A priori estimates for solution operators of diffusion equations

摘要

The diffusion equation d=Au is considered, where u=u(t,x), t0, and A is a second order uniformly elliptic differential operator in RmWhose coefficients are bounded. Other conditions are prescribed on A to generate known soiution operators. We derive growth estimates for these solution operators in certain function spaces together with estimates for their derivatives in t and also estimates on the products of the first two spatial derivatives with these solution operators. Bounds on the solution operators are given which depmd only upon the i.u.b.'s for the ternination coefficients of A and the formal adjoint A_ * of A : These estimates are best with respect to each function space considered in the sense that equality holds for a particular solution operator