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).


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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