Counterexamples related to high-frequency oscillation of Poisson's kernel

Abstract

Let

$$L_\varepsilon = \frac{\partial }{{\partial x^i }}a^{ij} \left( {\frac{x}{\varepsilon }} \right)\frac{\partial }{{\partial x^j }},L_0 = q^{ij} \frac{{\partial ^2 }}{{\partial x_i \partial x_j }},$$

wherea is a smooth periodic matrix andL ₀ is the homogenized operator corresponding to the family (L _ε). LetD be a nice domain, and letP _ε (x, y), P ₀ (x, y) be the Poisson kernels associated withL _ε andL ₀. We show that in generalP _ε (x, ·) does not converge strongly toP ₀ (x, ·) inL ^p, by exhibiting two counterexamples. This result has the following implication in the theory of boundary control of distributed systems: if

, withz given,u _ε (x) = ∫ P _ε (x, y)g(y) andu ₀ (x) = ∫P ₀ (x,y)g(y), then, in general,

.