Chapter 7 - Laplace and Poisson Equations

Summary

INTRODUCTION

The reasons for studying Laplace and Poisson equations are twofold.Primarily, these equations arise in a wide variety of physical contexts.Secondly, as mentioned earlier, Laplace operator is a prototype of a verygeneral class of linear elliptic operators. In fact, the Laplace operatorpossesses many features of the general class of elliptic operators. Recallthat the most general form of second-order linear partial differentialequations (PDE) in n variables is given by

where x ∈ Ω; an open set inℝⁿ,a_ij = a_ji. Theoperator L is said to be uniformlyelliptic if there exists an a > 0 suchthat for all. Recall that is the characteristic form associated with theoperator L. The ellipticity condition here implies that thecharacteristic variety is an empty set at all points in Ω.

Thus, the condition requires the uniform positive definiteness of thesymmetric matrix [a_ij(x)]. Ifwe take b_i = c_i =d = 0 and

there results the Laplace operator Δ. The ultimate interest is tostudy the classical solutions of the equationLu=f for a given dataf. The study of Laplace equationΔu=0 and Poisson equationΔu = f (potentialtheory) gives a starting point for the general theory ofLu = f. The Schaudertheory provides a general theory for Lu =f when the coefficients are smooth, that is,Hölder continuous and it is essentially an extension of the potentialtheory. The crucial result is the derivation of estimates (known asa-priori estimates), say, of the form: Anyu ∈ C²(Ω), asolution of Δu = f in adomainΩ ⊂ℝⁿ satisfies a uniform estimate:

where Ω′ ⊂⊂Ω; 0 < a < 1 andC is a constant depending only on a.Here is the standard Ḧolder space.

Such estimates eventually (with delicate analysis) lead to the solvability ofthe equation. The above estimate is an interior estimateand we also require boundary/global estimates that depend on the smoothnessof the boundary as well.