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Frank Duzaar

Publications and source records attributed to Frank Duzaar.

At least 19 recordsLinked to original sources
arXiv2026

Calderón-Zygmund estimates for parabolic $p$-Laplacian systems with non-divergence form right-hand sides

We establish local Calderón-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{μs}$, where the exponent $μ$ depends explicitly on $p$, $N$, and a prescribed target exponent $s>p$, then the spatial gradient of the solution enjoys improved integrability $Du \in L^s_{\rm{loc}}$. The result provides a sharp transfer of integrability from the data to the gradient, consistent with the natural parabolic scaling, and recovers the optimal exponents in the linear case $p=2$. The proof combines intrinsic scaling techniques with a Calderón-Zygmund type iteration scheme.

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arXiv2026

Sharp gradient integrability for $(s,p)$-Poisson type equations

We prove local $W^{1,q}$-regularity for weak solutions to fractional $p$-Laplacian type equations with right-hand side $f\in L^r_{\mathrm{loc}}(Ω)$. Assuming $p>1$, $s\in(0,1)$, and $sp'>1$, solutions belong to $W^{1,q}_{\mathrm{loc}}(Ω)$ for the optimal exponent $q=q(n,p,s,r)$. We obtain quantitative local gradient estimates involving nonlocal tail terms. The optimality of $q$ is confirmed by a counterexample.

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arXiv2026

Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients

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 $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1 \frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^\infty$-estimates. \emph{Higher integrability of the gradient:} $|Du|$ self-improves from $L^p_{\mathrm{loc}}$ to $L^{p(1+\varepsilon)}_{\mathrm{loc}}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms.

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arXiv2025

Schauder estimates for parabolic $p$-Laplace systems

We establish the local Hölder regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big( a(x,t)\big(μ^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $μ\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and is Hölder continuous in the space variable $x$. As an application, we prove Hölder estimates for the gradient of weak solutions to a doubly non-linear parabolic equation in the super-critical fast diffusion regime.

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arXiv2025

Parabolic PDEs with Dynamic Data under a Bounded Slope Condition

We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $$ \partial_tu-\text{div}_x \nabla_ξf(\nabla u)=0 $$ in a space-time cylinder $Ω_T=Ω\times (0,T)$, subject to time-dependent boundary data $g\colon \partial_{\mathcal{P}}Ω_T\to \mathbf{R}$ prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data $g$ along the lateral boundary $\partialΩ\times (0,T)$. More precisely, we require that for each fixed $t\in [0,T)$, the graph of $g(\cdot ,t)$ over $\partialΩ$ admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers.

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arXiv2025

Gradient estimates for the fractional $p$-Poisson equation

We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - Δ_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any $q>1$ that \begin{equation*} f\in L^{\frac{qp}{p-1}}_{\rm loc} \quad\Longrightarrow\quad \nabla u\in L^{qp}_{\rm loc}. \end{equation*} The qualitative result is accompanied by a local quantitative estimate.

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arXiv2024

Gradient regularity for $(s,p)$-harmonic functions

We study the local regularity properties of $(s,p)$-harmonic functions, i.e. local weak solutions to the fractional $p$-Laplace equation of order $s\in (0,1)$ in the case $p\in (1,2]$. It is shown that $(s,p)$-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power $q\geq 1$. As a result, $(s,p)$-harmonic functions are Hölder continuous to arbitrary Hölder exponent in $(0,1)$. In addition, the weak gradient of $(s,p)$-harmonic functions has certain fractional differentiability. All estimates are stable when $s$ reaches $1$, and the known regularity properties of $p$-harmonic functions are formally recovered, in particular the local $W^{2,2}$-estimate.

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arXiv2024

Regularity for the fractional $p$-Laplace equation

Higher Sobolev and Hölder 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. $$0<s\le\tfrac{p-2}{p}\quad \text{or} \quad\tfrac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev regularity estimates for the local $p$-Laplace are recovered. The case $p=2$ reproduces the almost $W^{1+s,2}_{\rm loc}$-regularity for the fractional Laplace equation of any order $s\in(0,1)$.

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arXiv2023

Hölder Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time cylinder. Hölder estimates are established for the gradient of its weak solutions in the super-critical fast diffusion regime $0<p-1< q<\frac{N(p-1)}{(N-p)_+}$ where $N$ is the space dimension. Moreover, decay estimates are obtained for weak solutions and their gradient in the vicinity of possible extinction time. Two main components towards these regularity estimates are a time-insensitive Harnack inequality that is particular about this regime, and Schauder estimates for the parabolic $p$-Laplace equation.

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arXiv2021

Boundary regularity for parabolic systems in convex domains

In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are assumed on that part of the lateral boundary.

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arXiv2021

On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II

We demonstrate two proofs for the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-Δ_p u=0,\quad p>2,\quad 0<q<p-1. \] The first proof takes advantage of the expansion of positivity for the degenerate, parabolic $p$-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value problems of Dirichlet type and Neumann type.

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arXiv2020

On the Hölder regularity of signed solutions to a doubly nonlinear equation

We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-Δ_p u=0,\quad p>1. \] The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain, up to the parabolic boundary. In addition, based on the expansion of positivity, we are able to give an alternative proof of Harnack's inequality for non-negative solutions. Moreover, as a consequence of the interior estimates, we also obtain a Liouville-type result.

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arXiv2018

Higher integrability for the singular porous medium system

In this paper we establish in the fast diffusion range the higher integrability of the spatial gradient of weak solutions to porous medium systems. The result comes along with an explicit reverse Hölder inequality for the gradient. The novel feature in the proof is a suitable intrinsic scaling for space-time cylinders combined with reverse Hölder inequalities and a Vitali covering argument within this geometry. The main result holds for the natural range of parameters suggested by other regularity results. Our result applies to general fast diffusion systems and includes both, nonnegative and signed solutions in the case of equations. The methods of proof are purely vectorial in their structure.

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arXiv2018

Higher integrability for doubly nonlinear parabolic systems

This paper proves a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems. The new feature of the argument is that the intrinsic geometry involves the solution as well as its spatial gradient. The main result holds true for a range of parameters suggested by other nonlinear parabolic systems.

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arXiv2013

The evolution of H-surfaces with a Plateau boundary condition

In this paper we consider the heat flow associated to the classical Plateau problem for surfaces of prescribed mean curvature. We show that an isoperimetric condition on H ensures the existence of a global weak solution. Moreover, we establish that these global solutions sub-converge, as time tends to infinity, to a conformal solution of the classical Plateau problem for surfaces of prescribed mean curvature.

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arXiv2009

Gradient estimates via non-linear potentials

We present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic equations.

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arXiv2009

Local Lipschitz regularity for degenerate elliptic systems

We start presenting an $L^{\infty}$-gradient bound for solutions to non-homogeneous $p$-Laplacean type systems and equations, via suitable non-linear potentials of the right hand side. Such a bound implies a Lorentz space characterization of Lipschitz regularity of solutions which surprisingly turns out to be independent of $p$, and that reveals to be the same classical one for the standard Laplacean operator. In turn, the a priori estimates derived imply the existence of locally Lipschitz regular solutions to certain degenerate systems with critical growth of the type arising when considering geometric analysis problems, as recently emphasized by Rivière

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