Abstract
Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF:\(F:u \to \smallint _{\rm T} f(u(t),t)d\mu (t)\) and byDom(F) its domain, it is the set {ueL p(T,E):ū(t)=f(u),t)eL 1(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL p(T, E).
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