Capacity and principal eigenvalues: The method of enlargement of obstacles revisited. (English) Zbl 0885.60063

Sznitman, Alain-Sol

Ann. Probab. 25, No. 3, 1180-1209 (1997).

From the introduction: “The motivation for the method we develop in the present article stemmed from a problem, which we now briefly describe. Consider a Poisson cloud of points \((x_i)\) in \(\mathbb{R}^d\), \(d\geq 2\), with constant intensity \(\nu>0\). Let \(W(\cdot)\) be a nonnegative, compactly supported, bounded measurable function, which is not a.e. equal to 0. It was shown that the principal Dirichlet eigenvalue \(\lambda_u\) of \(-\Delta/2+ \sum_i W (\cdot -x_i)\) in the box \((-u/2,u/2)^d\) has asymptotic behavior described by \[ \mathbb{P} \text{-a.s. } \lambda_u \sim c(d,\nu)/(\log u)^{2/d} \text{ as } u\to\infty, \tag{1.2} \] where \(c(d,\nu) \in (0, \infty )\) is a constant solely depending on \(d\) and \(\nu\). What can now be said about the fluctuations of the random variable \(\lambda_u\)? In particular, what bounds can be derived on the spread of the distribution of \(\lambda_u\) around a median? These questions, for instance, turn out to be of importance for the fine study of Brownian motion in a Poissonian potential. Some preliminary applications to the control of fluctuations of \(\lambda_u\) are developed in Section 3, but the main body of applications will be developed elsewhere.”
A new way of “enlargement of obstacles” given by the author is presented here in order to give a better way of estimating the following quantities: (i) the uniform upper-bounds one has on the possible upward shift caused by the replacement of the true obstacles by enlarged obstacles, (ii) the uniform controls on the volume of enlarged bad obstacles. This leads to the introduction of the notions of density and rarefaction boxes and to the estimation of the principal eigenvalue of the Laplacian in the given Euclidean domains. Section 2 is concerned with the notion of bad boxes and the control of their volume with the help of the capacity estimates on rarefaction boxes. The author presents in the last Section 3 some applications of his theory to the study of fluctuations of the above mentioned principal Dirichlet eigenvalue. It is noted that the paper is written in a self-contained way.

Reviewer: X.L.Nguyen (Hanoi)

MSC:

60J45	Probabilistic potential theory
35P15	Estimates of eigenvalues in context of PDEs
82D30	Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Keywords:

principal Dirichlet eigenvalue; Brownian motion; Poissonian potential

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