Displaying similar documents to “On convergence of gradient-dependent integrands”

Oscillations and concentrations in sequences of gradients

Martin Kružík, Agnieszka Kałamajska (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

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 > 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...

Evolutionary problems in non-reflexive spaces

Martin Kružík, Johannes Zimmer (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.

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

Similarity:

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...

Nonconcentrating generalized Young functionals

Tomáš Roubíček (1997)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

The Young measures, used widely for relaxation of various optimization problems, can be naturally understood as certain functionals on suitable space of integrands, which allows readily various generalizations. The paper is focused on such functionals which can be attained by sequences whose “energy” (= p th power) does not concentrate in the sense that it is relatively weakly compact in L 1 ( Ω ) . Straightforward applications to coercive optimization problems are briefly outlined.

Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes

Uldis Raitums (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

We consider the weak closure W Z of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems where Ω ⊂ 𝐑 n is a bounded Lipschitz domain, F s are strictly convex smooth functions with quadratic growth and S = { σ m e a s u r a b l e ∣ σ s ( x ) = 0 or 1 , s = 1 , ⋯ , s 0 , σ 1 ( x ) + ⋯ + σ s 0 ( x ) = 1 } . We show that W Z is the zero level set for an integral functional with the integrand Q ℱ being the 𝐀 -quasiconvex envelope for a certain function ℱ and the operator 𝐀 = ( curl,div ) m . If the functions F s are isotropic, then on the characteristic cone...

An existence theorem of positive solutions to a singular nonlinear boundary value problem

Gabriele Bonanno (1995)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this note we consider the boundary value problem y ' ' = f ( x , y , y ' ) ( x ∈ [ 0 , X ] ; X > 0 ) , y ( 0 ) = 0 , y ( X ) = a > 0 ; where f is a real function which may be singular at y = 0 . We prove an existence theorem of positive solutions to the previous problem, under different hypotheses of Theorem 2 of L.E. Bobisud [J. Math. Anal. Appl. 173 (1993), 69–83], that extends and improves Theorem 3.2 of D. O’Regan [J. Differential Equations 84 (1990), 228–251].