Jones, B. Frank2018-12-182018-12-181973Shapiro, Michael Richard. "On the existence of kernel functions for the heat equation in n dimensions." (1973) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/104723">https://hdl.handle.net/1911/104723</a>.https://hdl.handle.net/1911/104723Let 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).28 ppengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.ThesisRICE2359reformatted digitalThesis Math. 1973 Shapiro