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Applications of Fou...
Applications of Fourier Analysis in Homogenization of the Dirichlet Problem : L-p Estimates
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- Aleksanyan, Hayk (författare)
- KTH,Matematik (Avd.),The University of Edinburgh
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- Shahgholian, Henrik (författare)
- KTH,Matematik (Inst.)
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- Sjölin, Per (författare)
- KTH,Matematik (Inst.)
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KTH Matematik (Avd) (creator_code:org_t)
- 2014-08-08
- 2015
- Engelska.
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer. - 0003-9527 .- 1432-0673. ; 215:1, s. 65-87
- Relaterad länk:
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http://arxiv.org/pdf...
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https://urn.kb.se/re...
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https://doi.org/10.1...
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Abstract
Ämnesord
Stäng
- 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.
Ämnesord
- NATURVETENSKAP -- Matematik -- Annan matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Other Mathematics (hsv//eng)
Nyckelord
- Elliptic-Systems
- Green
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