Elliptic Partial Differential Equations of Second Order

Abstract

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by

$$ Lu{\text{: = }}{a_{ij}}\left( x \right){D_{ij}}u + {b_i}\left( x \right){D_i}u + c\left( x \right)u $$

is elliptic on Ω ⊂ ℝⁿ if the symmetric matrix [a _ij ] is positive definite for each x ∈ Ω. We have used the notation D _i u, D _ij u for partial derivatives with respect to x _i and x _i , x _j and the summation convention on repeated indices is used. A nonlinear operator Q,

$$ Q\left( u \right): = {a_{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) $$

[D u = (D ₁ u, ..., D _n u)], is elliptic on a subset of ℝⁿ × ℝ × ℝⁿ] if [a _ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝⁿ for Ω a domain in ℝⁿ. The function u will be in C ²(Ω) unless explicitly stated otherwise.