Related papers: H\"older regularity for anisotropic $p$-Laplace eq…

In this paper, we prove local H\"older continuity for the spatial gradient of weak solutions to $$u_t - \text{div} (|\nabla u|^{p-2}\nabla u) + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+ps}} \ dy…

We study here a new generalization of Caffarelli, Kohn and Nirenberg's partial regularity theory for weak solutions of the MHD equations. Indeed, in this framework some hypotheses on the pressure P are usually asked (for example P $\in$ L q…

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($\alpha< 1/2$) dissipation $(-\Delta)^\alpha$ : If a Leray-Hopf weak solution is H\"{o}lder continuous $\theta\in C^\delta({\mathbb…

We prove global H\"older regularity result for weak solutions $u\in N^{1,p}(\Omega, \mu)$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: \[ -\text{div}\!\left( |\nabla u|^{p-2}\nabla u \right)=\overline\nu, \] where…

We prove local Lipschitz regularity for weak solutions to a parabolic orthotropic $p$-Laplacian-type equation in the Heisenberg group $\Hn$, for the range $2\leq p\leq4$.

We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to…

We consider inhomogeneous $p$-Laplace type equations of the form $-\mathrm{div}\left(a(\nabla u)\right)=f$ in a possibly anisotropic setting. Under general assumptions on the source term $f$, we obtain quantitative Sobolev regularity…

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…

We give a proof of the H\"older continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincar\'e…

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…

We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions,…

We prove interior H\"older estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous $p$-Laplacian equation \[ u_t=|\nabla u|^{2-p} \mbox{ div} (|\nabla u|^{p-2}\nabla u), \] where $1<p<\infty$. This equation…

We obtain up to a flat boundary regularity results in parabolic H\"{o}lder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.

We prove existence and up to the boundary regularity estimates in $L^{p}$ and H\"{o}lder spaces for weak solutions of the linear system $$ \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text{ in…

This paper presents a detailed regularity analysis of anisotropic wavelet frames and subdivision. In the univariate setting, the smoothness of wavelet frames and subdivision is well understood by means of the matrix approach. In the…

In this note, we prove the boundary H\"{o}lder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the…

In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{align*} \partial_t u^q - \mbox{div}\, A(x,t,Du)=0 \qquad\mbox{in…

We give an elementary proof for the interior double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of the Euler Equations in a bounded $C^2$-domain $\Omega \subset \mathbb{R}^d$; $d\geq 3$. That is, for velocity $u…

We obtain an explicit H\"older regularity result for viscosity solutions of a class of second order fully nonlinear equations leaded by operator that are neither convex/concave nor uniformly elliptic.

In the elliptic theory for $p$-Laplacian-like problems, the H\"{o}lder continuity of solutions has been proven for problems arising as Euler--Lagrange equations of a convex potential with $p$-growth that additionally satisfies the splitting…