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Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-1316-7913
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2015 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 215, no 1, 65-87 p.Article in journal (Refereed) Published
Abstract [en]

Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.

Place, publisher, year, edition, pages
2015. Vol. 215, no 1, 65-87 p.
Keyword [en]
Elliptic-Systems, Green
National Category
Other Mathematics
URN: urn:nbn:se:kth:diva-159614DOI: 10.1007/s00205-014-0774-5ISI: 000347403500002OAI: diva2:787099
Swedish Research Council

QC 20150209

Available from: 2015-02-09 Created: 2015-02-05 Last updated: 2015-02-09Bibliographically approved

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Shahgholian, HenrikSjölin, Per
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