arXiv subjects

Fabian Bäuerlein

Publications and source records attributed to Fabian Bäuerlein.

3 recordsLinked to original sources

arXiv2026

Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition

We prove global higher integrability for functionals of double-phase type under a uniform local capacity density condition on the complement of the considered domain $Ω\subset \mathbb{R}^n$. In this context, we investigate a new natural notion of variational capacity associated to the double-phase integrand. Under the related fatness condition for the complement of $Ω$, we establish an integral Hardy inequality. Further, we show that fatness of $\mathbb{R}^n \setminus Ω$ is equivalent to a boundary Poincaré inequality, a pointwise Hardy inequality and to the local uniform $p$-fatness of $\mathbb{R}^n \setminus Ω$. We provide a counterexample that shows that the expected Maz'ya type inequality - a key intermediate step toward global higher integrability - does not hold with the notion of capacity involving the double-phase functional itself.

math.AP↗

arXiv2024

Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients

We consider vector valued weak solutions $u:Ω_T\to \mathbb{R}^N$ with $N\in \mathbb{N}$ of degenerate or singular parabolic systems of type \begin{equation*} \partial_t u - \mathrm{div} \, a(z,u,Du) = 0 \qquad\text{in}\qquad Ω_T= Ω\times (0,T), \end{equation*} where $Ω$ denotes an open set in $\mathbb{R}^{n}$ for $n\geq 1$ and $T>0$ a finite time. Assuming that the vector field $a$ is not of Uhlenbeck-type structure, satisfies $p$-growth assumptions and $(z,u)\mapsto a(z,u,ξ)$ is Hölder continuous for every $ξ\in \mathbb{R}^{Nn}$, we show that the gradient $Du$ is partially Hölder continuous, provided the vector field degenerates like that of the $p$-Laplacian for small gradients.

math.AP↗

arXiv2024

Gradient bounds for strongly singular or degenerate parabolic systems

We consider weak solutions $u:Ω_{T}\rightarrow\mathbb{R}^{N}$ to parabolic systems of the type \[ u_{t}-\mathrm{div}\,A(x,t,Du)=f \qquad \mathrm{in}\ Ω_{T}=Ω\times(0,T), \] where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $T>0$ and the datum $f$ belongs to a suitable Orlicz space. The main novelty here is that the partial map $ξ\mapsto A(x,t,ξ)$ satisfies standard $p$-growth and ellipticity conditions for $p>1$ only outside the unit ball $\{\vertξ\vert<1\}$. For $p>\frac{2n}{n+2}$ we establish that any weak solution \[ u\in C^{0}((0,T);L^{2}(Ω,\mathbb{R}^{N}))\cap L^{p}(0,T;W^{1,p}(Ω,\mathbb{R}^{N})) \] admits a locally bounded spatial gradient $Du$. Moreover, assuming that $u$ is essentially bounded, we recover the same result in the case $1<p\leq\frac{2n}{n+2}$ and $f=0$. Finally, we also prove the uniqueness of weak solutions to a Cauchy-Dirichlet problem associated with the parabolic system above. We emphasize that our results include both the degenerate case $p\geq2$ and the singular case $1<p<2$.

math.AP↗