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Remarks on the theory of elasticity

Sergio Conti, Camillo de Lellis (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the L 2 norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some L p -norm of the gradient with p > 2 is controlled...

Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization

Robert Lipton, Tadele Mengesha (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the L∞ norms of gradient fields. The...

Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization

Robert Lipton, Tadele Mengesha (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit...

Resonance of minimizers for n-level quantum systems with an arbitrary cost

Ugo Boscain, Grégoire Charlot (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem describing a laser-induced population transfer on a n -level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n = 2 and n = 3 ): instead of looking...

Resonance of minimizers for n-level quantum systems with an arbitrary cost

Ugo Boscain, Grégoire Charlot (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem describing a laser-induced population transfer on a n-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n=2 and n=3): instead...

Results on existence of solution for an optimal design problem.

Carmen Calvo Jurado, Juan Casado Díaz (2003)

Extracta Mathematicae

In this paper we study a control problem for elliptic nonlinear monotone problems with Dirichlet boundary conditions where the control variables are the coefficients of the equation and the open set where the partial differential problem is studied.

Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality

Liana Cioban, Ernö Csetnek (2013)

Open Mathematics

Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka...

Rigidity for the hyperbolic Monge-Ampère equation

Chun-Chi Lin (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Some properties of nonlinear partial differential equations are naturally associated with the geometry of sets in the space of matrices. In this paper we consider the model case when the compact set  K is contained in the hyperboloid ℋ - 1 , where ℋ - 1 ⊂ 𝕄 sym 2 × 2 , the set of symmetric 2 × 2 matrices. The hyperboloid ℋ - 1 is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation det ∇ 2 u = - 1 . For some compact subsets K ⊂ ℋ - 1 containing a rank-one connection, we show the rigidity property of K by imposing...

Saddle point criteria for second order η -approximated vector optimization problems

Anurag Jayswal, Shalini Jha, Sarita Choudhury (2016)

Kybernetika

The purpose of this paper is to apply second order η -approximation method introduced to optimization theory by Antczak [2] to obtain a new second order η -saddle point criteria for vector optimization problems involving second order invex functions. Therefore, a second order η -saddle point and the second order η -Lagrange function are defined for the second order η -approximated vector optimization problem constructed in this approach. Then, the equivalence between an (weak) efficient solution of the...

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