The Lp-intergrability of Green's functions andfundamental solutions for elliptic and parabolic equations

Abstract

Given d ≥ 1 and in (0,1) denote by Ad() the class of smooth, symmetric, d X d matrix-valued functions a (aij (x)) on Rd which satisfy I ≤ a(x) ≤ 1/ I x in Rd in the sense of nonnegative definiteness. Set La u = i,j=1d aij (x) ð2 u / ðxi ðxj (x) and let La* v = i,j=1d ð2 / ðyi ðyj (aij(y) v(y)) denote the adjoint of L. In the first part of this paper we study the interior behavior of nonnegative solutions, v, of the adjoint equation, La* v = 0, in a domain of Rd. Our main result is the establishment of an interior "backward Hölder inequality" for such solutions. In the second part we will use the estimate supx supa Ad () Ga (x,y) q dy < to study the integrability properties of the fundamental solution, a(t,x,y), (t,x,y) (0,) X Rd X Rd, to the parabolic initial-value problem: ðu / ðt (t,x) = La u(t,x), u(0,x) = f(x) (La u = i,j=1d aij (x) ð2 / ðxi ðxj).