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A compactness result for a second-order variational discrete model

Andrea Braides, Anneliese Defranceschi, Enrico Vitali (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions...

A comparison of solvers for linear complementarity problems arising from large-scale masonry structures

Mark Ainsworth, L. Angela Mihai (2006)

Applications of Mathematics

We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretisation of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretised using piecewise quadratic finite elements on a triangulation of the domain. At the discrete level, the variational inequality is reformulated as a classical linear complementarity system. We compare several state-of-art algorithms that have...

A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation

El-Hassan Benkhira, Rachid Fakhar, Youssef Mandyly (2021)

Applications of Mathematics

We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the...

A convergence result for evolutionary variational inequalities and applications to antiplane frictional contact problems

Mircea Sofonea, Mohamed Ait Mansour (2004)

Applicationes Mathematicae

We consider a class of evolutionary variational inequalities depending on a parameter, the so-called viscosity. We recall existence and uniqueness results, both in the viscous and inviscid case. Then we prove that the solution of the inequality involving viscosity converges to the solution of the corresponding inviscid problem as the viscosity converges to zero. Finally, we apply these abstract results in the study of two antiplane quasistatic frictional contact problems with viscoelastic and elastic...

A convergence result for the Gradient Flow of ∫ |A| 2 in Riemannian Manifolds

Annibale Magni (2015)

Geometric Flows

We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result...

A criterion of Γ-nullness and differentiability of convex and quasiconvex functions

Jaroslav Tišer, Luděk Zajíček (2015)

Studia Mathematica

We introduce a criterion for a set to be Γ-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Fréchet differentiable is Γ-null. Our criterion also implies a new result about Gâteaux (and Hadamard) differentiability of quasiconvex functions.

A critical point result for non-differentiable indefinite functionals

Salvatore A. Marano, Dumitru Motreanu (2004)

Commentationes Mathematicae Universitatis Carolinae

In this paper, two deformation lemmas concerning a family of indefinite, non necessarily continuously differentiable functionals are proved. A critical point theorem, which extends the classical result of Benci-Rabinowitz [14, Theorem 5.29] to the above-mentioned setting, is then deduced.

A deterministic affine-quadratic optimal control problem

Yuanchang Wang, Jiongmin Yong (2014)

ESAIM: Control, Optimisation and Calculus of Variations

A deterministic affine-quadratic optimal control problem is considered. Due to the nature of the problem, optimal controls exist under some very mild conditions. Further, it is shown that under some assumptions, the optimal control is unique which leads to the differentiability of the value function. Therefore, the value function satisfies the corresponding Hamilton–Jacobi–Bellman equation in the classical sense, and the optimal control admits a state feedback representation. Under some additional...

A duality-based approach to elliptic control problems in non-reflexive Banach spaces

Christian Clason, Karl Kunisch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...

A duality-based approach to elliptic control problems in non-reflexive Banach spaces*

Christian Clason, Karl Kunisch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...

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