Relaxation of free-discontinuity energies with 
obstacles

Matteo Focardi; Maria Stella Gelli

Displaying similar documents to “Relaxation of free-discontinuity energies with obstacles”

A relaxation result for energies defined on pairs set-function and applications

Andrea Braides, Antonin Chambolle, Margherita Solci (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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 We consider, in an open subset of $ℝ^{N}$ , energies depending on the perimeter of a subset $E \subset Ω$ (or some equivalent surface integral) and on a function which is defined only on $Ω ∖ E$ . We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” may collapse into a discontinuity of , whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an...

Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

Nicola Fusco, Virginia De Cicco, Micol Amar (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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New $L^{1}$ -lower semicontinuity and relaxation results for integral functionals defined in BV( $Ω$ ) are proved, under a very weak dependence of the integrand with respect to the spatial variable $x$ . More precisely, only the lower semicontinuity in the sense of the $1$ -capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to $x$ . Under this further BV dependence,...

An Campanato type regularity condition for local minimisers in the calculus of variations

Thomas J. Dodd (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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An Campanato type regularity condition is established for a class of WX local minimisers $\bar{u}$ of the general variational integral $\int_{Ω} F (\nabla u (x)) d x$ where $Ω \subset ℝ^{n}$ is an open bounded domain, is of class C, is strongly quasi-convex and satisfies the growth condition $F (ξ) \leq c (1 + | ξ |^{p})$ for a and where the corresponding Banach spaces X are the Morrey-Campanato space $ℒ^{p, μ} (Ω, ℝ^{N \times n})$ , < , Campanato space $ℒ^{p, n} (Ω, ℝ^{N \times n})$ and the space of bounded mean oscillation $BMO Ω, ℝ^{N \times n})$ . The admissible maps $u : Ω \to ℝ^{N}$ are of Sobolev class W, satisfying a Dirichlet boundary condition,...

Necessary conditions for weak lower semicontinuity on domains with infinite measure

Stefan Krömer (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in $ℝ^{N}$ . An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value $+ \infty$ .

Non-local approximation of free-discontinuity problems with linear growth

Luca Lussardi, Enrico Vitali (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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We approximate, in the sense of -convergence, free-discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls. The result extends to higher dimension what we already proved in the one-dimensional case.