EXISTENCE AND REGULARITY FOR THE CAUCHY PROBLEM IN FLAT FUNCTIONS FOR RADIALLY HYPERBOLIC OPERATORS
Abstract
We consider the Cauchy problem in flat functions for certain partial differential operators which become characteristic on submanifolds of codimension (GREATERTHEQ) 2. When the submanifold is just a point in (//R)('n) (let it be the origin), after changing to polar coordinates x = t(theta), we assume our operator P(x,(PAR-DIFF)(,x)) takes the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where P(,m-j) is a partial differential operator on the sphere S('n-1) of order m-j, with smooth coefficients in (t,(theta)) (ELEM) {0,T} x S('n-1), 0 < T (LESSTHEQ) (INFIN). Sufficient conditions on P of the form (*) are given for the well-posedness of the problem Pu = f, f (ELEM) C('(INFIN))({0,T} x S('n-1)), (PAR-DIFF)(,t)('j)f(VBAR)(,t=0) = 0 for all j (ELEM) u (ELEM) C('(INFIN))({0,T} x S('n-1)) (PAR-DIFF)(,t)('j)u(VBAR)(,t=0) = 0 for all j (ELEM) . If the submanifold (containing the origin in (//R)('n+(nu))) has dimension (nu) (GREATERTHEQ) 1, after changing variables and then using cylindrical coordinates (t,x,(theta)) (ELEM) {0,T} x (OMEGA) x S('n-1), 0 < T (LESSTHEQ) (INFIN), (OMEGA) (L-HOOK) (//R)('(nu)) an open set containing the origin, we assume our operator P has the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where P(,m-j) is a partial differential operator in t(PAR-DIFF)(,x) and (PAR-DIFF)(,(theta)), defined on (OMEGA) x S('n-1), of order m-j, with smooth coefficients in (t,x,(theta)) (ELEM) {0,T} x (OMEGA) x S('n-1). When P is a second order operator, i.e. m = 2 in (**), sufficient conditions are given for the well-posedness of the problem (***) Pu = f, f (ELEM) C('(INFIN))({0,T} x (OMEGA) x S('n-1)), (PAR-DIFF)(,t)('j)f(VBAR)(,t=0) = 0 for all j (ELEM) u (ELEM) C('(INFIN))({0,T} x (OMEGA) x S('n-1)) (PAR-DIFF)(,t)('j)u(VBAR)(,t=0) = 0 for all j (ELEM) . Sufficient conditions are also given for the well-posedness of first order systems similar to (***).
Degree
Ph.D.
Subject Area
Mathematics
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