Study of a singular equation set in the half-space

Chérif Amrouche (1) , Fabien Dahoumane (1) , Guy Vallet (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

This work is dedicated to the resolution of a singular equation set in the half-space, with a diffusion coe cient that blows up on the boundary. More precisely, for a datum g : R3 → R, our problem consists in seeking u : R3 → R formally solution to: −div(1/x3 grad u) = g in R3, u = 0 on Γ = R2 × {0} . We give existence and uniqueness results of weak and strong solutions in suitable weighted spaces, where the weight depends on x3 .

Mots clés: Elliptic equation; singular equation; half-space; weighted spaces; weak solutions; strong solutions; Hardy's inequality

Domaines: Equations aux dérivées partielles [math.AP]

Chérif Amrouche, Fabien Dahoumane, Guy Vallet. Study of a singular equation set in the half-space. 2010. ⟨hal-00629120⟩