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The Young measure representation for weak cluster points of sequences in M-spaces of measurable functions

Nguyen H. T., Pączka D.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

109-120

Let (X, Y) be a duality pair of M-spaces X, Y of measurable functions from Ω ⊂ R[sup]n into R[sup]d. The paper deals with Y-weak cluster points φ of the sequence φ(•, z[sub]j(•)) in X, where z[sub]j: Ω→ R[sup]m is measurable for j ∈ N and φ: Ω x R[sup]m → R[sup]d is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set Aφ, the integral I(φ,νx) := ∫R φ(x,λ) dνx(λ) exists for χ ∈ Ω \ Aφ and φ(x) = I(φ, νx) on Ω \ Aφ, where ν = {νx}x∈Ω is a measurable-dependent family of Radon probability measures on R[sup]m.

Existence theorems for the Dirichlet elliptic inclusion involving exponential-growth-type multivalued right-hand side

Nguyêñ H. T., Pączka D.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 4

361-375

We present two existence results for the Dirichlet elliptic inclusion with an upper semicontinuous multivalued right-hand side in exponential-type Orlicz spaces involving a vector Laplacian, subject to Dirichlet boundary conditions on a domain Ω ⊂ R2. The first result is obtained via the multivalued version of the Leray–Schauder principle together with the Nakano–Dieudonné sequential weak compactness criterion. The second result is obtained by using the nonsmooth variational technique together with a formula for Clarke's subgradient for Lipschitz integral functionals on “nonregular” Orlicz spaces.