A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals
Copyright policy )A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals
In the present paper we show that the integral functional\(I(y,u): = \int\limits_G {f(x,y(x),u(x))dx} \) is lower semicontinuous with respect to the joint convergence of yk to y in measure and the weak convergence of uk to u in L1. The integrand f: G × ℝN × ℝm → ℝ, (x, z, p) → f(x, z, p) is assumed to be measurable in x for all (z,p), continuous in z for almost all x and all p, convex in p for all (x,z), and to satisfy the condition f(x,z,p)≧Φ(x) for all (x,z,p), where Φ is some L1-function.
- University of Bonn Germany
- DigiZeitschriften Germany
Derivatives of functions in infinite-dimensional spaces, Lower Semicontinuity, Measurable Sets, Multiple Integrals, Weak Convergence, Existence theories in calculus of variations and optimal control, Article, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Selection Lemma, 510.mathematics, Integral Functional, Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Derivatives of functions in infinite-dimensional spaces, Lower Semicontinuity, Measurable Sets, Multiple Integrals, Weak Convergence, Existence theories in calculus of variations and optimal control, Article, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Selection Lemma, 510.mathematics, Integral Functional, Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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