Small perturbations in the type of boundary conditions for an elliptic operator

Abstract

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this "background" situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a "small" subset ωε of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a "small" subset ωε of the Dirichlet boundary. The relevant quantity that measures the "smallness" of the subset ωε differs in the two cases: while it is the harmonic capacity of ωε in the former case, we introduce a notion of "Neumann capacity" to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general ωε, under the sole assumption that it is "small" in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset ωε is a vanishing surfacic ball. Contents 1. General setting of the problem 2. Preliminary material 2.1. The Sobolev spaces H s (∂Ω), H s (Γ) and H s (Γ) 2.2. A short review of layer potentials 2.3. The fundamental solution N (x, y) to the background equation (1.3) 2.4. The capacity of a subset in R d 3. Replacing Neumann conditions by Dirichlet conditions on a "small set" 3.1. Some preliminary estimates 3.2. The representation formula 3.3. Properties of the limiting distribution µ 4. Replacing Dirichlet conditions by Neumann conditions on a "small set" 4.1. Preliminary estimates 4.2. The representation formula 5. An explicit asymptotic formula for the case of substituting Dirichlet conditions 5.1. Asymptotic expansion of the perturbed potential u ε in 2d 5.2. Adaptation to the three-dimensional case 6. An explicit asymptotic formula for the case of substituting Neumann conditions 7. Conclusion and future Directions Appendix A. A closer look to the quantity e(ω) A.1. Some differential geometry facts A.2. Derivation of "geometric" upper bounds for the quantity e(ω) Appendix B. The Peetre lemma Appendix C. Equilibrium distributions Appendix D. Some useful results about integral operators with homogeneous kernels

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

“…As an example, we mention the celebrated works of Cioranescu and Murat [10,11] and of Marčenko and Khruslov [33] and the more recent papers of Arrieta and Lamberti [4], Arrieta, Ferraresso, and Lamberti [3], and Ferraresso and Lamberti [22]. We also mention Bonnetier, Dapogny, and Vogelius [8] concerning small perturbations in the type of boundary conditions and Felli, Noris, and Ognibene [20,21] on disappearing Neumann or Dirichlet regions in mixed eigenvalue problems.…”