The nonlinear selfdual variational principle established in a preceeding paper [8] -though good enough to be readily applicable in many stationary nonlinear partial differential equations -did not however cover the case of nonlinear evolutions such as the Navier-Stokes equations. One of the reasons is the prohibitive coercivity condition that is not satisfied by the corresponding selfdual functional on the relevant path space. We show here that such a principle still hold for functionals of the formwhere L (resp., ℓ) is an anti-selfdual Lagrangian on state space (resp., boundary space), and Λ is an appropriate nonlinear operator on path space. As a consequence, we provide a variational formulation and resolution to evolution equations involving nonlinear operators such as the Navier-Stokes equation (in dimensions 2 and 3) with various boundary conditions. In dimension 2, we recover the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying u(0) = αu(T ) for any given α in (−1, 1). Our approach is quite general and does apply to many other situations.

We study the existence of positive solutions to the quasilinear elliptic problemwhere g has superlinear growth at infinity without any restriction from above on its growth.Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrödinger equations. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

For any given integer N ≥ 2, we show that every bounded measurable vector field from a bounded domain Ω into R d is N -cyclically monotone up to a measure preserving N -involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function on a product domain Ω N . The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N − 1 of them). In this case, we show that the supremum over all probability measures on Ω N which are invariant under cyclic permutations and with a given first marginal µ, is attained on a probability measure that is supported on the graph of a function of the form x → (x, Sx, S 2 x, ..., S N−1 x), where S is a µ-measure preserving transformation on Ω such that S N = I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N -cyclically antisymmetric.

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We study the concept and the calculus of Non-convex self-dual (Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of EulerLagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler-Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.