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Robert Haller-Dintelmann

Publications and source records attributed to Robert Haller-Dintelmann.

13 recordsLinked to original sources
arXiv2021

The Kato Square Root Problem follows from an Extrapolation Property of the Laplacian

On a domain $Ω\subseteq \mathbb{R}^d$ we consider second order elliptic systems in divergence form with bounded complex coefficients, realized via a sesquilinear form with domain $V \subseteq H^1(Ω)$. Under very mild assumptions on $Ω$ and $V$ we show that the Kato Square Root Problem for such systems can be reduced to a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends an earlier result of McIntosh to non-smooth coefficients.

math.FA↗
arXiv2021

The Kato Square Root Problem for Mixed Boundary Conditions

We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent $\frac{1}{2}$ its fractional power domains still coincide with suitable Sobolev spaces of optimal regularity. In combination with a reduction theorem recently obtained by the authors, this solves the Kato Square Root Problem for elliptic second order operators and systems in divergence form under the same geometric assumptions.

math.FA↗
arXiv2021

Rigorous Analysis and Dynamics of Hibler's sea ice model

This article develops for the first time a rigorous analysis of Hibler's model of sea ice dynamics. Identifying Hibler's ice stress as a quasilinear second order operator and regarding Hibler's model as a quasilinear evolution equation, it is shown that Hibler's coupled sea ice model, i.e., the model coupling velocity, thickness and compactness of sea ice, is locally strongly well-posed within the $L_q$-setting and also globally strongly well-posed for initial data close to constant equilibria.

math.AP↗
arXiv2021

Extendability of functions with partially vanishing trace

Let $Ω\subseteq \mathbb{R}^d$ be open and $D\subseteq \partialΩ$ be a closed part of its boundary. Under very mild assumptions on $Ω$, we construct a bounded Sobolev extension operator for the Sobolev space $\mathrm{W}^{k , p}_D (Ω)$, $1 \leq p < \infty$, which consists of all functions in $\mathrm{W}^{k , p} (Ω)$ that vanish in a suitable sense on $D$. In contrast to earlier work, this construction is global and \emph{not} using a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing $D$ and $\partial Ω\setminus D$. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on $D$.

math.CA↗
arXiv2020

The Kato Square Root Problem on locally uniform domains

We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet boundary part is Ahlfors--David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product we establish new regularity results for the fractional powers of the Laplacian with boundary conditions in our geometric setup.

math.FA↗
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↗
arXiv2011

Energy flow above the threshold of tunnel effect

We consider the Klein-Gordon equation on two half-axes connected at their origins. We add a potential that is constant but different on each branch. In a previous paper, we studied the L-infinity-time decay via Hörmander's version of the stationary phase method. Here we apply these results to show that for initial conditions in an energy band above the threshold of the tunnel effect a fixed portion of the energy propagates between group lines. Further we consider the situation that the potential difference tends to infinity while the energy band of the initial condition is shifted upwards such that the particle stays above the threshold of tunnel effect. We show that the total transmitted energy as well as the portion between the group lines tend to zero like the inverse of the square root of the higher potential in the corresponding branch if this potential tends to infinity. At the same time the cone formed by the group lines inclines to the t-axis while its aperture tends to zero.

math.AP↗
arXiv2011

The Influence of the Tunnel Effect on L-infinity-time decay

We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. Exploiting a spectral theoretic solution formula from a previous paper, we study the L-infinity-time decay via Hörmander's version of the stationary phase method. We analyze the coefficient c of the leading term c t^{-1/2} of the asymptotic expansion of the solution with respect to time. For two branches we prove that for an initial condition in an energy band above the threshold of tunnel effect, this coefficient tends to zero on the branch with the higher potential, as the potential difference tends to infinity. At the same time the incline to the t-axis and the aperture of the cone of t^{-1/2}-decay in the (t,x)-plane tend to zero.

math.AP↗
arXiv2010

Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation

We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. Further we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore the approach via the Sturm-Liouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the tunnel effect generalized eigenfunctions is justified for example by the perspective to study the influence of tunnel effect on the L-infinity-time decay.

math.AP↗
arXiv2009

The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: Expansions in generalized eigenfunctions

We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore the approach via the Sturm-Liouville theory for systems is not well-suited.

math.SP↗
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↗