Lp-Theory of degenerate-elliptic and parabolic operators of second order

Abstract

SynopsisWe consider the formal operator given byin the Banach space X =LP(Rn), 1<p<∞. The coefficients ajk(x),aj(x), and a(x) are real-valued functions, ajkε C2(Rn) has bounded second derivatives, ajε Cl(Rn) has bounded first derivatives, and aεL∞(Rn). Furthermore, we assume that the n × n matrix (ajk(x)) is symmetric and positive semidefinite (i.e. ajk(x)ξjξk≧0 for all (ξ1,…,ξn)εRnandxεRn). We prove that the degenerate-elliptic differential operator given by –Aand restricted to, the minimal realization of –A, is essentially quasi-m-dispersive inLp(Rn), (hence that the minimal realization of +Ais quasi-m-accretive) and that its closure coincides with the maximal realization of –A.