Displaying similar documents to “Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems”

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.


Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals

Irene Fonseca, Martin Kružík (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps { u k } k ∈ ℕ ⊂ L p ( Ω ; ℝ m ) satisfying a linear differential constraint 𝒜 u k = 0 . Applications to sequential weak lower semicontinuity of integral functionals on 𝒜 -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det ∇ ϕ k ⇀ * det ∇ ϕ in measures on the closure of Ω ⊂ ℝ n if ϕ k ⇀ ϕ in W 1 , n ( Ω ; ℝ n ) . This convergence holds, for...

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...

External approximation of first order variational problems estimates

Cesare Davini, Roberto Paroni (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving W - 1 , p norms obtained by Nečas and on the general framework of -convergence theory.

Equi-integrability results for 3D-2D dimension reduction problems

Marian Bocea, Irene Fonseca (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients ∇ α u ε | 1 ε ∇ 3 u ε bounded in L p ( Ω ; ℝ 9 ) , 1 < p < + ∞ . Here it is shown that, up to a subsequence, u ε may be decomposed as w ε + z ε , where z ε carries all the concentration effects, i.e. ∇ α w ε | 1 ε ∇ 3 w ε p is equi-integrable, and w ε captures the oscillatory behavior, i.e. z ε → 0 in measure. In addition, if { u ε } is a recovering sequence then z ε = z ε ( x α ) nearby ∂ Ω .