arXiv subjects

Michael Strunk

Publications and source records attributed to Michael Strunk.

5 recordsLinked to original sources

arXiv2026

A comparison principle for a class of doubly nonlinear parabolic fractional partial differential equations

In this paper, we establish a comparison principle for non-negative weak solutions to a class of doubly nonlinear parabolic fractional partial differential equations within a space-time cylinder $\Omega_T=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$. For the two solutions considered, we assume that at least one of them is time-independent outside the spatial domain, i.e. in $\Omega^{c}=\mathbb{R}^n\setminus\Omega$. As an application of this result, we readily infer the uniqueness of a non-negative weak solution to the corresponding Cauchy-Dirichlet problem.

math.AP↗

arXiv2026

Boundary regularity for parabolic systems with nonstandard $(p,q)$-growth conditions in smooth convex domains

We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form \begin{equation*} \partial_t u^i - \mathrm{div} \big( a(|Du|) Du^i \big)= f^i, \qquad i=1,\dots,N, \end{equation*} in a space-time cylinder $\Omega_T = \Omega \times (0,T)$, where $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded, convex $C^2$-domain and $T>0$. The inhomogeneity $f=(f^1,\dots,f^N)$ belongs to $L^{n+2+\sigma}(\Omega_T,\mathbb{R}^N)$ for some $\sigma>0$. The coefficients $a\colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ are of Uhlenbeck-type and satisfy a nonstandard $(p,q)$-growth condition with \[ 2 \leq p \leq q < p + \frac{4}{n+2}. \] Our main result establishes a local Lipschitz estimate up to the lateral boundary for any local weak solution that vanishes on the lateral boundary of the cylinder.

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arXiv2025

Gradient regularity for widely degenerate parabolic equations

In this paper, we are interested in the regularity of weak solutions $u\colon\Omega_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $\Omega_T$}, \end{equation*} where $\mathcal{F}$ is only elliptic for values of $Du$ outside a bounded and convex set $E\subset \mathbb{R}^n$ with the property that $0\in \mathrm{Int}{E}$. Here, $\Omega_T :=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$ denotes a space-time cylinder taken over a bounded domain $\Omega\subset\mathbb{R}^n$ for some finite time $T>0$. The function $\mathcal{F} : \Omega_T\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ present in the diffusion is assumed to satisfy: the partial mapping $\xi\mapsto \mathcal{F}(x,t,\xi)$ is regular whenever $\xi$ lies outside of $E$, and vanishes entirely whenever $\xi$ lies within this set. Additionally, the datum $f$ is assumed to be of class $L^{n+2+\sigma}(\Omega_T)$ for some parameter $\sigma > 0$. As our main result we establish that \begin{equation*} \mathcal{K}(Du)\in C^0(\Omega_T) \end{equation*} for any continuous function $\mathcal{K}\in C^0(\mathbb{R}^n)$ that vanishes on $E$. This article aims to extend the $C^1$-regularity result for the elliptic case to the parabolic setting.

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arXiv2025

Gradient regularity for widely degenerate elliptic partial differential equations

In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\Omega$}, \end{equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset\mathbb{R}^n$ with $0\in \mathrm{Int}\, E$. Here, $\Omega\subset\mathbb{R}^n$ denotes a bounded domain, and $\mathcal{F} \colon \Omega\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ is a function with the properties: for any $x\in\Omega$, the mapping $\xi\mapsto \mathcal{F}(x,\xi)$ is regular outside $E$ and vanishes entirely within this set. Additionally, we assume $f\in L^{n+\sigma}(\Omega)$ for some $\sigma > 0$, representing an arbitrary datum. Our main result establishes the regularity \begin{equation*} \mathcal{K}(Du)\in C^0(\Omega) \end{equation*} for any continuous function $\mathcal{K}\in C^0(\mathbb{R}^n)$ vanishing on $E$.

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arXiv2024

Gradient regularity for a class of doubly nonlinear parabolic partial differential equations

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 $\Omega_T$}, \end{align*} with $q>0$, $\Omega_T=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$ a space-time cylinder, and $A=A(x,t,\xi)$ a vector field satisfying standard $p$-growth conditions. Our main result establishes the local H\"older continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime $$0<p-1<q<\frac{n(p-1)}{(n-p)_+}.$$ This result is achieved by utilizing a time-insensitive Harnack inequality and Schauder estimates that are developed for equations of parabolic $p$-Laplacian type. Additionally, we establish a local $L^{\infty}$-bound for the spatial gradient.

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