Abstract
We derive interior L p-estimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our approach relies on BMO-estimates and an interpolation argument, gradients are treated with the help of finite differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an L p-corrector result for approximate solutions.
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Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)
Avellaneda, M., Lin, F.H.: Compactness methods in the theory of homogenization. Commun. Pure Appl. Math. 40(6), 803–847 (1987)
Avellaneda, M., Lin, F.H.: Counterexamples related to high-frequency oscillation of Poisson’s kernel. Appl. Math. Optim. 15(2), 109–119 (1987)
Avellaneda, M., Lin, F.H.: Compactness methods in the theory of homogenization. II. Equations in nondivergence form. Commun. Pure Appl. Math. 42(2), 139–172 (1989)
Avellaneda, M., Lin, F.H.: L p bounds on singular integrals in homogenization. Commun. Pure Appl. Math. 44(8–9), 897–910 (1991)
Astala, K., Faraco, D., Székelyhidi, L.: Convex integration and the L p theory of elliptic equations. MPI-MIS (2004, preprint)
Caffarelli, L.A., Peral, I.: On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51(1), 1–21 (1998)
Campanato, S.: Su un teorema di interpolatione di G Stampacchia. Ann. Sc. Norm. Sup. Pisa 20, 649–652 (1966)
Campanato, S.: Equazioni ellittiche del secondo ordine e spazi \({\mathcal{L}^{2,\lambda}}\) , Ann. Mat. Pura Appl. 69, 321–380 (1965)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)
Giaquinta, M.: Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (1993)
Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)
Kristensen, J., Melcher, C.: Regularity in oscillatory nonlinear elliptic systems. Math. Z. (2008, in press)
Li, Y., Nirenberg, L.: Estimates for elliptic systems from composite material. Commun. Pure Appl. Math. 56(7), 892–925 (2003)
Lipton R. Homogenization theory and the assessment of extreme field values in composites with random microstructure. SIAM J. Appl. Math. 65(2), 475–493, 2004/2005 (electronic)
Morrey, C.B. Jr. : Multiple Integrals in the Calculus of Variations. Springer Heidelberg, New York (1966)
Meyers, N.G.: An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17(3), 189–206 (1963)
Schweizer, B.: Uniform estimates in two periodic homogenization problems. Commun. Pure Appl. Math. 53(9), 1153–1176 (2000)
Stampacchia, G.: The spaces L p,λ,N p,λ and interpolation. Ann. Sc. Norm. Sup. Pisa 19, 443–462 (1965)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Space. Princeton University Press, Princeton (1975)
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Melcher, C., Schweizer, B. Direct approach to L p estimates in homogenization theory. Annali di Matematica 188, 399–416 (2009). https://doi.org/10.1007/s10231-008-0078-1
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DOI: https://doi.org/10.1007/s10231-008-0078-1