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 establish a range of isomorphism for the Laplace operator in weighted Sobolev spaces. These spaces are similar to standard Sobolev spaces, but they are endowed with weights prescribing the 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. We provide a complete framework in order to fill this gap. The weights, which arise naturally from Hardy inequalities, permit to prove fundamental Poincaré inequalities. The results presented here provide a unified framework for microscopic problems in the boundary layers theory. Computing a periodic Green function for the Laplace operator, we construct explicit weak solutions for a wide range of weights. We identify these convolutions with variational solutions as well.

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. 2012. ⟨hal-00728408v1⟩