Searcharxiv⌕ Search

arXiv subjects

Joachim Rehberg

Publications and source records attributed to Joachim Rehberg.

At least 19 recordsLinked to original sources
arXiv2025

Sharp angle estimates for second order divergence operators

This article is about the (minimal) sector containing the numerical range of the principal part of a linear second-order elliptic differential operator defined by a form on closed subspaces V of the first-order Sobolev space $W^{1,2}(Ω)$ incorporating mixed boundary conditions. We collect a comprehensive array of results on the angle of sectoriality and the $H^\infty$-angle attached to realizations of the elliptic operator. We thereby consider the operator in several scales of Banach spaces: the Lebesgue space, the negative Sobolev space, and their interpolation scale. For the latter two types of spaces, we rely on recent results regarding the Kato square root property. We focus on minimal assumptions on geometry, and we consider both real and complex coefficients. Not all results presented are new, but we strive for a streamlined and comprehensive overall picture from several branches of operator theory, and we complement the existing results with several new ones, in particular aiming at explicit estimates built on readily accessible problem data. This concerns for example a new estimate on the angle of the sector containing the numerical range of a linear, continuous and coercive Hilbert space operator, but also an explicit estimate for the angle of sectoriality for the elliptic operator on $L^p(Ω)$ with complex coefficients without any assumptions on geometry and a general transfer principle for the Crouzeix-Delyon theorem from bounded operators to sectorial ones, keeping the explicit constant.

math.FA↗
arXiv2025

Optimal Sobolev Regularity for Second Order Divergence Elliptic Operators on Domains with Buried Boundary Parts

We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $Ω$ in $\mathbb R^3$. The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on one part $D$ of the boundary and Neumann on the complementing part. The peculiarity is that $D$ is partly `buried' in $Ω$ in the sense that the topological interior of $Ω\cup D$ properly contains $Ω$. The main result is that the singularity of the solution along the border of the buried contact behaves exactly as the singularity for the solution of a mixed boundary value problem along the border between the Dirichlet and the Neumann boundary part.

math.AP↗
arXiv2024

Bounded functional calculus for divergence form operators with dynamical boundary conditions

We consider divergence form operators with complex coefficients on an open subset of Euclidean space. Boundary conditions in the corresponding parabolic problem are dynamical, that is, the time derivative appears on the boundary. As a matter of fact, the elliptic operator and its semigroup act simultaneously in the interior and on the boundary. We show that the elliptic operator has a bounded $\mathrm{H}^\infty$-calculus in $\mathrm{L}^p$ if the coefficients satisfy a $p$-adapted ellipticity condition. A major challenge in the proof is that different parts of the spatial domain of the operator have different dimensions. Our strategy relies on extending a contractivity criterion due to Nittka and a non-linear heat flow method recently popularized by Carbonaro-Dragičević to our setting.

math.AP↗
arXiv2022

Hölder Regularity for Domains of Fractional Powers of Elliptic Operators with Mixed Boundary Conditions

This work is about global Hölder regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the realization of an elliptic differential operator in a negative Sobolev space with integrability $q > d$ embeds into a space of Hölder continuous functions, then so do the domains of suitable fractional powers of this operator. The second main result then establishes that the premise of the first is indeed satisfied. The proof goes along the classical techniques of localization, transformation and reflection which allows to fall back to the classical results of Ladyzhenskaya or Kinderlehrer. One of the main features of our approach is that we do not require Lipschitz charts for the Dirichlet boundary part, but only an intriguing metric/measure-theoretic condition on the interface of Dirichlet- and Neumann boundary parts. A similar condition was posed in a related work by ter Elst and Rehberg in 2015, but the present proof is much simpler, if only restricted to space dimension up to 4.

math.AP↗
arXiv2020

On the numerical range of second order elliptic operators with mixed boundary conditions in $L^p$

We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on $L^p$ in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [Chill et al. 2006]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin -- instead of classical Neumann -- boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions.

math.AP↗
arXiv2020

Extrapolated Elliptic Regularity and Application to the van Roosbroeck system of Semiconductors

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order $X^{s-1,q}_D(Ω)$ for $s > 0$ small, including mixed boundary conditions and with a fully nonsmooth geometry of $Ω$ and the Dirichlet boundary part $D$. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.

math.AP↗
arXiv2019

On the numerical range of sectorial forms

We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper $\mathcal{H}^\infty$-angle for the $\mathcal{H}^\infty$-calculus on $L_p$ for all $p \in (1,\infty)$ if the coefficients are real valued.

math.AP↗
arXiv2018

$L^\infty$-estimates for the Neumann problem on general domains

Let $Ω\subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(Ω) \subset L^r(Ω)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on $Ω$. Let $q > d$ with $\frac{1}{2}-\frac{1}{q} > \frac{1}{r}$. If $p$ is the dual exponent of $q$, then we show that the pre-image of the space $(W^{1,p}(Ω))^*$ under the map $A$ is contained in the space of bounded functions on $Ω$. The considerations are complemented by results on optimal Sobolev regularity for $A$.

math.AP↗
arXiv2018

The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

math.AP↗
arXiv2017

The full Keller-Segel model is well-posed on nonsmooth domains

In this paper we prove that the full Keller-Segel system, a quasilinear strongly coupled reaction-crossdiffusion system of four parabolic equations, is well-posed in space dimensions 2 and 3 in the sense that it always admits an unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. The proof is done via an abstract solution theorem for nonlocal quasilinear equations by Amann and is carried out for general source terms. It is fundamentally based on recent nontrivial elliptic and parabolic regularity results which hold true even on rather general nonsmooth spatial domains. This enables us to work in a nonsmooth setting which is not available in classical parabolic systems theory. Apparently, there exists no comparable existence result for the full Keller-Segel system up to now. Due to the large class of possibly nonsmooth domains admitted, we also obtain new results for the "standard" Keller-Segel system consisting of only two equations as a special case.

math.AP↗
arXiv2016

On maximal parabolic regularity for non-autonomous parabolic operators

We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

math.AP↗
arXiv2016

Optimal Control of the Thermistor Problem in Three Spatial Dimensions

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

math.OC↗
arXiv2015

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.

math.AP↗
arXiv2014

The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$

We show that, under general conditions, the operator $\bigl (-\nabla \cdot μ\nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(Ω)$ and $L^p(Ω)$, for $p \in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $Ω$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $\overline {\partial Ω\setminus D}$ the existence of bi-Lipschitzian boundary charts is required.

math.CA↗
arXiv2013

Parabolic equations with dynamical boundary conditions and source terms on interfaces

We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable $L^p$-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.

math.AP↗
arXiv2009

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.

math.AP↗