Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates

作者：Aleksanyan Hayk^*; Shahgholian Henrik; Sjolin Per

来源：Archive for Rational Mechanics and Analysis, 2015, 215(1): 65-87.

DOI：10.1007/s00205-014-0774-5

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.

• 出版日期2015-1