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Heat kernels on metric measure spaces and an application to semilinear elliptic equations. (English) Zbl 1018.60075

Let \((M, d, \mu)\) be a metric measure space and let \(p_t(x, y)\) be a heat kernel on \(M\). It is assumed that the heat kernel satisfies the estimates \[ \frac{1}{t^{\alpha/\beta}} \Phi_1\left(\frac{d(x,y)}{t^{1/\beta}}\right)\leq p_t(x,y)\leq \frac{1}{t^{\alpha/\beta}} \Phi_2\left(\frac{d(x,y)}{t^{1/\beta}}\right) \] for \(\mu\)-almost every \(x,y\in M\) and all \(t > 0\); here \(\alpha\) and \(\beta\) are positive constants and \(\Phi_1\) and \(\Phi_2\) are nonnegative decreasing functions on \([0,\infty)\) with \(\Phi_1(1) > 0\). If \(\Phi_2\) satisfies a specific growth condition, it is shown that \(\alpha\) is the Hausdorff dimension of \((M, d,\mu)\) and \(\beta\), termed the walk dimension, is also an invariant of the space. The parameter \(\beta\) is defined in terms of the Besov spaces \(W^{\sigma,2}\) on \(M\) that serve in this general context as a generalization for classical Sobolev spaces on \(\mathbb{R}^n\). Moreover, it is shown that \(2\leq\beta\) and, if the metric space \((M,d)\) satisfies a natural chain condition, then \(\beta\leq \alpha +1\). The authors investigate and prove slightly weaker results under more mild growth conditions on \(\Phi_2\).
The heat kernel \(p_t\) determines a semigroup with infinitesimal general \(L\). The authors obtain existence results for semilinear equations of the form \(-Lu+ f(x,u) = g(x)\). These theorems are obtained by the use of embedding results for the Besov spaces \(W^{\beta/2,2}\) which are also proved in the present paper. Several examples of metric measure spaces and heat kernels satisfying the relevant hypotheses are given, including the fractal domain known as the generalized Sierpiński carpet in \(\mathbb{R}^n\).

MSC:

60J35 Transition functions, generators and resolvents
28A80 Fractals
35J60 Nonlinear elliptic equations
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