## 35B65 Smoothness and regularity of solutions

We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a $p(x)$-Laplace-type problem with piecewise constant exponent.
We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincar\'e inequalities, and a Gehring-type Lemma for the $p(x)$-Laplacian. Finally, Schauder's fixed-point theorem is used to show the existence of solutions.

We study the expansion of the eigenfunctions of Schrödinger operators with smooth confinement potentials in Hermite functions; confinement potentials are potentials that become unbounded at infinity. The key result is that such eigenfunctions and all their derivatives decay more rapidly than any exponential function under some mild growth
conditions to the potential and its derivatives. Their expansion in Hermite functions converges therefore very fast, super-algebraically.

The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces
(2012)

The paper continues the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schr\"odinger operators and their
exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order $\vartheta$ for $\vartheta<3/4$. The bound $3/4$ is best possible and can neither be reached nor surpassed. Such results are important for the study
of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons.

Global higher integrability of minimizers of variational problems with mixed boundary conditions
(2012)

We consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order $p+\epsilon$ for a uniform $\epsilon >0$. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.

Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints
(2012)

A global higher differentiability result in Besov
spaces is proved for the displacement fields of linear elastic models
with self contact.
Domains with cracks are studied, where nonpenetration
conditions/Signorini conditions are imposed on the crack faces.
It is shown that
in a neighborhood of crack tips (in 2D) or
crack fronts (3D) the displacement fields are
$B^{3/2}_{2,\infty}$ regular.
The proof relies on a difference
quotient argument for the directions tangential to the crack. In order
to obtain the regularity estimates also in the normal direction, an
argument due to
Ebmeyer/Frehse/Kassmann is modified.
The methods are then applied to further examples like
contact problems with nonsmooth rigid foundations, to a model with
Tresca friction and
to minimization problems with
nonsmooth energies and constraints as they occur for instance in the modeling of
shape memory alloys.
Based on Falk's approximation Theorem for variational
inequalities, convergence rates for FE-discretizations of contact
problems are derived relying on the proven regularity properties.
Several numerical examples illustrate the theoretical results.

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an Lp function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.

In this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural'tseva, formulating sharper hypotheses for the existence of a classical solution.
Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higher-order estimates, and for the existence result.

We investigate the regularity of the weak solution to elliptic transmission problems that involve two layered anisotropic materials separated by a boundary intersecting interface. Under a compatibility condition for the angle of contact of the two surfaces and the boundary data, we prove the existence of square-integrable second derivatives, and the global Lipschitz continuity of the solution. We show that the second weak derivatives remain integrable to a certain power less than two if the compatibility condition is violated.

In this paper we will consider elliptic boundary value problems with
oscillatory diffusion coefficient, say A. We will derive regularity
estimates in Sobolev norms which are weighted by certain derivatives of A.
The constants in the regularity estimates then turn out to be independent of
the variations in A.
These regularity results will be employed for the derivation of error
estimates for hp-finite element discretizations which are explicit with
respect to the local variations of the diffusion coefficient.

We study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and
Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the
remaining faces. Under natural smoothness assumptions on the data we obtain
$u\in L^\infty((0,T);H^{3/2-\delta}(\Omega))$ for the displacements and
$z\in L^\infty((0,T);H^{1/2-\delta}(\Omega))$ for the internal variables.
The proof is based on a difference quotient technique and a reflection argument.