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On nonhomogeneous reinforcements of varying shape and different exponents

Mohamed Boutkrida, Jacqueline Mossino, Gonoko Moussa (1999)

Bollettino dell'Unione Matematica Italiana

Studiamo un problema ellittico quasilineare concernente un dominio circondato da un rinforzo sottile di spessore variabile, in cui il coefficiente dell'equazione è (localmente) non costante. Esso concerne due diversi esponenti, uno nel dominio e l'altro nel rinforzo, una condizione di Dirichlelet sulla frontiera esterna e una condizione di trasmissione. Prediciamo il comportamento asintotico della soluzione quando lo spessore, insieme con il coefficiente nel rinforzo, tende a zero perché essi siano...

On nonlinear, nonconvex evolution inclusions

Nikolaos S. Papageorgiou (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider a nonlinear evolution inclusion driven by an m-accretive operator which generates an equicontinuous nonlinear semigroup of contractions. We establish the existence of extremal integral solutions and we show that they form a dense, G δ -subset of the solution set of the original Cauchy problem. As an application, we obtain “bang-bang”’ type theorems for two nonlinear parabolic distributed parameter control systems.

On optimal matching measures for matching problems related to the Euclidean distance

José Manuel Mazón, Julio Daniel Rossi, Julián Toledo (2014)

Mathematica Bohemica

We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a p -Laplacian system. We prove...

On periodic homogenization in perfect elasto-plasticity

Gilles A. Francfort, Alessandro Giacomini (2014)

Journal of the European Mathematical Society

The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.

On reduced pairs of bounded closed convex sets.

Jerzy Grzybowski, Ryszard Urbanski (2003)

Revista Matemática Complutense

In this paper certain criteria for reduced pairs of bounded closed convex set are presented. Some examples of reduced and not reduced pairs are enclosed.

On relations among the generalized second-order directional derivatives

Karel Pastor (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In the paper, we deal with the relations among several generalized second-order directional derivatives. The results partially solve the problem which of the second-order optimality conditions is more useful.

On robustness of set-valued maps and marginal value functions

Armin Hoffmann, Abebe Geletu (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7,26], and further developed by Shi, Zheng, Zhuang [18,19,20], Phú, Hoffmann and Hichert [8,9,10,17] to weaken up the semi-continuity requirements of certain global optimization algorithms. The robust analysis, along with the measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the...

On second–order Taylor expansion of critical values

Stephan Bütikofer, Diethard Klatte, Bernd Kummer (2010)

Kybernetika

Studying a critical value function ϕ in parametric nonlinear programming, we recall conditions guaranteeing that ϕ is a C 1 , 1 function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of D ϕ . Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization....

On semiconvexity properties of rotationally invariant functions in two dimensions

Miroslav Šilhavý (2004)

Czechoslovak Mathematical Journal

Let f be a function defined on the set 𝐌 2 × 2 of all 2 by 2 matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function f can be represented as a function f ˜ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of f in terms of its representation f ˜ .

On sets of non-differentiability of Lipschitz and convex functions

Luděk Zajíček (2007)

Mathematica Bohemica

We observe that each set from the system 𝒜 ˜ (or even 𝒞 ˜ ) is Γ -null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ℝ n is σ -strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex...

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