Relations Between Solutions to a Discrete and Continuous Dirichlet Problem

Abstract

Consider the Dirichlet problem on the planar region W consisting of the unit disc minus a sector:$W = \left\{ {x = \left( {{x_1},{x_2}} \right):0 < |x| < 1, - {a_2} < \arg ument\left( x \right) < {a_1}} \right\}$ for some $0 < {a_1},{a_2} \leqslant \pi ,{a_1} + {a_2} >\pi$ (see Fig. 1). We denote by L _i , i = 1,2, the line segment $\left\{ {\operatorname{x} :argument\left( x \right) = {{\left( { - 1} \right)}^{i - 1}}{a_i},0 \leqslant |x| \leqslant 1} \right\}$ in ∂W. Let ϕ be a piecewise continuous function on ∂W which vanishes on L ₁ U L ₂ and let u(ϕ,•) be the harmonic function on W with boundary values ϕ. Thus △ϕ = O. The discrete analogue, or the analogue for a simple random walk with step size h, is a function u _h (ϕ, •) on hℤ ² which satisfies

$${\Delta _h}{\mu _h}\left( {\varphi ,\chi } \right): = \frac{1}{4}\sum\limits_{i = 1}^2 {\left[ {{\mu _h}\left( {\varphi ,\chi + {h_{ei}}} \right) + {\mu _h}\left( {\varphi ,\chi - {h_{ei}}} \right)} \right]} - {\mu _h}\left( {\varphi ,\chi } \right) = 0$$

for x ∈ hℤ² ∩ W, where e _i is the i — th coordinate vector. Moreover u _h (ϕ, •) should be “close to ϕ on the discrete boundary of W”. We prove an asymptotic relation (as h ↓ 0) between u _h ϕ, hv) and u(ϕ,hv) for fixed υ ∈ ℤ². This can be used to find the asymptotic behavior (as n → ∞) of the probability of a simple random walk “starting at ∞” to hit the “triangular” region $\left\{ {x \in {Z^2}:|x| \leqslant n, - {\alpha _2} \leqslant \operatorname{argument} \left( x \right) \leqslant {\alpha _1}} \right\}$ near the origin. This kind of estimate is useful for a modified version of Diffusion—Limited Aggregation.

Partially supported by a National Science Foundation grant to Cornell University