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The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

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Nguyêñ H., Pączka D.

Bulletin of the Polish Academy of Sciences. Mathematics

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2008

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tom 56

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nr 2

109-120

EN

Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into $ℝ^d$. The paper deals with Y-weak cluster points ϕ̅ of the sequence $ϕ(·,z_{j}(·))$ in X, where $z_j:Ω → ℝ^m$ is measurable for j ∈ ℕ and $ϕ:Ω×ℝ^m → ℝ^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_ϕ$, the integral $I(ϕ,ν_x):= ∫_{ℝ^m} ϕ(x,λ) dν_x(λ)$ exists for $x ∈ Ω∖ A_ϕ$ and $ϕ̅(x) = I(ϕ,ν_x)$ on $Ω∖ A_ϕ$, where $ν={ν_x}_{x ∈ Ω}$ is a measurable-dependent family of Radon probability measures on $ℝ^m$.

2

Existence Theorems for the Dirichlet Elliptic Inclusion Involving Exponential-Growth-Type Multivalued Right-Hand Side

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Nguyêñ H., Pączka D.

Bulletin of the Polish Academy of Sciences. Mathematics

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2005

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tom 53

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nr 4

361-375

EN

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.

3

The Euler-Lagrange inclusion in Orlicz-Sobolev spaces

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Nguyêñ H., Pączka D.

Banach Center Publications

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2014

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tom 101

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nr 1

127-131

EN

We establish the Euler-Lagrange inclusion of a nonsmooth integral functional defined on Orlicz-Sobolev spaces. This result is achieved through variational techniques in nonsmooth analysis and an integral representation formula for the Clarke generalized gradient of locally Lipschitz integral functionals defined on Orlicz spaces.

4

Generalized gradients for locally Lipschitz integral functionals on non-$L^p$-type spaces of measurable functions

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Nguyêñ H., Pączka D.

Banach Center Publications

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2008

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tom 79

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nr 1

135-156

EN

Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E*_{ω*}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G(x): = ∫_{Ω} g(s,x(s))dμ(s)$. Consider the integral functional G defined on some non-$L^{p}$-type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke's generalized gradient (multivalued C-subgradient) $∂_{C}G(x)$ has the representation $γ(v) = ∫_{Ω} ⟨ζ(s),v(s)⟩dμ(s) (v ∈ X)$ via some measurable function $ζ: Ω → E*_{w*}$ of the associate space X' such that $ζ(s) ∈ ∂_{C}g(s,x(s))$ for almost all s ∈ Ω. Here, given a fixed s ∈ Ω, $∂_{C}g(s,u₀)$ denotes Clarke's generalized gradient for the function g(s,·) at u₀ ∈ E. What concerning X, we suppose that it is either a so-called non-solid Banach M-space (in particular, non-solid generalized Orlicz space) or Köthe-Bochner space (solid space).

Ograniczanie wyników

2 Banach Center Publications

2 Bulletin of the Polish Academy of Sciences. Mathematics

4 Nguyêñ H.

4 Pączka D.

1 2014

2 2008

1 2005

The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

Existence Theorems for the Dirichlet Elliptic Inclusion Involving Exponential-Growth-Type Multivalued Right-Hand Side

The Euler-Lagrange inclusion in Orlicz-Sobolev spaces

Generalized gradients for locally Lipschitz integral functionals on non-$L^p$-type spaces of measurable functions