The boundary behavior of heat kernels of Dirichlet Laplacians (Q1614750)

It is obtained qualitative sharp description of heat kernel \(G\) of Dirichlet Laplacian on bounded \(C^{1,1}\) domain \(D.\) There exists positive constants \(c_1, c_2\) such that, for \(\rho(x)=\text{dist}(x,\partial D)\) \[ \left[{\rho(x)\rho(y)\over t}\wedge 1\right]{c_1\over t^{n/2} } e^{-c_2|x-y|^2/t} \leq G(x,t;y,0)\leq \left[{\rho(x)\rho(y)\over t}\wedge 1\right]{1\over c_1 t^{n/2} } e^{-|x-y|^2/(c_2t)} \] for all \(x,y\in D\) and \(0<t\leq T.\)