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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.), Mathematics (Div.). The University of Edinburgh.
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, p. 65-87Article 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
Springer, 2015. Vol. 215, no 1, p. 65-87
##### Keywords [en]
Elliptic-Systems, Green
##### National Category
Other Mathematics
##### Identifiers
ISI: 000347403500002OAI: oai:DiVA.org:kth-159614DiVA, id: diva2:787099
##### Funder
Swedish Research Council
##### Note

QC 20150209

Available from: 2015-02-09 Created: 2015-02-05 Last updated: 2017-06-20Bibliographically approved

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#### Authority records BETA

Shahgholian, Henrik

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