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Zbl 0922.31008
Balzano, Michele; Notarantonio, Lino
On the asymptotic behavior of Dirichlet problems in a Riemannian manifold less small random holes. (English)
[J] Rend. Semin. Mat. Univ. Padova 100, 249-282 (1998). ISSN 0041-8994

From the authors' abstract: ``In a Riemannian manifold with boundary, $\overline{M}= M\cup\partial M$, the authors study sequences of Dirichlet problems $$-\Delta u_h= f\text{ in }M\setminus E_h, \quad u_h\in H_0^1 (M\setminus E_h),$$ where $\Delta$ is the Laplace-Beltrami operator, and $E_h$ is the union of closed geodesic balls, $E_h:= \bigcup_{i=1}^h \overline{B_{r_h} (x_h^i)}$, $r_h>0$, $h\in \bbfN$, the family $\{x_h^i$; $1\leq i\leq h\}$ consists of independent, identically distributed random variables whose distribution is given by a Radon measure $\beta$ with finite energy. By means of a capacitary method and under assumptions on the asymptotic behaviour of the sequence $(r_h)_{h\in\bbfN}$, the limit problem has the form $$-\Delta u+\nu u=f \text{ in }M, \quad u\in H_0^1(M),$$ where $\nu$ is a Radon measure depending on $\beta$. The proof rests on estimates of the harmonic capacity of concentric geodesic balls''.
[N.Jacob (Erlangen)]

MSC 2000:: *31C12 Potential theory on Riemannian manifolds
31C15 Generalizations of potentials, etc.
53C21 Methods of Riemannian geometry (global)
58J37 Perturbations; asymptotics
58J32 Boundary value problems on manifolds
58J60 Relations with special manifold structures

Keywords: Dirichlet problem; Laplace-Beltrami operator; random domain; capacities

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