Introduction

Abstract

One of the main purposes of this book is to study the Dirichlet problem

$$ \left\{ {\begin{array}{*{20}{c}} {{F_i}\left( {x.u(x),Du(x)} \right) = 0,a.e.x \in \Omega ,i = 1, \ldots ,I} \\ {u(x) = \varphi (x),x \in \partial \Omega ,} \end{array}} \right. $$

(1.1)

Where $\Omega \subset {\mathbb{R}^n}$ is an open set, $u:\Omega \to {\mathbb{R}^m}$ and therefore $Du \in {\mathbb{R}^{m \times n}}$ (if m = 1 we say that the problem is scalar and otherwise we say that it is vectorial), ${F_i}:\Omega \times {\mathbb{R}^m} \times {\mathbb{R}^{m \times n}} \to \mathbb{R},{F_i} = {F_i}(x,s,\xi ),i = 1, \ldots ,I,$ are given. The boundary condition rp is prescribed (depending of the context it will be either continuously differentiable or only Lipschitz-continuous).