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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
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