$L_\infty$\protect\unboldmath\ Estimates on the Solutions of Nonselfadjoint Elliptic and Parabolic Equations in Bounded Domains
Adrian T. Hill · SIAM Journal on Mathematical Analysis · 1998
This paper considers explicit upper bounds in $L_{\infty}$ 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 \begin{displaymath} Lu=-\sum_{i=1}^{N}a_{i}(x,\,t)\frac{\partial^{2} u}{\partial x_{i}^{2}} +\sum_{i=1}^{N}b_{i}(x,\,t)\frac{\partial u}{\partial x_{i}}, \end{displaymath} where $a_{i}\geq d_{i}>0$, $|b_{i}|\leq M_{i}$, $i=1,\ldots,\,N$, 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 $L_{\infty}$. 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 $\|(\lambda I+L)^{-1}\|_{\infty}$, $\lambda\geq 0$, are obtained in $(-1,\,1)^{N}$ 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 $\mathbb{R}^{N}$.