Displaying similar documents to “Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals”

Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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For a fixed bounded open set Ω ⊂ ℝ N , a sequence of open sets Ω n ⊂ Ω and a sequence of sets Γ n ⊂ ∂ Ω ∩ ∂ Ω n , we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on Ω n , satisfying Neumann boundary conditions on Γ n and Dirichlet boundary conditions on  ∂ Ω n ∖ Γ n . We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on Ω n and Γ n locally.


DiPerna-Majda measures and uniform integrability

Martin Kružík (1998)

Commentationes Mathematicae Universitatis Carolinae

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The purpose of this note is to discuss the relationship among Rosenthal's modulus of uniform integrability, Young measures and DiPerna-Majda measures. In particular, we give an explicit characterization of this modulus and state a criterion of the uniform integrability in terms of these measures. Further, we show applications to Fatou's lemma.

Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems

Stephan Luckhaus, Yoshie Sugiyama (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

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We consider the following reaction-diffusion equation: ( KS ) u t = ∇ · ∇ u m - u q - 1 ∇ v , x ∈ ℝ N , 0 < t < ∞ , 0 = Δ v - v + u , x ∈ ℝ N , 0 < t < ∞ , u ( x , 0 ) = u 0 ( x ) , x ∈ ℝ N , where N ≥ 1 , m > 1 , q ≥ max { m + 2 N , 2 } .
In [Sugiyama, Nonlinear Anal. 63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of q ≥ max { m + 2 N , 2 } , the above problem (KS) is solvable globally in time for “small L N ( q - m ) 2 data”. Moreover, the decay of the solution (u,v)...

Weak solutions for elliptic systems with variable growth in Clifford analysis

Yongqiang Fu, Binlin Zhang (2013)

Czechoslovak Mathematical Journal

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In this paper we consider the following Dirichlet problem for elliptic systems: D A ( x , u ( x ) , D u ( x ) ) ¯ = B ( x , u ( x ) , D u ( x ) ) , x ∈ Ω , u ( x ) = 0 , x ∈ ∂ Ω , where D is a Dirac operator in Euclidean space, u ( x ) is defined in a bounded Lipschitz domain Ω in ℝ n and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the...

Oscillations and concentrations in sequences of gradients

Martin Kružík, Agnieszka Kałamajska (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, { ∇ u k } , bounded in L p ( Ø ; ℝ m × n ) if p &gt; 1 and Ω ⊂ ℝ n is a bounded domain with the extension property in W 1 , p . Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity...