Browsing by Author "Shapiro, Michael Richard"

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    On the existence of kernel functions for the heat equation in n dimensions
    (1973) Shapiro, Michael Richard; Jones, B. Frank
    Let be a bounded open set in Fnx (tQ,t^) such that each cross section t = nfl(Rnx {t}) is star-like. We define the lateral boundary ST Q = U SO,. L t€(t,tl) C and the parabolic boundary S^O = ô^O U t where fl. denotes the base of * c Theorem 1.1: Let be as above, then there exists a function u such that u is continuous in OU S^O* u > in Q, u = on S^O, and u is caloric in . Theorem 122; Suppose the boundary of extends continuously to a point (x',tQ) in the boundary of the base. Then there exists a kernel function in at the point (x',tQ). Theorem 1.3: There exists a kernel function at an interior point (XQ,tg) of the base of . If we restrict our attention somewhat we obtain the 2 following asymptotic relations Suppose a e c (,1], aa." e L^(,1), a(t) •> as t -* , and a > on (,T).