Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L p Estimates
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| 008 | 210128e20150101xx s 000 0 eng | ||
| 024 | 7 | 0 | |a 10.1007/s00205-014-0774-5 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00205-014-0774-5 | ||
| 245 | 0 | 0 | |a Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L p Estimates |h [Elektronische Daten] |c [Hayk Aleksanyan, Henrik Shahgholian, Per Sjölin] |
| 520 | 3 | |a Let u ɛ be a solution to the system $$\rm div(A_\varepsilon(x)\nabla u_\varepsilon(x)) = 0 \quad\text{in}\, D,\quad u_\varepsilon(x) = g(x,x/\varepsilon)\quad\text{on} \,\partial\, D,$$ div ( A ε ( x ) ∇ u ε ( x ) ) = 0 in D , u ε ( x ) = g ( x , x / ε ) on ∂ D , where $${D \subset \mathbb{R}^d (d \geqq 2)}$$ D ⊂ R d ( d ≧ 2 ) , is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A ɛ 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 $${A_\varepsilon(x) = A(x)}$$ A ε ( x ) = A ( x ) , with the following convergence rate for all $${1\leqq p < \infty}$$ 1 ≦ p < ∞ $$\left.\begin{array}{ll}\parallel\,u_\varepsilon - u_0\parallel{L^p(D)} \leqq C_p \left\{\begin{array}{ll}\varepsilon^{1/2p}, \quad\quad\quad\quad d=2, \\(\varepsilon \mid {\rm ln} \varepsilon \mid)^{1/p}, \quad d = 3, \\ \varepsilon^{1/p}, \quad\quad\quad\quad d \geqq 4,\end{array}\right.\end{array}\right.$$ ‖ u ε - u 0 ‖ L p ( D ) ≦ C p ε 1 / 2 p , d = 2 , ( ε ∣ ln ε ∣ ) 1 / p , d = 3 , ε 1 / p , d ≧ 4 , which we prove is (generically) sharp for $${d \geqq 4}$$ 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 $${d \geqq 3}$$ d ≧ 3 ) $$\parallel\,u_\varepsilon - u_0\parallel\,L^p(D) \leqq C_p [\varepsilon ({\rm ln}(1/ \varepsilon))^2 ]^{1/p}$$ ‖ u ε - u 0 ‖ L p ( D ) ≦ C p [ ε ( ln ( 1 / ε ) ) 2 ] 1 / p for both the oscillating operator and boundary data. For this case, we take $${A_\varepsilon = A(x/ \varepsilon)}$$ A ε = A ( x / ε ) , 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. | |
| 540 | |a Springer-Verlag Berlin Heidelberg, 2014 | ||
| 700 | 1 | |a Aleksanyan |D Hayk |u School of Mathematics, The University of Edinburgh, JCMB The King's Buildings, Mayfield Road, EH9 3JZ, Edinburgh, UK |4 aut | |
| 700 | 1 | |a Shahgholian |D Henrik |u Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden |4 aut | |
| 700 | 1 | |a Sjölin |D Per |u Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden |4 aut | |
| 773 | 0 | |t Archive for Rational Mechanics and Analysis |d Springer Berlin Heidelberg |g 215/1(2015-01-01), 65-87 |x 0003-9527 |q 215:1<65 |1 2015 |2 215 |o 205 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00205-014-0774-5 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00205-014-0774-5 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Aleksanyan |D Hayk |u School of Mathematics, The University of Edinburgh, JCMB The King's Buildings, Mayfield Road, EH9 3JZ, Edinburgh, UK |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Shahgholian |D Henrik |u Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Sjölin |D Per |u Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Archive for Rational Mechanics and Analysis |d Springer Berlin Heidelberg |g 215/1(2015-01-01), 65-87 |x 0003-9527 |q 215:1<65 |1 2015 |2 215 |o 205 | ||
| 986 | |a SWISSBIB |b 56086518X | ||