At. Hill, L-INFINITY ESTIMATES ON THE SOLUTIONS OF NONSELFADJOINT ELLIPTIC AND PARABOLIC EQUATIONS IN BOUNDED DOMAINS, SIAM journal on mathematical analysis, 29(3), 1998, pp. 720-735
This paper considers explicit upper bounds in L-infinity on the soluti
on operator of a class of second-order parabolic Dirichlet problems de
fined 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 th
an or equal to M-i, i = 1,...,N, uniformly across the domain. Symmetry
and the maximum principle are used to identify those coefficients, ob
eying 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 c
oefficient problems in one space dimension. This representation is mad
e quantitatively explicit by Laplace transform evaluation of the one-d
imensional problems. Similar sharp quantitative estimates on the resol
vent parallel to (lambda I + L)(-1)parallel to(infinity), lambda great
er 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, techniqu
e 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.