Effective Laws for the Poisson Equation on Domains with Curved Oscillating Boundaries
نویسندگان
چکیده
In this article, we derive approximations and effective boundary laws for solutions uε of the Poisson equation on a domain Ωε ⊂ Rn whose boundary differs from the smooth boundary of a domain Ω ⊂ R n by rapid oscillations of size ε. First, we construct a boundary layer correction which yields an O(ε3/2) approximation in the energy norm, and an O(ε2) approximation in the L2-norm if Ω is bounded. Then, we show that for 1 ≤ p ≤ 2 an O(ε1+1/p)-approximation in the Lp-norm can already be obtained by solving an effective equation on Ω satisfying a boundary condition of Robin type.
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