8.1 Introduction

We briefly recall some of the definitions, formulas and results from the book Ref.[45] which are relevant for solving the exercises of this chapter. A reader can refer the above-cited book or any other book with similar content for more detailed proofs and discussion. The most general form of a second-order linear partial differential equation (PDE) in n variables is given by

where x ∈ Ω, an open set in ℝn, aij = aji. The operator L is said to be uniformly elliptic if there exists an α > 0 such that for all x ∈ Ω and ξ ∈ ℝn. Recall that is the characteristic form, also called the principal symbol associated with the operator L. An important elliptic operator is the Laplace operator. This operator has many interesting properties − mean value property (MVP) and minimum and maximum principles. There is also a notion of a fundamental solution. This is not specific to the Laplace operator Δ, but every constant coefficient differential operator possesses a fundamental solution. A restricted definition is the following. A locally integrable function E is called a fundamental solution of L if for all smooth functions ψ with compact support, where L′ is the adjoint operator of L. Symbolically, this is written as LE = δ, the Dirac delta function. Since the operator Δ is self-adjoint, we have. Interestingly, the fundamental solution is not a solution of the Laplace equation ΔE = 0 in a strict sense but very useful in the construction/representation of solutions to the Laplace and Poisson equations. Note that ΔE = 0 in ℝn\﹛0﹜, and hence E indeed has a singularity at the origin. A fundamental solution is not unique.