相关论文: Partial regularity for degenerate parabolic system…

We study solutions ${\bf v}$ of the parabolic system of PDE $$ \partial_t\left(D\psi({\bf v})\right)=\text{div}DF(D{\bf v}). $$ Here $\psi$ and $F$ are convex functions, and this is a model equation for more general doubly nonlinear…

In this article, we obtain higher H\"older regularity results for weak solutions to nonlocal problems driven by the fractional double phase operator \begin{align*} \mc L u(x):=&2 \; {\rm P.V.} \int_{\mathbb R^N}…

We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{\"o}lder continuous Lp valued functions…

We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the $p$-Laplacian type and in non-divergence form. We provide local H\"older and Lipschitz estimates for the solutions. In the degenerate…

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau…

We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-\Delta_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $\Delta_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the…

In this paper we prove a H\"older partial regularity result for weak solutions $u:\Omega\to \mathbb{R}^N$, $N\geq 2$, to non-autonomous elliptic systems with general growth of the type: \begin{equation*} -\rm{div}\, a(x, u, Du)= b(x, u, Du)…

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 establish the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad 1<p<2,\quad 0<p-1<q. \] The proof…

We study local boundedness and H\"older continuity of a parabolic equation involving the fractional $p$-Laplacian of order $s$, with $0<s<1$, $2\leq p < \infty$, with a general right hand side. We focus on obtaining precise H\"older…

This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. A model equation is \begin{equation*}\partial_t u-\operatorname{div}(u^{m-1}|Du|^{p-2}Du)=f\qquad…

This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the…

A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic $p$-Laplacian system is locally H\"older continuous in the interior. However, the boundary regularity is not yet well understood. In…

We study local H\"older regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form \[ (|u|^{q-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} dy = 0, \] with…

In this paper, we study the mathematical properties of the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ to the degenerate parabolic system \begin{equation*} \bold{u}_t=\nabla\cdot\left(\left|\nabla\bold{u}\right|^{p-2}\nabla…

In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-\div\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in…

We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ D\psi({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $\psi$ and $F$ are convex functions. This is a prototype for more general doubly…

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…

We carry on the investigation started in [2] about the regularity of weak solutions to the strongly degenerate parabolic equation \[ u_{t}-\mathrm{div}\left[(\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert…

A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. H\"older modulus of continuity is then deduced under a slightly stronger tail…