PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS TO SEVERAL COMPLEX VARIABLES

CHIN-HUEI CHANG, Purdue University

Abstract

In chapter one we consider the linear partial differential operator (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) defined in an open connected subset (OMEGA) of (//R)('n). We assume that the coefficients of P are real-analytic and its principal part is the power of a vector field X. Suppose M is a real-analytic submanifold of any codimension contained in (OMEGA) such that the vector field X is totally characteristic on M. We can construct a local flat solution to P and thus show the non-analytic hypoellipticity of P. In chapter two we first review briefly the hypoanalytic structures of a C('(INFIN)) locally integrable Frobenius algebra T'('(PERP)) defined on an open connected subset (OMEGA) in (//R)('N). For x(,0) (ELEM) (OMEGA) we denote by (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) the characteristic set of T'('(PERP)) at x(,0). For a distribution h defined in (OMEGA) such that Lh = 0 for all C('(INFIN)) section L of T'('(PERP)) we denote by WF(,ha)h the hypoanalytic wave-front set of h. We prove the following: Suppose (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) and there exists L (ELEM) T'('(PERP)) such that <(xi)('0),Im{L,L} (VBAR)(,x(,0))> > 0 then (x(,0),(xi)('0)) (NOT ELEM) WF(,ha)h for any distribution solution h of T'('(PERP)). Also for (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) if for all L (ELEM) T'('(PERP)) with L(VBAR)(,x(,0)) (NOT=) 0 we have <(xi)('0),Im{L,L} (VBAR)(,x(,0))> < 0 then there exists a C('1) solution h of T'('(PERP)) such that WF(,ha)h = {(x(,0),(mu)(xi)('0)); (mu) > 0}. The final result is that for (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) if for all L (ELEM) T'('(PERP)) <(xi)('0),Im{L,L} (VBAR)(,x(,0))> = 0 and there exists L (ELEM) T'('(PERP)) such that <(xi)('0),Re{L,{L,L}} (VBAR)(,x(,0))> (NOT=) 0 then (x(,0),(xi)('0)) (NOT ELEM) WF(,ha)h for any distribution solution h of T'('(PERP)).

Degree

Ph.D.

Subject Area

Mathematics

COinS