## Compactness methods in the theory of homogenization.(English)Zbl 0632.35018

We consider the boundary value problem $(1)\quad L_{\epsilon}u_{\epsilon}=div f\quad in\quad D,\quad u_{\epsilon}=g\quad on\quad \partial D,$ where $$(L_{\epsilon}u)_ i\equiv (\partial /\partial x^{\alpha})[A_{ij}^{\alpha \beta}(x/\epsilon)\partial u^ j/\partial x^{\beta}]$$ and D is a $$C^{1,\gamma}$$ domain. The coefficients $$A_{ij}^{\alpha \beta}(\cdot)$$ are periodic, Hölder continuous and the operator is strongly elliptic. We establish the following a-priori estimates:
(2) $$[u_{\epsilon}]_{C^{\mu}(-D)}\leq C_ p^{(1)}[\| f\| _{L^ p(D)}+[g]_{C^{\gamma}(\partial D)}]$$ where $$\mu =\min (\gamma,1-n/p);$$
(3) $$\| \nabla u_{\epsilon}\| _{L^{\infty}(D)}\leq C_ p^{(2)}[\| div f\| _{L^ p(D)}+[g]_{C^ 1,\gamma (\partial D)}]$$, $$p>n,$$
(4) if $$f\equiv 0$$, $$\| u_{\epsilon}\| _{L^ p(D)}\leq C_ p^{(3)}\| g\| _{L^ p(\partial D)}$$, $$1\leq p\leq \infty,$$
(5) if $$g\equiv 0$$, $$\| u_{\epsilon}\| _{L^ q(D)}\leq C_{pq}^{(4)}\| f\| _{L^ p(D)}$$, 1/q$$\leq 1/n-1/p;$$
(6) if $$g\equiv 0$$, $$\| \nabla u_{\epsilon}\| _{L^ q(D)}\leq C_{pq}^{(5)}\| div f\| _{L^ p(D)}$$, 1/q$$\leq 1/n-1/p.$$
In estimates (2)-(6) the constants $$C_ p^{(i)}$$ are independent of $$\epsilon$$. We derive from this result several estimates on Green’s function and the Poisson kernel of $$L_{\epsilon}$$, as well as some new convergence results and error estimates for (1).

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35B45 A priori estimates in context of PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B40 Asymptotic behavior of solutions to PDEs
Full Text:

### References:

  Avellaneda, J. Appl. Math. and Optimiz. 15 pp 93– (1987)  Avellaneda, J. Appl. Math. and Optimiz. 15 pp 109– (1987)  , and , Asymptotic Analysis of Periodic Structures, North-Holland Publ., 1978.  and , Interpolation Spaces, Springer-Verlag, 1976.  Campanato, Ann. Scuolla, Norm. Sup. Pisa 17 pp 175– (1963)  Campanato, Ann. Mat. Pura et. Appl. 69 pp 321– (1965)  , and , Some smoothness properties of linear laminates, Preprint #199, Institute for Math. and its Applications, Minneapolis, 1985.  Multiple Integrals in the Calculus of Variations, Study 105, Annals of Math. Studies, Princeton Univ. Press, 1983.  and , Elliptic Partial Differential Equation of Second Order, Springer-Verlag, Heidelberg, New York, 1977.  and , An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.  Kozlov, Math. USSR, Sbornik 35 pp 481– (1979)  Asymptotic problems in distributed systems, Preprint #147, Institute for Mathematics and its Applications, Minneapolis, 1985.  Mortola, Comm. in P.D.E. 7 pp 645– (1982)  Fichera, Arch. Ratl. Mech. and Analysis 7 pp 373– (1961)  Lemrabet, J. Math. Pures et Appl. 56 pp 1– (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.