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Heat kernels on metric measure spaces and an application to semilinear elliptic equations

Authors:	Alexander Grigor'yan Jiaxin Hu Ka-Sing Lau

Institution:	Department of Mathematics, Imperial College, London, SW7 2BZ, United Kingdom and The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong ; Department of Mathematical Sciences, Tsinghua University, Beijing 100084 China and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong ; Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Abstract:	We consider a metric measure space $(M,d,\mu )$ and a heat kernel $p_{t}(x,y)$ on satisfying certain upper and lower estimates, which depend on two parameters $\alpha$ and $\beta$ . We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space $(M,d,\mu )$ . Namely, $\alpha$ is the Hausdorff dimension of this space, whereas $\beta$ , called the walk dimension, is determined via the properties of the family of Besov spaces $W^{\sigma ,2}$ on . Moreover, the parameters $\alpha$ and $\beta$ are related by the inequalities $2\leq \beta \leq \alpha +1$ . We prove also the embedding theorems for the space $W^{\beta /2,2}$ , and use them to obtain the existence results for weak solutions to semilinear elliptic equations on of the form $\begin{displaymath}-\mathcal{L}u+f(x,u)=g(x), \end{displaymath}$ where $\mathcal{L}$ is the generator of the semigroup associated with $p_{t}$ . The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in ${\mathbb{R}^{n}}$ .

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