Praca ma charakter przeglądowy i jest skierowana do młodych matematyków i doktorantów. Dotyczy problematyki omawianej przeze mnie na Zimowej Szkole Centrum Badań Nieliniowych im. J.P. Schaudera w Toruniu w roku 2009. Zawarłam w niej wybrane, znane wyniki dotyczące problemu minimalizacji funkcjonałów całkowych w przestrzeniach Sobolewa funkcji jednej zmiennej.

In this paper we study some properties of anisotropic Orlicz and Orlicz–Sobolev spaces of vector valued functions for a special class of G-functions. We introduce a variational setting for a class of Lagrangian Systems. We give conditions which ensure that the principal part of variational functional is finitely defined and continuously differentiable on Orlicz–Sobolev space.

We consider the elliptic partial differential equation in the divergence form $$-\div(\nabla G(\nabla u(x))) t + F_u (x, u(x)) = 0,$$ where $G$ is a convex, anisotropic function satisfying certain growth and ellipticity conditions We prove that weak solutions in $W^{1,G}$ are in fact of class $W^{2,2}_{loc}\cap W^{1,\infty}_{loc}$.

Using the Mountain Pass Theorem, we establish the existence of periodic solution for Euler–Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part and a forcing term. We consider two situations: G satisfying at infinity and globally. We give conditions on the growth of the potential near zero for both situations.

A surrogate-based technique for efficient multi-objective antenna optimization is discussed. Our approach exploits response surface approximation (RSA) model constructed from low-fidelity antenna model data (here, obtained through coarse-discretization electromagnetic simulations). The RSA model enables fast determination of the best available trade-offs between conflicting design goals. The cost of RSA model construction for multi-parameter...