Anisotropic Orlicz–Sobolev spaces of vector valued functions and Lagrange equations

Abstract

In this paper we study some properties of anisotropic Orlicz and anisotropic 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.2010 Mathematics Subject Classification. 46B10 , 46E30 , 46E40.

Clarke duality for Hamiltonian systems with nonstandard growth

“…Next, we will be concerned with the notion of G-function and Orlicz spaces. We refer the reader to [13,17] for more comprehensive information about convex functions and to [2,3,4,18,20] for more information on anisotropic G-functions and Orlicz spaces.…”

Mountain pass solutions to Euler-Lagrange equations with general anisotropic operator

“…In this section we briefly recall the notion of anisotropic Orlicz-Sobolev spaces. For more details we refer the reader to [10,1] and references therein. We assume that (G) G : R N → [0, ∞) is a continuously differentiable G-function (i.e.…”

“…It is proved in [10,Theorem 4.5] that for every u ∈ W 1 0 L G the following form of Poincaré inequality holds…”

“…The same is true if F (t, u, v) = G(v), since in this case R G (u n ) → R G (u). The last condition implies desired convergence for {u n } (see [10,Lemma 3.16] and [1, p. 593]).…”

“…(t, u n ,u n ) + F (t, u n , −u) 2 = (t, u,u) + F (t, u, −u) (t, u,u) + F (t, u, −u) 2 dt ≤ (t, u,u) + F (t, u, −u) 2 dt−lim sup I F t, u n ,u n −u 2 dtThusu n →u in L G by[10, Theorem 3.13]. Mountain Pass geometry.…”