The mixed problem for the Lam\'e system in two dimensions

TL;DR

This paper investigates the mixed boundary value problem for the Lamé system in two dimensions, establishing conditions for existence and uniqueness of solutions with boundary data in various function spaces.

Contribution

It provides a scale-invariant condition on the boundary partition and identifies a range of p-values for which solutions exist uniquely with controlled boundary behavior.

Findings

Unique solutions exist for 1<p<p_0 with boundary data in L^p and Sobolev spaces.

Solutions are also unique when data is in Hardy and Hardy-Sobolev spaces for p in (p_1,1].

The paper establishes a scale-invariant boundary condition for the problem.

Abstract

We consider the mixed problem for $L$ the Lam\'e system of elasticity in a bounded Lipschitz domain $ \Omega\subset\reals ^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 $p_1 <1$.