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

Abstract:

We consider a metric measure space $(M,d,\mu )$ and a heat kernel $p_{t}(x,y)$ on $M$ 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 $M$. 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 $M$ of the form \begin{equation*} -\mathcal {L}u+f(x,u)=g(x), \end{equation*} 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 Sierpiński carpet in ${\mathbb {R}^{n}}$.