DOMAIN CONTINUITY FOR AN ELLIPTIC OPERATOR OF FOURTH ORDER

Abstract

In this paper, we deal with the continuity with respect to the domain of the solutions of a first boundary value problem of fourth order in dimension 2 and 3. These dimensions are those involved in applications and are critical for this question of continuity. Indeed, continuity holds in dimension 1 thanks to Sobolev embeddings while homogenization may occur in dimensions higher than or equal to 4. Specific phenomena appear in dimensions 2 and 3. Here, we provide a necessary and sufficient condition for continuity with respect to the complementary Hausdorff metric. A main point is that the condition involves only the regularity of the limit domain and not the sequence of approximating domains. We then study various sufficient conditions for continuity in terms of the H2-capacity and we analyze a discontinuous case on an explicit example.