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 $\Omega \subset {\mathbb{R}}^2$. We suppose that the boundary is written as the union of two disjoint sets, $\partial \Omega =D\cup N$. We take traction data from the space $L^p(N)$ and Dirichlet data from a Sobolev space $W^{1,p}(D)$ and look for a solution $u$ of $Lu=0$ with the given boundary conditions. We give a scale invariant condition on $D$ and find an exponent $p_0>1$ so that for $1<p<p_0$, we have a unique solution of this boundary value problem with the non-tangential maximal function of the gradient of the solution in $L^p(\partial \Omega )$. We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with $p$ in $(p_1,1]$ for some $p_1<1$.

References in corpus (3)

• The Green function for elliptic systems in two dimensions
• The mixed problem in Lipschitz domains with general decompositions of the boundary
• The L^p Boundary Value Problems on Lipschitz Domains

Cited by in corpus (3)

• The Green function for elliptic systems in two dimensions
• The Green function for the mixed problem for the linear Stokes system in domains in the plane
• Estimates for the $L^q$-mixed problem in $C^{1,1}$-domains