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Search results for: ANISOTROPIC ORLICZ–SOBOLEV SPACE EULER–LAGRANGE EQUATIONS VARIATIONAL FUNCTIONAL MOUNTAIN PASS THEOREM PALAIS–SMALE CONDITION
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Mountain pass type periodic solutions for Euler–Lagrange equations in anisotropic Orlicz–Sobolev space
PublicationUsing 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.
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Existence of Two Periodic Solutions to General Anisotropic Euler-Lagrange Equations
PublicationAbstract. This paper is concerned with the following Euler-Lagrange system d/dtLv(t,u(t), ̇u(t)) =Lx(t,u(t), ̇u(t)) for a.e.t∈[−T,T], u(−T) =u(T), Lv(−T,u(−T), ̇u(−T)) =Lv(T,u(T), ̇u(T)), where Lagrangian is given by L=F(t,x,v) +V(t,x) +〈f(t),x〉, growth conditions aredetermined by an anisotropic G-function and some geometric conditions at infinity.We consider two cases: with and without forcing termf. Using a general version...
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Mountain pass solutions to Euler-Lagrange equations with general anisotropic operator
PublicationUsing the Mountain Pass Theorem we show that the problem \begin{equation*} \begin{cases} \frac{d}{dt}\Lcal_v(t,u(t),\dot u(t))=\Lcal_x(t,u(t),\dot u(t))\quad \text{ for a.e. }t\in[a,b]\\ u(a)=u(b)=0 \end{cases} \end{equation*} has a solution in anisotropic Orlicz-Sobolev space. We consider Lagrangian $\Lcal=F(t,x,v)+V(t,x)+\langle f(t), x\rangle$ with growth conditions determined by anisotropic G-function and some geometric conditions...
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Anisotropic Orlicz–Sobolev spaces of vector valued functions and Lagrange equations
PublicationIn 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.
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The Palais–Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications
PublicationWe show that the Hamiltonian action satisfies the Palais-Smale condition over a “mixed regular- ity” space of loops in cotangent bundles, namely the space of loops with regularity H^s, s ∈ (1/2, 1), in the baseand H^{1−s} in the fiber direction. As an application, we give a simplified proof of a theorem of Hofer-Viterbo on the existence of closed characteristic leaves for certain contact type hypersufaces in cotangent bundles.
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On the Peano Theorem for Some Functional Differential Equations on Time Scale
PublicationThe Peano Theorem for some functional differential equations on time scale is proved. Assumptions are of Caratheodory type. Two counter examples for false Peano theorems in the literature are presented.
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On the Fenchel–Moreau conjugate of G-function and the second derivative of the modular in anisotropic Orlicz spaces
PublicationIn this paper, we investigate the properties of the Fenchel–Moreau conjugate of G-function with respect to the coupling function c(x, A) = |A[x]2 |. We provide conditions that guarantee that the conjugate is also a G-function. We also show that if a G-function G is twice differentiable and its second derivative belongs to the Orlicz space generated by the Fenchel–Moreau conjugate of G then the modular generated by G is twice differentiable...
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A convergence result for mountain pass periodic solutions of perturbed Hamiltonian systems
PublicationIn this work, we study second-order Hamiltonian systems under small perturbations. We assume that the main term of the system has a mountain pass structure, but do not suppose any condition on the perturbation. We prove the existence of a periodic solution. Moreover, we show that periodic solutions of perturbed systems converge to periodic solutions of the unperturbed systems if the perturbation tends to zero. The assumption on...
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Variational principles for bound states of Schrödinger and Dirac equations allowing the use of discontinuous trial functions
PublicationWe present systematic constructions of variational principles for energies of bound states of the Schroedinger and Dirac equations. The principles allow the use of discontinuous trial functions. The method employed is based on a generalized Lagrange procedure. Relationships between our variational principles and those available in the literature are established.
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Regularity of weak solutions for aclass of elliptic PDEs in Orlicz-Sobolev spaces
PublicationWe 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 $W^{2,2}_{loc}\cap W^{1,\infty}_{loc}$.