Directional homogenization of elliptic equations in non-divergence form (Q2304440)

The article under consideration is concerned with the homogenisation problem associated to a family of operators \((L_\varepsilon)_\varepsilon\) in non-divergence form, given by \[ L_\varepsilon u_\varepsilon = \sum_{i,j\in \{1,\dots,d\}} a_{ij}(x',x''/\varepsilon) \partial_i\partial_j u_\varepsilon = f, \] for \(x=(x',x'')=(x'_1,\dots,x'_q,x''_{1},\dots,x''_{d-q})\in \Omega\subseteq \mathbb{R}^d\) open, \(d\geq 2\), \(0 < q < d\). The authors address the limit as \(\varepsilon\to 0\). The coefficient matrix \(a\) is assumed to be peridiodic with respect to the latter \(d-q\) variables, elliptic and symmetric. Moreover, \(a\) is assumed to be uniformly (in \(x'\)) Hölder continuous in \(x''\). Furthermore, uniformly in \(y''\), \(a(\cdot,y'')\) is supposed to be Hölder continuous and continuous with a certain modulus of continuity, which in turn satisfies some integrability condition at zero (Dini condition). The main results of the paper at hand consist of local \(L_p\) estimates for the second derivatives of \(u_\varepsilon\) in terms of the \(L_p\)-norm of \(u_\varepsilon\) and \(f\). For results with \(p=\infty\), additional continuity assumptions of \(f\) need to be warranted. These estimates and their proofs form the major part of the contribution at hand. If \(\Omega\) is a bounded \(C^2\)-domain, the authors also provide a corresponding homogenisation result and provide an explicit homogenised coefficient in terms of solutions of local problems involving the original coefficent \(a\).