Towards a two-scale calculus

\magnification=1080 \centerline{\bf Towards a Two-Scale Calculus} \medskip \noindent\centerline{ESAIM Control Optim. Calc. Var. 12 (2006), 371--397} \medskip \noindent\centerline{A. Visintin} \medskip \noindent{\bf A.M.S.\ Subject Classification (2000):} 35B27, 35J20, 74Q, 78M40. \medskip \noindent{\bf Keywords:} Two-scale convergence, Two-scale decomposition, Sobolev spaces, Homogenization. \medskip \medskip \noindent\centerline{\bf Abstract} \medskip \noindent More than twenty years ago first Nguetseng and then Allaire introduced the concept of {\sl two-scale convergence,\/} as a tool for homogenization. Two-scale convergence also offers a new point of view for {\sl multiscaling\/} --- a classical topic of applied mathematics that plays a fundamental role in physics and engineering and nowadays is attracting a renewed interest among mathematicians. Two-scale convergence may be regarded as a mathematical object in itself, and may also be equipped with a calculus of its own. The purpose of this work is to illustrate some possibilities in this direction. In Sect.\ 1 we reformulate this notion via the procedure of {\sl periodic unfolding.\/} We represent weak and strong two-scale convergence in $L^p({\bf R}^N\!\times\! Y)$ for any $p\in[1,+\infty]$, and in the Fr\'echet space $C^0({\bf R}^N \!\times\! Y)$ (here $Y= [0,1[^N$). In Sect.\ 2 we derive some properties of two-scale convergence. Some of these results are already known. Here we organize their derivation by using the tool of two-scale decomposition, and also deal with two-scale convergence in $C^0$ and in ${\cal D}'$, with the Fourier transform, and with two-scale convolution. In Sect.\ 3 we study weak and strong two-scale compactness. We prove a two-scale version of a result of Chacon, known as {\sl the biting lemma.\/} We characterize strong two-scale compactness in $L^p$ and in $C^0$, generalizing classic criteria of Riesz and Ascoli-Arzel\`a. Along the same lines, we also extend Vitali's convergence theorem. Differential properties of two-scale convergence are the main concern of this paper. The two-scale limit of sequences bounded in $H^1(\Omega)$ had already been studied by Nguetseng and then Allaire; the present analysis moves towards a different direction. In Sect.\ 4 we show that it is possible to express the gradient of the two-scale limit without the need of evaluating the limit itself, via what we name {\sl approximate two-scale derivatives.\/} More specifically, we define an approximate gradient $\Lambda_\varepsilon$ such that, denoting the weak two-scale limit by $\lim_{\varepsilon\to 0}{}^{(2)}$, $$ \lim_{\varepsilon\to 0}{}^{(2)} \Lambda_\varepsilon u_\varepsilon = (\nabla_x, \nabla_y) \lim_{\varepsilon\to 0}{}^{(2)} u_\varepsilon \qquad\hbox{ in } L^p({\bf R}^N \!\times\! Y)^{2N}. $$ By means of these two-scale approximate derivatives, in Sect.\ 5 we define two-scale convergence in spaces of differentiable functions: $W^{m,p}$, $C^m$, $C^{m,\lambda}$, ${\cal D}$. We then derive two-scale versions of the Rellich compactness theorem and of the Sobolev and Morrey imbedding theorems. Indeed several classic results have a two-scale counterpart, which does not concern single functions but sequences of functions. \end