Failure of Fredholm solvability for the Dirichlet problem corresponding to weakly elliptic systems

Abstract

It is known (cf. Martell et al. in Rev Mat Iberoam 32(3):913–970, 2016) that the $$L^p$$ -Dirichlet problem for constant coefficient second-order systems satisfying the Legendre-Hadamard (strong) ellipticity condition is well posed in the upper half-space. However, this may fail if only weak ellipticity for the system in question is assumed. In this paper we shall show that the aforementioned failure is at a fundamental level, in the sense that there exist systems which are weakly elliptic (i.e., their characteristic matrix is invertible) for which the $$L^p$$ -Dirichlet problem in the upper half-space is not even Fredholm solvable.