Convergence of minima of integral functionals and multiplicative perturbations of the integrands
Considered is a sequence of functions \[ f_ h: (x,z)\in R^ n\times R^ n\to f_ h(x,z)\in [0,+\infty [,\quad h=1,2,...,\infty \] measurable in x, convex in z such that \[ | z|^ p\leq f_ h(x,z)\leq \Lambda (1+| z|^ p),\quad \Lambda \geq 1,\quad p>1 \] and verifying a standard consequence of \(\Gamma\)-convergence theory, i.e., for every bounded open set \(\Omega\) of \(R^ n\), \(\phi\) in \(L^{\infty}(\Omega)\) the minimum values of the problems \[ \min \{\int_{\Omega}f_ h(x,Du)+\int_{\Omega}\phi u,\quad u=0\quad on\quad \partial \Omega \} \] converge to the minimum value of the problem \[ \min \{\int_{\Omega}f_{\infty}(x,Du)+\int_{\Omega}\phi u,\quad u=0\quad on\quad \partial \Omega \}. \] Conditions on a measurable function w are given so that the above convergence continues to hold after a multiplication of the integrands for w. Therefore it is proved that for every bounded open set \(\Omega\), \(\phi\) in \(L^{\infty}(\Omega)\) and every nonnegative function w with w, \(w^{- 1/(p-1)}\) in \(L^ 1_{loc}(R^ n)\) the minimum values of the problems \[ \min \{\int_{\Omega}w(x)f_ h(x,Du)+\int_{\Omega}\phi u,\quad u=0\quad on\quad \partial \Omega \} \] converge to the minimum value of \[ \min \{\int_{\Omega}w(x)f_{\infty}(x,Du)+\int_{\Omega}\phi u,\quad u=0\quad on\quad \partial \Omega \}. \] The techniques used are related to the study of the continuity properties of the Hardy maximal operators between weighted \(L^ p\) spaces and applying typical techniques of \(\Gamma\)-convergence theory.
scientific article; zbMATH DE number 4056408