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Effective energy integral functionals for thin films with bending moment in the Orlicz-Sobolev space setting

100%
Laskowski W., Nguyêñ H.
Banach Center Publications
|
2014
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tom 102
|
nr 1
143-167
EN
In this paper we deal with the energy functionals for the elastic thin film ω ⊂ ℝ² involving the bending moments. The effective energy functional is obtained by Γ-convergence and 3D-2D dimension reduction techniques. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions Δ₂ and ∇₂ (that is equivalent to the reflexivity of Orlicz and Orlicz-Sobolev spaces generated by M). These results extend results of G. Bouchitté, I. Fonseca and M. L. Mascarenhas for the case $M(t) = |t|^p$ for some p ∈ (1,∞).
2
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The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

100%
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 $ℝ^m$.
3
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Existence Theorems for the Dirichlet Elliptic Inclusion Involving Exponential-Growth-Type Multivalued Right-Hand Side

100%
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.
4
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The Euler-Lagrange inclusion in Orlicz-Sobolev spaces

100%
Nguyêñ H., Pączka D.
Banach Center Publications
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2014
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tom 101
|
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.
5
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Generalized gradients for locally Lipschitz integral functionals on non-$L^p$-type spaces of measurable functions

100%
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).
6
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CM-Selectors for pairs of oppositely semicontinuous multivalued maps with $𝕃_{p}$-decomposable values

81%
Nguyêñ H., Juniewicz M., Ziemińska J.
Studia Mathematica
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2001
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tom 144
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nr 2
135-152
EN
We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_{p}(T,E)$-decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_{p}(T,E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x) ∩ G(x) ≠ ∅ for all x ∈ X, then there exists a CM-selector for the pair (F,G), i.e. a continuous selector for G (as in the theorem of H. Antosiewicz and A. Cellina (1975), A. Bressan (1980), S. Łojasiewicz, Jr. (1982), generalized by A. Fryszkowski (1983), A. Bressan and G. Colombo (1988)) which is simultaneously an ε-approximate continuous selector for F (as in the theorem of A. Cellina, G. Colombo and A. Fonda (1986), A. Bressan and G. Colombo (1988)).
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