Linear Elliptic Operators

Fundamental solutions are very useful in the theory of partial differential equations. Among many applications, they are used, e.g., in solving non-homogeneous equations like (1.4.13), and in telling us about the regularity and growth of solutions. We have proved a general existence and uniqueness theorem (Theorem 1.4.2). We will now prove that every partial differential operator L(D) with constant coefficients has a fundamental solution. This result which was a conjecture until 1954 when it was established independently by Ehrenpreis (1954) and Malgrange (1955), together with the definition of hypoellipticity, implies that a differential operator with constant coefficients is hypoelliptic iff it has a fundamental solution which belongs to the class C ∞ in a region that does not contain the origin. The existence of a tempered fundamental solution for a partial differential operator with constant coefficients was proved by Hörmander (1958). We will derive fundamental solutions for the classical elliptic differential operators, like the Laplace, Helmholtz, and Cauchy-Riemann operators, and also a method for constructing fundamental solutions for homogeneous elliptic operators, and discuss maximum principle. Specific applications in the area of boundary element methods will be discussed in Chapter 10.