Boundary value problems for the biharmonic operator in the unit ball

Abstract

This paper deals with several boundary value problems (bvp) for the biharmonic operator in the unit ball in Rn, n ≥ 2. Some of them satisfy respectively violate Shapiro-Lopatinskii conditions. Depending on the structure of the boundary differential operators the corresponding bvp are either of Fredholm type or not. By putting some extra conditions we obtain in the latter case Fredholm bvp. In both cases subelliptic estimates for the solutions with loss of regularity 2−1k+1,k∈N are obtained. At the end of the paper degenerate Steklov bvp is studied and subelliptic estimate with loss of regularity 2 is found. Certainly, it also violates Shapiro-Lopatinskii conditions.This paper deals with several boundary value problems (bvp) for the biharmonic operator in the unit ball in Rn, n ≥ 2. Some of them satisfy respectively violate Shapiro-Lopatinskii conditions. Depending on the structure of the boundary differential operators the corresponding bvp are either of Fredholm type or not. By putting some extra conditions we obtain in the latter case Fredholm bvp. In both cases subelliptic estimates for the solutions with loss of regularity 2−1k+1,k∈N are obtained. At the end of the paper degenerate Steklov bvp is studied and subelliptic estimate with loss of regularity 2 is found. Certainly, it also violates Shapiro-Lopatinskii conditions.