On the Dirichlet Problem for Fully Nonlinear Elliptic Hessian Systems

TL;DR

This paper establishes existence and uniqueness of strong solutions for a class of fully nonlinear elliptic Hessian systems with Dirichlet boundary conditions, using a new ellipticity condition and extending classical inequalities.

Contribution

It introduces a weaker ellipticity condition for Hessian systems and proves well-posedness results that extend previous work under less restrictive hypotheses.

Findings

Proves well-posedness of the Dirichlet problem for fully nonlinear Hessian systems.

Extends the Miranda-Talenti inequality to vector-valued second order systems.

Introduces a new ellipticity condition weaker than classical Campanato's.

Abstract

We consider the problem of existence and uniqueness of strong solutions $u: \Omega \subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$ in $(H^{2}\cap H^{1}_0)(\Omega)^N$ to the problem \[\label{1} \tag{1} \left\{ \begin{array}{l} F(\cdot,D^2u ) \,=\, f, \ \ \text{ in }\Omega,\\ \hspace{31pt} u\,=\, 0, \ \ \text{ on }\partial \Omega, \end{array} \right. \] when $ f\in L^2(\Omega)^N$, $F$ is a Carath\'eodory map and $\Omega$ is convex. \eqref{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 $\mathbb{R}^n$, we prove well-posedness of \eqref{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.