A criterion for continuity of an integral functional on a sequence of functions
The author studies the continuity properties of the functional \[ J(u) = \int_\Omega L \bigl( x,u(x), \nabla u(x) \bigr) dx \] on the class \(W^1_1 (\Omega, \mathbb{R}^m)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^d\) with smooth boundary. In connection with the well-known Lavrentiev phenomenon the necessary and sufficient conditions for convergence of the sequence \(J(u_n)\) to \(J(u)\) under the condition \(|u_n - u |_{L_1 (\Omega, \mathbb{R}^m)} \to 0\), \(n \to \infty\), are constructed. One of the main results reads as follows. Let \(u_n,u \in W_1^1 (\Omega; \mathbb{R}^m);\) \(|u_n - u |_{L_1 (\Omega, \mathbb{R}^m)} \to 0\), \(n \to \infty\), \(J(u) < \infty\) and for almost all \(x \in \Omega\) the function \(L (x, u(x), \cdot)\) is strictly subdifferentiable at \(v_0 = \nabla u(x)\), i.e., \[ L \bigl( x,u(x), v \bigr) - L \bigl( x,u(x), v_0 \bigr) > (f,v - v_0)\quad \forall v \neq v_0,\quad f \in \partial L \bigl( x,u(x), v \bigr) |_{v = v_0}. \] Then the following statements are equivalent: \[ J(u_n) \to J(u),\;n \to \infty, \tag{1} \] \[ |u_n - u |_{W^1_1 (\Omega, \mathbb{R}^m)} \to 0,\;n \to \infty, \tag{2} \] and for each \(\varepsilon > 0\) there exists \(\delta > 0\) such that \(|L (\cdot, u_n (\cdot),\nabla u_n (\cdot)) |_{L_1 (\widetilde \Omega)} < \varepsilon\) whenever \(n \in \mathbb{N}\) and \(\widetilde \Omega \subset \Omega\) with \(\text{mes} \widetilde \Omega < \delta\), \[ \biggl |L \bigl( \cdot, u_n (\cdot), \nabla u_n (\cdot) \bigr) - L \bigl( \cdot, u (\cdot), \nabla u (\cdot) \bigr) \biggr |_{L_1 (\Omega)} \to 0,\;n \to \infty. \tag{3} \] A generalization of this result for the functional \(J(u, \xi) = \int_\Omega L(x,u(x), \xi (x)) dx\) considered on \(L_1 (\Omega, \mathbb{R}^m) \times L_1 (\mathbb{R}^\ell)\) is also presented.
scientific article; zbMATH DE number 923491
scientific article