Gaussian estimates for second order elliptic operators with boundary conditions. (English) Zbl 0879.35041

The authors derive Gaussian estimates for the kernel of the semigroup generated by the operator \[ {\mathcal A}u= -\sum_{i,j=1}^{d} D_{j}(a_{ij}D_{i}u)+ \sum_{i=1}^{d} b_{i}D_{i}u- \sum_{i=1}^{d} D_{i}(c_{i}u)+c_{0}u, \] with real (not necessary symmetric) coefficients \(a_{ij}\in L^{\infty}(\Omega)\) satisfying a uniform ellipticity condition; \(b_{i}\), \(c_{i}\in W^{1,\infty}(\Omega)\) and \(c_{0}\in L^{\infty}(\Omega)\), \(\Omega\in {\mathbb{R}}^{d}\).
It is studied the realization \(A\) of \(\mathcal A\) in \(L^{2}(\Omega)\) obtained by quadratic form methods. It is shown that, in the cases of Dirichlet, Neumann, and Robin boundary conditions, \(A\) generates a semigroup \(S=(e^{-tA})_{t>0}\) given by a kernel \((K_{t})_{t>0}\) which satisfies a Gaussian estimate \(|K_{t}(x;y)|\leq ct^{-d/2}e^{-b|x-y|^{2}t^{{-1}}}e^{{\omega t}}\), \((x;y)\) almost everywhere for all \(t>0\). Moreover, the authors show that \(A+\omega I\) has a bounded \(H^{\infty}\)-calculus and it has bounded imaginary powers if \(\omega\) is large enough.


35J25 Boundary value problems for second-order elliptic equations
47A60 Functional calculus for linear operators
47D06 One-parameter semigroups and linear evolution equations