Gaussian estimates for degenerate diffusion
The author shows Gaussian estimates for the kernel of the semigroup generated by the operator \(m \Delta\) on the weighted Lebesgue space \(L^p(\Omega, m^{-1}(x) \, dx)\) for \(p \in [1,\infty)\) and bounded measurable functions \(m: \Omega \to (0,\infty)\) on arbitrary open sets \(\Omega \subset \mathbb{R}^N\). In the main result it is shown that if \(m^{-1} \in L^q(\Omega)\) for some \(q > \max(N/2,1)\), the operator \(m\Delta\) generates a strongly continuous semigroup on \(L^p(\Omega, m^{-1}(x) \, dx)\) for all \(p \in [1, \infty)\). Furthermore, the semigroup \((e^{m\Delta t})_{t \geq 0}\) is then given by a non-negative bounded measurable kernel \(K(t,x,y)\) which satisfies NEWLINE\[NEWLINEK(t,x,y) \leq C_{\hat{N},q} \cdot t^{-\frac{\hat{N}(q-1)}{2q-\hat{N}}} e^{-\frac{\|x-y\|^2}{4\tilde{c}t}} \qquad \text{for all } t > 0,NEWLINE\]NEWLINE where the constants \(\tilde{c}\) and \(\hat{N}\) depend on \(\|m\|_{\infty}\) and \(N\) respectively. The proof mainly relies on the form methods introduced by Kato and Lions.
scientific article; zbMATH DE number 6204266
scientific article