On the Dirichlet problem for fully nonlinear elliptic hessian systems

Abstract

We consider the problem of existence and uniqueness of strong solutions u : Ω ⊂ Rn −→ RN in (H2 ∩H1 0 )(Ω)N to the problem (1) { F (·, D2u) = f, in Ω, u = 0, on ∂Ω, when f ∈ L2(Ω)N , F is a Caratheodory map and Ω is convex. (1) has been considered by several authors, firstly by Campanato and under Campanato’s ellipticity condition. By employing a new weaker notion of ellipticity introduced in recent work of the author [K2] for the respective global problem on Rn, we prove well-posedness of (1). Our result extends existing ones under hypotheses weaker than those known previously. An essential part of our analysis in an extension of the classical Miranda-Talenti inequality to the vector case of 2nd order linear hessian systems with rank-one convex coefficients.