Regularity of solutions to characteristic boundary value problem for symmetric systems

Abstract

Let Ω be a bounded open set in R ⁿ with smooth boundary ∂Ω. In Ω we study a first order symmetric system

$$ Lu = \sum\limits_{j = 1}^n {A_j \left( x \right)\partial _j u + B\left( x \right)u,\,A_j \left( x \right),B\left( x \right) \in C^\infty \left( {\bar \Omega } \right),\,A_j^* \left( x \right) = A_j \left( x \right)} $$

where u = (u ₁,…,u _N) and ∈_j = ∈/∈x _j. Recall that the boundary matrix is given by

$$ A_b \left( x \right) = \sum\limits_{j = 1}^n {v_j \left( x \right)A_j \left( x \right),\,x \in \partial \Omega } $$

where v(x) = (v ₁(x),…,v _n(x)) is the unit outward normal to Ω. Let be symmetric positive definite. We study the following boundary value problem

$$ \left\{ {\begin{array}{*{20}c} {\left( {L + \lambda H} \right)u = f\,{\text{in}}\,\Omega } \\ {u \in M\,{\text{at}}\,\partial \Omega } \\ \end{array} \,} \right. $$

((BVP))

where M(x) is a linear subspace of C ^N. We are concerned with the case in which the rank of A _b is not constant.