By Francois Treves

Targeting the archetypes of linear partial differential equations, this article for upper-level undergraduates and graduate scholars employs nontraditional ways to clarify classical fabric. subject matters comprise the Cauchy challenge, boundary worth difficulties, and combined difficulties and evolution equations. approximately four hundred workouts let scholars to reconstruct proofs. 1975 variation.

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**Example text**

Let 52 be an open subset of R3,bounded, whose boundary is a smooth surface S. The Laplace equation Ah = 0 in R has infinitely many linearly independent solutions. But we may consider only those which take preassigned values on S . Given a function f defined on S (and not necessarily anywhere else), we may seek a harmonic function h in R which is equal tof on S. Under suitable conditions, there will be one and only one such function. 2. 7) (a/ax)2u- (ajay)*U = o 16 BASIC EXAMPLES AND FUNDAMENTAL SOLUTIONS [Chap.

I BASIC EXAMPLES AND FUNDAMENTAL SOLUTIONS Now let K denote an arbitrary distribution with values in the space of p x p matrices or, equivalently, a p x p matrix whose entries are distributions. We have, by the Leibniz formula, L ( U K ) = ( L U ) K + UK' = UK'. Thus, if we wish to solve L ( U K ) = SI, which corresponds to finding a rightfundamental solution of L,we must solve K' = 6u-I = = 61. 5) = 61. 5) are well known as we have pointed out: They are the distributions of the form H(x) C, where H is the Heaviside function and C an arbitrary constant.

But there is more: Iff is an analytic function, all the solutions will also be analytic. We recall that a complex-valued function u in an open set R c R"is analytic if its Taylor expansion about any point of R converges to the function in a full neighborhood of the point (cf. 2 below). An equivalent definition is that u can be extended to an open neighborhood of R in C" as an analytic function of the complex variables zl, . , z", or, as we shall often say, as a holomorphic function. Obviously, this property that all the solutions are analytic when the righthand side is an analytic function does not hold for an arbitrary linear PDE with constant coefficients: It does not hold for the wave equations (cf.