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Singular points of order k of Clarke regular and arbitrary functions

Luděk Zajíček (2012)

Commentationes Mathematicae Universitatis Carolinae

Let X be a separable Banach space and f a locally Lipschitz real function on X . For k ∈ ℕ , let Σ k ( f ) be the set of points x ∈ X , at which the Clarke subdifferential ∂ C f ( x ) is at least k -dimensional. It is well-known that if f is convex or semiconvex (semiconcave), then Σ k ( f ) can be covered by countably many Lipschitz surfaces of codimension k . We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove...

Slice convergence : stabilité et optimisation dans les espaces non réflexifs

Khalid El Hajioui, Driss Mentagui (2004)

ESAIM: Control, Optimisation and Calculus of Variations

Il est démontré par Mentagui [ESAIM : COCV 9 (2003) 297-315] que, dans le cas des espaces de Banach généraux, la convergence d’Attouch-Wets est stable par une classe d’opérations classiques de l’analyse convexe, lorsque les limites des suites d’ensembles et de fonctions satisfont certaines conditions de qualification naturelles. Ceci tombe en défaut avec la slice convergence. Dans cet article, nous établissons des conditions de qualification uniformes assurant la stabilité de la slice convergence...

Slice convergence: stabilité et optimisation dans les espaces non réflexifs

Khalid El Hajioui, Driss Mentagui (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Il est démontré par Mentagui [ESAIM: COCV9 (2003) 297-315] que, dans le cas des espaces de Banach généraux, la convergence d'Attouch-Wets est stable par une classe d'opérations classiques de l'analyse convexe, lorsque les limites des suites d'ensembles et de fonctions satisfont certaines conditions de qualification naturelles. Ceci tombe en défaut avec la slice convergence. Dans cet article, nous établissons des conditions de qualification uniformes assurant la stabilité de la slice convergence...

Smooth bifurcation for a Signorini problem on a rectangle

Jan Eisner, Milan Kučera, Lutz Recke (2012)

Mathematica Bohemica

We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof...

Smoothing a polyhedral convex function via cumulant transformation and homogenization

Alberto Seeger (1997)

Annales Polonici Mathematici

Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family g ₜ t > 0 which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family g ₜ t > 0 involves the concept of cumulant transformation and a standard homogenization procedure.

Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality

Mark A. Peletier, Robert Planqué, Matthias Röger (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We derive a new criterion for a real-valued function u to be in the Sobolev space W 1 , 2 ( ℝ n ) . This criterion consists of comparing the value of a functional ∫ f ( u ) with the values of the same functional applied to convolutions of u with a Dirac sequence. The difference of these values converges to zero as the convolutions approach u , and we prove that the rate of convergence to zero is connected to regularity: u ∈ W 1 , 2 if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization...

Solution for a classical problem in the calculus of variations via rationalized Haar functions

Mohsen Razzaghi, Yadollah Ordokhani (2001)

Kybernetika

A numerical technique for solving the classical brachistochrone problem in the calculus of variations is presented. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Application of this method results in the transformation of differential and integral expressions into some algebraic equations to which Newton-type methods can be applied. The method is general, and yields accurate results.

Solution of degenerate parabolic variational inequalities with convection

Jozef Kacur, Roger Van Keer (2003)

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

Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard’s equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.

Solution of degenerate parabolic variational inequalities with convection

Jozef Kacur, Roger Van Keer (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard's equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.

Solution of Signorini's contact problem in the deformation theory of plasticity by secant modules method

Jindřich Nečas, Ivan Hlaváček (1983)

Aplikace matematiky

A problem of unilateral contact between an elasto-plastic body and a rigid frictionless foundation is solved within the range of the so called deformation theory of plasticity. The weak solution is defined by means of a variational inequality. Then the so called secant module (Kačanov's) iterative method is introduced, each step of which corresponds to a Signorini's problem of elastoplastics. The convergence of the method is proved on an abstract level.

Solution set in a special case of generalized Nash equilibrium games

Josef Cach (2001)

Kybernetika

A special class of generalized Nash equilibrium problems is studied. Both variational and quasi-variational inequalities are used to derive some results concerning the structure of the sets of equilibria. These results are applied to the Cournot oligopoly problem.

Solvability and numerical algorithms for a class of variational data assimilation problems

Guri Marchuk, Victor Shutyaev (2002)

ESAIM: Control, Optimisation and Calculus of Variations

A class of variational data assimilation problems on reconstructing the initial-value functions is considered for the models governed by quasilinear evolution equations. The optimality system is reduced to the equation for the control function. The properties of the control equation are studied and the solvability theorems are proved for linear and quasilinear data assimilation problems. The iterative algorithms for solving the problem are formulated and justified.

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