Abstract
Abstract This article is devoted to the Friedrichs inequality, where the domain is periodically perforated along the boundary. It is assumed that the functions satisfy homogeneous Neumann boundary conditions on the outer boundary and that they vanish on the perforation. In particular, it is proved that the best constant in the inequality converges to the best constant in a Friedrichs-type inequality as the size of the perforation goes to zero much faster than the period of perforation. The limit Friedrichs-type inequality is valid for functions in the Sobolev space H 1. AMS 2010 Subject Classification: 39A10; 39A11; 39A70; 39B62; 41A44; 45A05.
Highlights
1 Introduction This article deals with Friedrichs-type inequalities for functions defined on domains which have a periodic perforation along the boundary
The size, shape and distribution of the perforation are described by a small parameter
We consider the case where the functions satisfy a homogeneous Neumann condition on the part of the boundary corresponding to the domain without perforation and vanish on the perforation
Summary
This article deals with Friedrichs-type inequalities for functions defined on domains which have a periodic perforation along the boundary. In [1] (see [2]), domains with a periodical perforation along the boundary were considered and the precise asymptotics of the best constant in a Friedrichs-type inequality was established. In [3], a Friedrichs-type inequality was proved for functions vanishing on small periodically alternating pieces of the boundary. The limit problem for (2) depends on how fast the size of the perforation goes to zero relative the length of the period. In the case with p = ∞ (Dirichlet boundary conditions in the limit problem) the smallest eigenvalue λ10 for the limit problem is related to the best constant in the Friedrichs inequality for functions in H01( ). The goal of this article is to answer the following questions, in the case 0
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