The oblique derivative problem for general elliptic systems in Lipschitz domains

Abstract

Parenthetically, we observe that if L is strongly elliptic, then L− λ, λ ∈ R, satisfies the non-singularity hypothesis (3) relative to any subdomain Ω ⊆ M provided λ is sufficiently large. This is a consequence of Garding’s inequality, which is valid in our setting (even though V may be unbounded). Also, clearly, if L is strongly elliptic and negative semidefinite, then L−λ satisfies (3) for any λ > 0. A concrete example of an elliptic, formally self-adjoint operator satisfying (1)–(3) is the Hodge-Laplacian corresponding to a Riemannian metric with coefficients in H2,r, r > m. Let Ω be a Lipschitz subdomain of M , and let ν ∈ T ∗M be the unit outward conormal to ∂Ω. In order to formalize the partial derivative operator u 7→ ∇wu+Au, where A ∈ L∞(M, Hom (E , E)) and w is a vector field on M transversal to ∂Ω (that is, essinf 〈ν, w〉 > 0 on ∂Ω), we work with a first-order differential operator P = P (x, D) on E such that