Laplace operators related to self-similar measures on \(\mathbb R^{d}\). (English) Zbl 1109.47038
The authors study a Laplace-type operator \(\Delta_\mu\), the Dirichlet Laplacian with respect to \(\mu\), that extends the classical Laplacian for a bounded open subset \(\Omega\) of \(\mathbb R^d\) with \( d \geq 1\) and a positive finite Borel measure \(\mu\) supported on \(\overline \Omega\) with \(\mu (\Omega)>0\). In fact, they prove that the Sobolev space \(H_0^1 (\Omega)\) is compactly embedded in \(L^2 (\Omega, \mu)\), which leads to the existence of an orthonormal basis of \(L^2 (\Omega , \mu )\) consisting of eigenfunctions of \(\Delta_\mu\) under the condition \(\underline {\dim}_\infty (\mu) > d-2\), where \(\underline{\dim}_\infty (\mu)\) denotes the lower \(L^\infty\)-dimension of \(\mu\). Under the same conditions, they also show that the Green operator associated with \(\mu\) exists and is the inverse of \(-\Delta_\mu\). Using the multifractal \(L^q\)-spectrum of the measure, the authors investigate several equivalent conditions to \(\underline {\dim}_\infty (\mu) > d-2\) for an invariant measure, especially for self-similar measures \(\mu\) defined by iterated function systems (IFS) satisfying the open set condition(OSC). Finally, using the same multifractal spectrum, they scrutinize the condition \(\underline {\dim}_\infty (\mu) > d-2\) for the measure \(\mu\) defined by an IFS satisfying the weak separation condition* (WSC*) but not the OSC.
Reviewer: Kun Soo Chang (Seoul)
MSC:
| 47F05 | General theory of partial differential operators |
| 28A80 | Fractals |
| 28A75 | Length, area, volume, other geometric measure theory |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P05 | General topics in linear spectral theory for PDEs |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
Keywords:
Laplacian; self-similar measure; eigenvalue; eigenfunction; \(L^q\)-spectrum; \(L^\infty \)-dimension; Green operatorReferences:
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