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Boundary value problems on non-Lipschitz uniform domains: Stability, compactness and the existence of optimal shapes

Michael Hinz, Anna Rozanova-Pierrat, Alexander Teplyaev

Abstract

We study boundary value problems for bounded uniform domains in $\mathbb{R}^n$, $n\geq 2$, with non-Lipschitz (and possibly fractal) boundaries. We prove Poincaré inequalities with trace terms and uniform constants for uniform $(\varepsilon,\infty)$-domains within bounded common confinements. We then introduce generalized Dirichlet, Robin and Neumann problems for Poisson type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, and this also implies the norm convergence of the associated resolvents and the convergence of the corresponding eigenvalues and eigenfunctions. Based on our earlier work, we prove compactness results for parametrized classes of admissible domains, energy functionals and weak solutions. Using these results, we can verify the existence of optimal shapes in these classes.

Boundary value problems on non-Lipschitz uniform domains: Stability, compactness and the existence of optimal shapes

Abstract

We study boundary value problems for bounded uniform domains in , , with non-Lipschitz (and possibly fractal) boundaries. We prove Poincaré inequalities with trace terms and uniform constants for uniform -domains within bounded common confinements. We then introduce generalized Dirichlet, Robin and Neumann problems for Poisson type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, and this also implies the norm convergence of the associated resolvents and the convergence of the corresponding eigenvalues and eigenfunctions. Based on our earlier work, we prove compactness results for parametrized classes of admissible domains, energy functionals and weak solutions. Using these results, we can verify the existence of optimal shapes in these classes.

Paper Structure

This paper contains 8 sections, 33 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Traces on closed subsets
  3. Poincaré inequalities with uniform constants
  4. Generalized boundary value problems
  5. Stability under convergence of domains and measures
  6. Norm resolvent convergence and eigenvalues
  7. Shape admissible domains and compactness
  8. Existence of optimal shapes

Key Result

Theorem 2.1

Let $0< d\leq n$, $(n-d)/p<\beta$ and suppose that $\mu$ is a Borel measure with support $\Gamma=\mathop{\mathrm{supp}}\nolimits\mu$ and satisfying E:upperdreg. Then defines a linear map $\operatorname{Tr}_\Gamma$ from $H^{\beta,p}(\mathbb{R}^n)$ into $L^0(\Gamma,\mu)$. If in addition $\beta\leq n/p$ and $p<q<pd/(n-p\beta)$, then $\operatorname{Tr}_\Gamma$ is a bounded linear operator from $H^{\

Theorems & Definitions (74)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • Proposition 2.1
  • Remark 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.2
  • Corollary 2.3
  • ...and 64 more