ISSN:
1432-0606
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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, .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01442649
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