We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear hamiltonian seems to be the...

We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be...

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

We extend a regularity theorem of Hildebrandt and Widman [3] to certain degenerate systems of variational inequalities and prove Hölder-continuity of solutions which are in some sense stationary.

We discuss regularity results concerning local minimizers $u:{ℝ}^n\supset \Omega \to {ℝ}^n$ of variational integrals like $$\begin{array}{c}\hfill {\int }_{\Omega }\{F(·,\epsilon \left(w\right))-f·w\}dx\end{array}$$ defined on energy classes of solenoidal fields. For the potential $F$ we assume a $(p,q)$-elliptic growth condition. In the situation without $x$-dependence it is known that minimizers are of class $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of $\Omega$ with full measure if $q<p\frac{n+2}{n}$ (for $n=2$ we have ${\Omega }_0=\Omega$). In this article we extend this to the case of nonautonomous integrands. Of course our result extends to weak solutions of the corresponding nonlinear...

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}\left({ℝ}^n\right)$. This criterion consists of comparing the value of a functional $\int f\left(u\right)$ 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: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization...

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.

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.

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.

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.

Given a family of curves constituting the general solution of a system of ordinary differential equations, the natural question occurs whether the family is identical with the totality of all extremals of an appropriate variational problem. Assuming the regularity of the latter problem, effective approaches are available but they fail in the non-regular case. However, a rather unusual variant of the calculus of variations based on infinitely prolonged differential equations and systematic use of...

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.