caa a22 4500 605515212 CHVBK 20210128105158.0 cr unu---uuuuu 210128e20150101xx s 000 0 eng 10.1007/s00205-014-0774-5 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0774-5 Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L p Estimates [Elektronische Daten] [Hayk Aleksanyan, Henrik Shahgholian, Per Sjölin] 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. Springer-Verlag Berlin Heidelberg, 2014 Aleksanyan Hayk School of Mathematics, The University of Edinburgh, JCMB The King's Buildings, Mayfield Road, EH9 3JZ, Edinburgh, UK aut Shahgholian Henrik Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden aut Sjölin Per Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 215/1(2015-01-01), 65-87 0003-9527 215:1<65 2015 215 205 https://doi.org/10.1007/s00205-014-0774-5 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0774-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Aleksanyan Hayk School of Mathematics, The University of Edinburgh, JCMB The King's Buildings, Mayfield Road, EH9 3JZ, Edinburgh, UK aut NATIONALLICENCE 700 1- Shahgholian Henrik Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden aut NATIONALLICENCE 700 1- Sjölin Per Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 215/1(2015-01-01), 65-87 0003-9527 215:1<65 2015 215 205 SWISSBIB 56086518X