Weighted Sobolev spaces for the Laplace equation in periodic infinite strips

Vuk Milisic (1) , Ulrich Razafison (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

This paper establishes isomorphism for the Laplace operator in weighted Sobolev spaces (WSS). These spaces are similar to standard Sobolev spaces, but they are endowed with weights prescribing functions' growth or decay at infinity. Although well established in the whole space, these weighted results do not apply in the specific hypothesis of periodicity. This kind of problem appears studying singularity perturbed domains (roughness, sieves, porous media, etc). When zooming on a single pertubation pattern, one often ends with a periodic problem set on an infinite strip. We present a unified framework that enables a systematic treatment of such problems. We provide existence and uniqueness of solutions in our WSS. This gives a refined description of solutions behavior at infinity which is of importance in the multi-scale context. These isomorphism results hold for any weight exponent and any regularity index m. We then identify these solutions with the convolution of a Green function (specific to periodical infinite strips) and the given data.

Mots clés: Weighted Sobolev spaces; Hardy inequalities; isomorphism of the Laplace operator; infinite strip; Green function; convolution

Domaines: Equations aux dérivées partielles [math.AP]

Vuk Milisic, Ulrich Razafison. Weighted Sobolev spaces for the Laplace equation in periodic infinite strips. 2014. ⟨hal-00728408v3⟩