assumptions that the flux function $a$ is strictly…

We propose a high-order finite element method for linear fourth-order elliptic problems that is both nodally bound-preserving and mass-conservative, based on a variational inequality formulation. The method admits an equivalent strictly…

We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary, and assuming local boundedness,…

In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.

In this paper, we mainly derive monotonicity formula of generalized map using conservation law, including $\phi$-$F$ harmonic map coupled with $\phi$-$F$ symphonic map with $m$ form and potential from metric measure space, $ p $ harmonic…

We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…

In recent years, the use of entropy-regularized optimal transport with $L^p$-type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling,…

High-order methods for convex and nonconvex optimization, particularly $p$th-order Adaptive Regularization Methods (AR$p$), have attracted significant research interest by naturally incorporating high-order Taylor models into adaptive…

We consider the homogeneous Landau equation with Coulomb potential and general initial data $f_{in} \in L^p$, where $p$ is arbitrarily close to $3/2$. We show the local-in-time existence and uniqueness of smooth solutions for such initial…

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the $L^p_tL_x^q(Q_1)$ norm of the velocity is sufficiently…

\small{In 2004, Del Pino and Dolbeault \cite{DPDo} and Gentil \cite{G} investigated, independently, best constants and extremals associated to sharp Euclidean $L^p$-entropy inequalities. In this work, we present some important advances in…

In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach…