The Method of Enlargement of Obstacles
The object of this chapter is to present a coarse graining method, which provides lower bounds on the bottom of the Dirichlet spectrum of the Laplacian in regions receiving many small and possibly random obstacles. This technique will for instance turn ou
- PDF / 4,509,924 Bytes
- 73 Pages / 439.37 x 666.142 pts Page_size
- 89 Downloads / 180 Views
The object of this chapter is to present a coarse graining method, which provides lower bounds on the bottom of the Dirichlet spectrum of the Laplacian in regions receiving many small and possibly random obstacles. This technique will for instance turn out to be very helpful in the study of various problems related to Poissonian obstacles. The method is outlined in Section 1. The results are then developed in Section 2 (eigenvalue estimates) and Section 3 (capacity and volume estimates). Section 4 gives several applications of the method to the study of principal Dirichlet eigenvalues of the Laplacian with Poissonian obstacles, in arbitrary dimension. In Section 5 we apply the method to investigate the long time behavior of certain quenched and annealed Wiener expectations.
4.1 Orientation In this section we shall somewhat informally outline the principle of this 'method of enlargement of obstacles', and motivate the developments of the subsequent sections. We begin with some notations. We let Jl stand for the set of locally finite simple pure point measures on IRd, d 2': 1. An element w of Jl has the form
w (1.1)
w(K) w({x})
< oo, Oorl,
forK C IRd compact, for x E !Rd.
Keeping in mind that these cloud configurations will typically be random in our applications, we shall endow Jl with the canonical a-algebra, which is generated by the applications:
w E Jl--+ w(A) E IN U {oo }, A E B(!Rd) . Very much in the spirit of Chapter 3 §3, we attach to each point of supp wan obstacle. We shall either be interested in the case of soft or hard obstacles, which will be suitably scaled. A.-S. Sznitman, Brownian Motion, Obstacles and Random Media © Springer-Verlag Berlin Heidelberg 1998
144
4. The Method of Enlargement of Obstacles
In the case of soft obstacles, we choose a fixed function W(·), nonnegative bounded measurable, compactly supported and not a.e. equal to 0. It is the (unsealed) model for the obstacle attached to each point of supp w. We then define for f E (0, 1), wED, x E IR.d: (1.2)
~(x,w) = I>- 2 w(X~Xi) =
f
€- 2
w(x~y) w(dy).
t
The nonnegative potential
~ ( ·, w)
is the soft obstacle attached to w and
€.
The model in the case of hard obstacles will be a given fixed nonpolar subset C of IR.d. The hard obstacle (or trap configuration) attached tow = I:i Ox, E [! and E E (0, 1) is the closed set:
sf= Uxi + EC.
(1.3)
i
We shall impose Dirichlet conditions on Sf. Informally, the hard obstacle case corresponds to the singular potential W (·) = oo · 1c (-). We also denote by a the positive number (1.4)
a(W) = inf{s > 0; W = 0 on B(O, s)c},
in the soft obstacle case, and (1.5)
a( C)= inf{s > 0, C
s;; B(O, s)},
in the hard obstacle case. The basic objects of interest are the 'principal eigenvalues' (although they need not be eigenvalues) (1.6)
..\~(U) ~f .Av,(-,w}(U),
f
E (0, 1), wED, Us;; IR.d open,
in the soft obstacle case and: (1.7)
..\~(U) ~ ..\(U\Sf),
in the hard obstacle case, using the notation (3.1.3) to denote the bottom of the Dirichlet spectrum of - ~ .:1 in a given open set
Data Loading...
