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On the Newton partially flat minimal resistance body type problems

M. Comte, Jesus Ildefonso Díaz (2005)

Journal of the European Mathematical Society

We study the flat region of stationary points of the functional ∫ Ω F ( | ∇ u ( x ) | ) d x under the constraint u ≤ M , where Ω is a bounded domain in ℝ 2 . Here F ( s ) is a function which is concave for s small and convex for s large, and M > 0 is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when Ω is a ball. We also analyze some other qualitative properties. Moreover, we show the...

Quasi-minima

Mariano Giaquinta, Enrico Giusti (1984)

Annales de l'I.H.P. Analyse non linéaire

Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach

Filippo Cagnetti, Rodica Toader (2011)

ESAIM: Control, Optimisation and Calculus of Variations

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [G. Dal Maso and C. Zanini, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253–279] is recovered. In this case, the convergence of the discrete time approximations is improved....

Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach

Filippo Cagnetti, Rodica Toader (2011)

ESAIM: Control, Optimisation and Calculus of Variations

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [G. Dal Maso and C. Zanini, Proc. Roy. Soc. Edinburgh Sect. A137 (2007) 253–279] is recovered. In this case, the convergence of the discrete time approximations is improved. ...

Second-order sufficient condition for ℓ ˜ -stable functions

Dušan Bednařík, Karel Pastor (2007)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The aim of our article is to present a proof of the existence of local minimizer in the classical optimality problem without constraints under weaker assumptions in comparisons with common statements of the result. In addition we will provide rather elementary and self-contained proof of that result.

Semi-smooth Newton methods for the Signorini problem

Kazufumi Ito, Karl Kunisch (2008)

Applications of Mathematics

Semi-smooth Newton methods are analyzed for the Signorini problem. A proper regularization is introduced which guarantees that the semi-smooth Newton method is superlinearly convergent for each regularized problem. Utilizing a shift motivated by an augmented Lagrangian framework, to the regularization term, the solution to each regularized problem is feasible. Convergence of the regularized problems is shown and a report on numerical experiments is given.

Shape and topological sensitivity analysis in domains with cracks

Alexander Khludnev, Jan Sokołowski, Katarzyna Szulc (2010)

Applications of Mathematics

The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural...

Some concepts of regularity for parametric multiple-integral problems in the calculus of variations

M. Crampin, D. J. Saunders (2009)

Czechoslovak Mathematical Journal

We show that asserting the regularity (in the sense of Rund) of a first-order parametric multiple-integral variational problem is equivalent to asserting that the differential of the projection of its Hilbert-Carathéodory form is multisymplectic, and is also equivalent to asserting that Dedecker extremals of the latter ( m + 1 ) -form are holonomic.

Some geometric aspects of the calculus of variations in several independent variables

David Saunders (2010)

Communications in Mathematics

This paper describes some recent research on parametric problems in the calculus of variations. It explains the relationship between these problems and the type of problem more usual in physics, where there is a given space of independent variables, and it gives an interpretation of the first variation formula in this context in terms of cohomology.

Sufficient conditions for infinite-horizon calculus of variations problems

Joël Blot, Naïla Hayek (2010)

ESAIM: Control, Optimisation and Calculus of Variations

After a brief survey of the literature about sufficient conditions, we give different sufficient conditions of optimality for infinite-horizon calculus of variations problems in the general (non concave) case. Some sufficient conditions are obtained by extending to the infinite-horizon setting the techniques of extremal fields. Others are obtained in a special qcase of reduction to finite horizon. The last result uses auxiliary functions. We treat five notions of optimality. Our problems are essentially motivated...

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