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On a class of variational problems with linear growth and radial symmetry

Michael Bildhauer, Martin Fuchs (2021)

Commentationes Mathematicae Universitatis Carolinae

We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.

On asymptotic exit-time control problems lacking coercivity

M. Motta, C. Sartori (2014)

ESAIM: Control, Optimisation and Calculus of Variations

The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem....

On second order Hamiltonian systems

Dana Smetanová (2006)

Archivum Mathematicum

The aim of the paper is to announce some recent results concerning Hamiltonian theory. The case of second order Euler–Lagrange form non-affine in the second derivatives is studied. Its related second order Hamiltonian systems and geometrical correspondence between solutions of Hamilton and Euler–Lagrange equations are found.

On the continuity of minimizers for quasilinear functionals

David Cruz-Uribe, Patrizia Di Gironimo, Luigi D'Onofrio (2012)

Czechoslovak Mathematical Journal

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log ( 1 / | x | ) - 1 . Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann....

On the regularity of local minimizers of decomposable variational integrals on domains in 2

Michael Bildhauer, Martin Fuchs (2007)

Commentationes Mathematicae Universitatis Carolinae

We consider local minimizers u : 2 Ω N of variational integrals like Ω [ ( 1 + | 1 u | 2 ) p / 2 + ( 1 + | 2 u | 2 ) q / 2 ] d x or its degenerate variant Ω [ | 1 u | p + | 2 u | q ] d x with exponents 2 p < q < which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. 16 (2003), 177–186. We prove interior C 1 , α - respectively C 1 -regularity of u under the condition that q < 2 p . For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. 31 (2006), 349–362.

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

We prove a regularity result for weak minima of integral functionals of the form Ω F ( x , D u ) d x where F(x,ξ) is a Carathéodory function which grows as | ξ | p with some p > 1.

Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients

Hengyan Li, Shuguan Ji (2011)

Open Mathematics

This paper is concerned with an optimal control problem governed by the nonlinear one dimensional periodic wave equation with x-dependent coefficients. The control of the system is realized via the outer function of the state. Such a model arises from the propagation of seismic waves in a nonisotropic medium. By investigating some important properties of the linear operator associated with the state equation, we obtain the existence and regularity of the weak solution to the state equation. Furthermore,...

Optimal feedback control of Ginzburg-Landau equation for superconductivity via differential inclusion

Yuncheng You (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Slightly below the transition temperatures, the behavior of superconducting materials is governed by the Ginzburg-Landau (GL) equation which characterizes the dynamical interaction of the density of superconducting electron pairs and the exited electromagnetic potential. In this paper, an optimal control problem of the strength of external magnetic field for one-dimensional thin film superconductors with respect to a convex criterion functional is considered. It is formulated as a nonlinear coefficient...

Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem

Giuseppe Buttazzo, Eugene Stepanov (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In the paper the problem of constructing an optimal urban transportation network in a city with given densities of population and of workplaces is studied. The network is modeled by a closed connected set of assigned length, while the optimality condition consists in minimizing the Monge-Kantorovich functional representing the total transportation cost. The cost of trasporting a unit mass between two points is assumed to be proportional to the distance between them when the transportation is carried...

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