Semicontinuité des fonctionnelles intégrales
The problems of lower semicontinuity of integral functionals in general multidimensional setting are studied. Namely, the functionals of the form \(I(u,v)=\int_{T}f(t,u(t),v(t))\mu (dt)\) are considered, where, for instance, (T,\(\mu)\) is a compact metrizable space supplied with a Radon positive measure, the integrand \(f:T\times F\times E\to [0,+\infty]\) is approximatively semicontinuous, F is a Suslin locally convex space and E is a Suslin space. Several general theorems concerning lower semicontinuity and some specifications of these theorems are given. For instance, it is proved that under suitable assumptions \(\lim \inf_{n\to \infty}I(u_ n,v_ n)\geq I(u_ 0,v_ 0),\) when \(u_ n\to u_ 0\) in the weak topology \(\sigma (L_ F,L_{F'})\) and \(v_ n\) converges to a function \(v_ 0\) pointwise or in measure. The methods applied for getting such kind of theorems are those of the theory of duality of convex integrals and approximation of convex normal integrands by more regular ones.
lower semicontinuity of integral functionals
scientific article; zbMATH DE number 3888398