Justin L. Taylor | Searcharxiv

arXiv2012

Convergence of Dirichlet Eigenvalues for Elliptic Systems on Perturbed Domains

We consider the eigenvalues of an elliptic operator for systems with bounded, measurable, and symmetric coefficients. We assume we have two non-empty, open, disjoint, and bounded sets and add a set of small measure to form the perturbed domain. Then we show that the Dirichlet eigenvalues corresponding to the family of perturbed domains converge to the Dirichlet eigenvalues corresponding to the unperturbed domain. Moreover, our rate of convergence is independent of the eigenvalues. In this paper, we consider the Lamé system, systems which satisfy a strong ellipticity condition, and systems which satisfy a Legendre-Hadamard ellipticity condition.

math.AP↗

arXiv2011

The mixed problem in Lipschitz domains with general decompositions of the boundary

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $Ω\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partialΩ=D\cup N$, $D$ and $N$ disjoint. We let $Λ$ denote the boundary of $D$ (relative to $\partialΩ$) and impose conditions on the dimension and shape of $Λ$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $Ω$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev space $W^ {1,p}(D) $ has a unique solution with the non-tangential maximal function of the gradient of the solution in $L^p(\partialΩ)$. We also obtain results for $p=1$ when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.

math.AP↗