Horst Heck

L p-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle

Abstract Consider the equations of Navier-Stokes in the exterior of a rotating domain. It is shown that, after rewriting the problem on a fixed domain Ω, the solution of the corresponding Stokes equation is governed by a C 0-semigroup on L σ p (Ω), 1 < p < ∞, with generator . Moreover, for and initial data u 0 ∈ L σ p (Ω), we prove the existence of a unique local mild solution to the Navier-Stokes problem.

Stability estimates for the inverse boundary value problem by partial Cauchy data

In this paper we study the inverse boundary value problem for the Schrodinger equation with a potential and the conductivity equation using partial Cauchy data. We derive stability estimates for these inverse problems.

Mikhlin's Theorem for Operator–Valued Fourier Multipliers in n Variables

An operator–valued Mikhlin theorem is proved for multipliers of the form M : ℝn ℒ(X, Y) where X and Y are UMD spaces. The usual norm bounds of the classical Mikhlin condition are replaced by R–bounds. Furthermore, the concept of R–bounded variation is introduced to generalize the Marcinkiewicz Fourier multiplier Theorem to the operator–valued setting.

On the equation div u=g and Bogovskii's operator in Sobolev spaces of negative order

Consider the divergence problem with homogeneous Dirichlet data on a Lipschitz domain. Two approaches for its solutions in the scale of Sobolev spaces are presented. The first one is based on Calderon-Zygmund theory, whereas the second one relies on the Stokes equation with inhomogeneous data.

Weak Neumann implies Stokes

Abstract Consider a domain Ω ⊂ ℝn with possibly non compact but uniform C3-boundary and assume that the Helmholtz projection P exists on Lp(Ω) for some 1 < p < ∞. It is shown that the Stokes operator in Lp(Ω) generates an analytic semigroup on admitting maximal Lq-Lp-regularity. Moreover, for there exists a unique local mild solution to the Navier–Stokes equations on domains of this form provided p > n.

Optimal stability estimate of the inverse boundary value problem by partial measurements

In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem in which log-log-type estimates were derived.

Linear sampling method for identifying cavities in a heat conductor

We consider an inverse problem of identifying the unknown cavities in a heat conductor. Using the Neumann-to-Dirichlet map as measured data, we develop a linear sampling-type method for the heat equation. A new feature is that there is a freedom to choose the time variable, which suggests that we have more data than the linear sampling methods for the inverse boundary value problem associated with EIT and inverse scattering problem with near field data.

Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates

Abstract In this article, we establish logarithmic stability estimates for the determination of the perturbation of the biharmonic operator from partial data measurements when the inaccessible part of the domain is flat and homogeneous boundary conditions are assumed on this part. This is an improvement to a log-log type stability estimate proved earlier for the partial data case.

Stability Estimates for the Inverse Conductivity Problem for Less Regular Conductivities

A log-type stability estimate for the inverse conductivity problem in space dimension n ≥ 3, if the conductivity has C 3/2+ϵ regularity is proven.

Increasing stability for the inverse problem for the Schrödinger equation

In this article, we study the increasing stability property for the determination of the potential in the Schrodinger equation from partial data. We shall assume that the inaccessible part of the boundary is flat, and homogeneous boundary condition is prescribed on this part. In contrast to earlier works, we are able to deal with the case when potentials have some Sobolev regularity and also need not be compactly supported inside the domain.