1.1 Introduction

Let u = u ( x ₁, , x _n) be a function of n independent variables x ₁, , x _n. A Partial Differential Equation (PDE for short) is an equation that contains the independent variables x ₁, , x _n, the dependent variable or the unknown function u and its partial derivatives up to some order. It has the form

(1.1)

where F is a given function and are the partial derivatives of u. The order of a PDE is the order of the highest derivative which appears in the equation.

A set ? in the n-dimensional Euclidean space R ⁿ is called a domain if it is an open and connected set. A region is a set consisting of a domain plus, perhaps, some or all of its boundary points. We denote by C ( ?) the space of continuous functions in ? and by C ^k ( ?) the space of continuously differentiable functions up to the order k in ?. Suppose (1.1) is a PDE of order m. By a solution of the equation (1.1) we mean a function u ? C ^m ( ?) such that the substitution of u and its derivatives up to the order m in (1.1) makes it an identity in ( x ₁