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.

The asymptotic behaviour is studied for minima of regular variational problems with Neumann boundary conditions on noncompact part of boundary.

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 $loglog{(1/|x\left|\right)}^{-1}$. 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....

We consider local minimizers $u:{ℝ}^2\supset \Omega \to {ℝ}^N$ of variational integrals like ${\int }_{\Omega }\left[\right(1+|{\partial }_1{u|}^2{)}^{p/2}+(1+|{\partial }_2{u|}^2{)}^{q/2}]dx$ or its degenerate variant ${\int }_{\Omega }\left[\right|{\partial }_1{u|}^p+|{\partial }_{2}u{|}^q]dx$ with exponents $2\le p<q<\infty$ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. 16 (2003), 177–186. We prove interior $C^{1,\alpha }$- respectively $C^1$-regularity of $u$ under the condition that $q<2p$. 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.

Minimizers of a functional with exponential growth are shown to be smooth. The techniques developed for power growth are not applicable to the exponential case.

We prove a regularity result for weak minima of integral functionals of the form ${\int }_{\Omega }F(x,Du)dx$ where F(x,ξ) is a Carathéodory function which grows as ${\left|\xi \right|}^p$ with some p > 1.

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,...

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...

We prove partial regularity with optimal Hölder exponent of vector-valued minimizers u of the quasiconvex variational integral $\int F(x,u,Du)\mathrm{d}x$ under polynomial growth. We employ the indirect method of the bilinear form.

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...


We prove a $C^{k,\alpha }$ partial regularity result for local minimizers of variational integrals of the type $I\left(u\right)={\int }_{\Omega }f\left(D^{k}u\left(x\right)\right)\mathrm{d}x$, assuming that the integrand f satisfies (p,q) growth conditions.