Potential theory in conical domains. III (Q2760486)

The main subject of the paper are estimates on the heat kernel of a second order elliptic operator with periodic coefficients with Dirichlet boundary conditions in a conical domain in \({\mathbb R}^d\). Analogous problems for the Laplace operator were treated in the first part [ibid. 125, 335-384 (1999; Zbl 0918.31008)]. NEWLINENEWLINENEWLINEMore precisely, let \(Q=\sum_{i,j}{\partial \over \partial x_i}a_{ij}(x) {\partial \over \partial x_j}\), \(x\in {\mathbb R}^d\), be an elliptic, formally self adjoint operator with sufficiently smooth, periodic coefficients. Let \(z(t)\in {\mathbb R}^d\) be the diffusion generated by \(Q\). Let \(\Omega \subset {\mathbb R}^d\) be a conical domain with vertex at 0, and let \(\tau=\inf\{t: z(t)\notin \Omega\}\) be the exit time. Denote NEWLINE\[NEWLINE p_t(x,y) dy={\mathbb P}(t<\tau,\;z(t)\in dy), \quad P(t,x)=\int_{\Omega}p_t(x,y) dy ={\mathbb P}(t<\tau). NEWLINE\]NEWLINE Moreover, let \(u\) be the first eigenfunction of the \(d-1\) dimensional spherical Laplacian, \(\Delta_{d-1}u=-\lambda u\), let \(\alpha \) satisfy \(\alpha(\alpha+d-2)= \lambda\), and extend \(u\) to \(\Omega\) by \(u(x)=r^{\alpha}u(x/|x|)\). Note that \(u\), known as the reduite, is harmonic in \(\Omega\) with zero boundary condition. NEWLINENEWLINENEWLINEThe main result of the paper is the following upper estimate (proved under certain smoothness assumptions on \(\Omega\)): There exists \(C>0\) such that NEWLINE\[NEWLINE p_t(x,y)\leq C{u(x)u(y) \over t^{\alpha+d/2}} \exp\left(-{d^2(x,y) \over Ct}\right), \quad x,y\in \Omega,\;d(x,\partial \Omega), d(y,\partial \Omega)\geq 1 , NEWLINE\]NEWLINE where \(d(\cdot, \cdot) \) is some translation invariant distance on \({\mathbb R}^d\). Integration gives the estimate for \(P(t,x)\). The author also obtains lower estimate for \(p_t(x,y)\) which is not optimal. The key to the proof of the upper estimate is the result saying that the reduite of \(Q\) and the reduite of \(\Delta\), \(u\), are comparable. NEWLINENEWLINENEWLINEThat result relies on the techniques coming from homogenization theory. Analogous results are discussed and proved for cases in which \({\mathbb R}^d\) is replaced by more general spaces like \(M={\mathbb R}^d\times K\), where \(K\) is a compact smooth manifold, or a covering manifold.