Richard Beals

On Local Solvability of Linear Partial Differential Equations

The title indicates more or less what the talk is going to be about. I t is going to be about the problem which is probably the most primitive in partial differential equations theory, namely to know whether an equation does, or does not, have a solution. Even this is meant in the most primitive terms. I would like to begin by explaining what the terms are. As you all know, the really difficult analysis these days, and perhaps always, is the global analysis. Well, the problem that I am going to discuss is purely local—in the strictest possible sense: we would like to find out if a linear partial differential equation, with coefficients as smooth as you wish, admits locally a solution. Obviously, in this connection, negative results are very important: and negative results about local solvability have global implications. But of course positive results have also their importance. Let us state precisely what is the problem. The partial differential equation under study will be

Semigroups and abstract Gevrey spaces

Abstract Conditions are found under which a closed linear operator A in a Banach space X generates a continuous semigroup in a linear topological space Y which is dense in X . The space Y is an abstract Gevrey space associated with the operator A . This is an abstract setting for some results for hyperbolic systems with data in spaces of Gevrey functions.

On the abstract Cauchy problem

Conditions are found under which the abstract Cauchy problem u′(t) = Au(t), u(0) = u0 has a unique solution for each u0 in a dense subspace of a Banach space X. These conditions are shown to be best possible. In the Hilbert space case, conditions are found under which there is a unique weak solution for each u0in D(A). Corresponding results are obtained for the inhomogeneous problem. Application is made to some simple hyperbolic systems with multiple characteristics.

Periodic Functions and Periodic Distributions

Suppose u is a complex-valued function defined on the real line ℝ. The function u is said to be periodic with period a ≠ 0 if

The Laplace Transform

u\left( {x + a} \right) = u\left( x \right)

Foundations of multidimensional scaling

for each x ∊ ℝ.

Analysis: An Introduction

Suppose u and v are in C, the space of continuous complex-valued periodic functions. The inner product of u and v is the number (u,v) defined by