The mixed problem for the Lamé system in two dimensions

Abstract

We consider the mixed problem for L the Lamé system of elasticity in a bounded Lipschitz domain Ω⊂R2. We suppose that the boundary is written as the union of two disjoint sets, ∂Ω=D∪N. We take traction data from the space Lp(N) and Dirichlet data from a Sobolev space W1,p(D) and look for a solution u of Lu=0 with the given boundary conditions. In our main result, we find a scale-invariant condition on D and an exponent p0>1 so that for 1<p<p0, we can find a unique solution of this boundary value problem with the non-tangential maximal function of the gradient in Lp(∂Ω). We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy–Sobolev spaces with p in (p1,1] for some p1<1.