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.
|Number of pages||28|
|Journal||Journal of Differential Equations|
|State||Published - Jun 15 2013|
Bibliographical noteFunding Information:
E-mail address: email@example.com (R.M. Brown). 1 Katharine Ott is partially supported by a grant from the US National Science Foundation, DMS 1201104. 2 Russell Brown is partially supported by a grant from the Simons Foundation (#195075).
ASJC Scopus subject areas
- Applied Mathematics