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

Abstract

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