L∞ estimates on the solutions of nonselfadjoint elliptic and parabolic equations in bounded domains

This paper considers explicit upper bounds in L-infinity on the solution operator of a class of second-order parabolic Dirichlet problems defined in (-1, 1)(N). The elliptic part of the operator L is given by [GRAPHICS] where a(i) greater than or equal to d(i) > 0, \b(i)\ less than or equal to M-i, i = 1,...,N, uniformly across the domain. Symmetry and the maximum principle are used to identify those coefficients, obeying these bounds, which result in the largest possible value for the norm of the solution operator in L-infinity. The norm of this optimal case is found in terms of (d(i)) and (M-i) and a family of constant coefficient problems in one space dimension. This representation is made quantitatively explicit by Laplace transform evaluation of the one-dimensional problems. Similar sharp quantitative estimates on the resolvent parallel to (lambda I + L)(-1)parallel to(infinity), lambda greater than or equal to 0, are obtained in N (-1, 1)(N) as a corollary of the parabolic results. For comparison, a related, but direct, technique is used to derive optimal bounds on the resolvent of a slightly more general class of elliptic operators defined on the unit ball in R-N.