On a homogenization technique for variational problems
Let \(\mu\) be a nonnegative Borel measure on \(\mathbb{R}^N\) that is periodic with respect to each of the variables \(x_1,\dots, x_N\) and is normalized by the condition \(\int_\square d\mu= 1\), where \(\square= [0,1)^N= T^N\) is the torus or the cell of periodicity. Define a measure \(\mu_\varepsilon\) by the relation \(\mu_\varepsilon(A)= \varepsilon^N \mu(\varepsilon^{-1} A)\), where \(A\) is an arbitrary Borel set in \(\mathbb{R}^N\) and \(\varepsilon^{-1}A= \{\varepsilon^{-1}x, x\in A\}\). The measure \(\mu_\varepsilon\) weakly converges to the Lebesgue measure. Let a Lagrangian \(f(x,s)\) be also given. Assume that it is 1-periodic and \(\mu\)-measurable with respect to \(x\in \mathbb{R}^N\) and convex with respect \(s\in\mathbb{R}^N\) and satisfies the power-law estimate \(|s|^p\leq f(x,s)\leq c|s|^p+1\) with an exponent \(p>1\) and the condition \(f(x,0)\equiv 0\). Let us consider the Dirichlet problem \[ m_\varepsilon= \inf_{u\in C^\infty_0(\Omega)} \int_\Omega (f(\varepsilon^{-1} x,\nabla u)+|u|^p- g\cdot u) d\mu_\varepsilon,\;g\in C^\infty(\overline\Omega),\tag{1} \] and the Dirichlet problem related to the Lebesgue measure \[ m_0= \inf_{u\in C^\infty_0(\Omega)} \int_\Omega (f_0(\nabla u)+ |u|^p- g\cdot u) dx,\tag{2} \] where \(\Omega\) is a bounded Lipschitz domain in \(\mathbb{R}^N\) and \(\overline\Omega\) is its closure, \(f_0(s)\) is the homogenized (average) Lagrangian defined by means of the variational problem on the cell periodicity, \[ f_0(s)= \inf_{u\in C^\infty_{\text{per}}(\square)} \int_\square f(x,s+\nabla u) d\mu, \] where \(C^\infty_{\text{per}}(\square)\) is the set of all infinitely differentiable periodic functions. Definition: A periodic measure \(\mu\) is said to be \(p\)-connected on the torus of periodicity if \(u\equiv \text{const.}\) \(\mu\)-almost everywhere whenever there are \(u_n\in C^\infty_{\text{per}}(\square)\) such that \[ \lim_{n\to\infty} \int_\square|u_n- u|^p d\mu= 0,\quad \lim_{n\to\infty} \int_\square|\nabla u_n|^p d\mu= 0. \] The main result of the paper is the following Theorem (homogenization theorem): Assume that \(\mu\) is a \(p\)-connected measure, \(u^\varepsilon\) is a solution of problem (1), and \(u^0\) is a solution of the homogenized problem (2). In this case, as \(\varepsilon\to 0\), we have (i) the convergence of solutions, \[ \int_\Omega \varphi\cdot u^\varepsilon d\mu_\varepsilon\to \int_\Omega \varphi\cdot u^0 dx,\quad \forall\varphi\in C^\infty(\overline\Omega), \] (ii) the convergence of energies, \(m_\varepsilon\to m_0\), and the validity of the relations \[ \int_\Omega f(\varepsilon^{-1} x,\nabla u^\varepsilon) d\mu_\varepsilon\to \int_\Omega f_0(\nabla u^0) dx, \] \[ \int_\Omega|u^\varepsilon|^p d\mu_\varepsilon\to \int_\Omega|u^0|^p dx. \] In this case, the homogenized Lagrangian \(f_0\) is not coercive in general.
scientific article; zbMATH DE number 1407610