We study a variable exponent model for image restoration in the case that the exponent attains the critical value one. We prove existence and Γ-convergence. The results answer an open question by Li, Li and Pi [Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (2010), no. 3, 870–882].

Received (Day Month Year) Communicated by (xxxxxxxxxx) Let Ω be an open bounded set of R 3 and let W and V be two non-negative continuous functions vanishing at α, β and α ′ , β ′ , respectively. We analyze the asymptotic behavior as ε → 0, in terms of Γ-convergence, of the following functional Fε(u) := ε p−2 Z Ω |Du| p dx + 1 ε p−2 p−1 Z Ω W (u)dx + 1 ε Z ∂Ω V (T u)dH 2 (p > 2), where u is a s...

We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence. We prove that the Γ-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γlimit is also stable under volume constraint and various type of boundary conditions.

We provide an approximation result in the sense of Γ-convergence for energies of the form ˆ Ω Q1(e(u)) dx+ aH(Ju) + b ˆ Ju Q 1/2 0 ([u] νu) dH n−1, where Ω ⊂ Rn is a bounded open set with Lipschitz boundary, Q0 and Q1 are coercive quadratic forms on Mn×n sym , a, b are positive constants, and u runs in the space of fields SBD2(Ω) , i.e., it’s a special field with bounded deformation such that i...

The energy of a Von Kármán circular plate is described by a nonlocal nonconvex one-dimensional functional depending on the thickness ε. Here we perform the asymptotic analysis via Γ-convergence as the parameter ε goes to zero.

We show the Γ-convergence of a family of discrete functionals to the Mumford and Shah image segmentation functional. The functionals of the family are constructed by modifying the elliptic approximating functionals proposed by Ambrosio and Tortorelli. The quadratic term of the energy related to the edges of the segmentation is replaced by a nonconvex functional.

The formation of microdomains, also called rafts, in biomembranes can be attributed to the surface tension of the membrane. In order to model this phenomenon, a model involving a coupling between the local composition and the local curvature was proposed by Seul and Andelman in 1995. In addition to the familiar Cahn-Hilliard/Modica-Mortola energy, there are additional ‘forces’ that prevent larg...

In this paper we study the rigorous sharp interface limit of a diffuse interface model related to the dynamics of tumor growth, when a parameter ε, representing the interface thickness between the tumorous and non tumorous cells, tends to zero. More in particular, we analyze here a gradient-flow type model arising from a modification of the recently introduced model for tumor growth dynamics in...

We define the total curvature of a semialgebraic graph Γ ⊂ R as an integral K(Γ) = R Γ dμ, where μ is a certain Borel measure completely determined by the local extrinsic geometry of Γ. We prove that it satisfies the Chern-Lashof inequality K(Γ) ≥ b(Γ), where b(Γ) = b0(Γ) + b1(Γ), and we completely characterize those graphs for which we have equality. We also prove the following unknottedness r...

Ni2+ supported on hydroxyapatite-core@shell γ-Fe2O3 nanoparticles (γ-Fe2O3@HAp-Ni2+) was found to be a useful catalyst for the synthesis of benzimidazole derivatives from o-phenylenediamine and aldehydes under solvent and solvent-free conditions at 80 °C. This reaction affords the corresponding benzimidazole derivatives compared with the classical reactions this method consistently gives a high...