Non-differentiable embedding of Lagrangian systems and partial differential equations

We develop the non-differentiable embedding theory of differential operators and Lagrangian systems using a new operator on non-differentiable functions. We then construct the corresponding calculus of variations and we derive the associated non-differentiable Euler-Lagrange equation, and apply this formalism to the study of PDEs. First, we extend the characteristics method to the non-differentiable case. We prove that non-differentiable characteristics for the Navier-Stokes equation correspond to extremals of an explicit non-differentiable Lagrangian system. Second, we prove that the solutions of the Schrödinger equation are non-differentiable extremals of the Newton?s Lagrangian.     

H-measures and variants applied to parabolic equations

Since their introduction H-measures have been mostly used in problems related to propagation effects for hyperbolic equations and systems. In this study we give an attempt to apply the H-measure theory to other types of equations. Through a number of examples we present how do the differences between parabolic and hyperbolic equations reflect in the properties of H-measures corresponding to the solutions. Secondly, we apply the H-measures to the Schrödinger equation, where we succeed in proving a propagation property. However, our conclusion is that a variant of H-measures should be sought which would be better suited to parabolic problems. We propose such a variant, show some fundamental properties and illustrate its applicability by some examples. In particular, we show that the variant provides new information in a number of situations where the original H-measures did not. Finally, we describe how the new variant can be used in small amplitude homogenisation of parabolic equations.     

Function variational principles and coercivity

The function type extension of Ekeland's variational principle [J. Math. Anal. Appl. 47 (1974) 324-353] due to Zhong [Nonlinear Anal. 29 (1997) 1421-1431] is deductible in a simplified manner and in a larger functional context. This is also true for his (normed) coercivity result, based on Palais-Smale techniques.     

Nonnegative solution for quasilinear Schrödinger equations that include supercritical exponents with nonlinearities that are indefinite in sign

In this study, we establish the existence of solutions for quasilinear Schrödinger equations involving supercritical growth with nonlinearities that are indefinite in sign. By changing the variables, the quasilinear equations are reduced to semilinear and variational methods, which are then used to obtain existence results.     

Study of Some Noncooperative Linear Elliptic Systems

Using an approximation method, we show the existence of solutions for some noncooperative elliptic systems defined on an unbounded domain.     

The Maslov Complex Germ in the Case of a Degenerate Rest Point of the Hamiltonian

Mathematical Notes -     

Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions

In [T. Duyckaerts, F. Merle, Dynamic of threshold solutions for energy-critical NLS, preprint, arXiv:0710.5915 [math.AP]], T. Duyckaerts and F. Merle studied the variational structure near the ground state solution W of the energy critical NLS and classified the solutions with the threshold energy E(W) in dimensions d=3,4,5 under the radial assumption. In this paper, we extend the results to all dimensions d?6. The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W.     

Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with E being a critical frequency in the sense that . We show that if the zero set of W−E has several isolated connected components Z_i(i=1,…,m) such that the interior of Z_i is not empty and ∂Z_i is smooth, then for ?>0 small there exists, for any integer k,1?k?m, a standing wave solution which is trapped in a neighborhood of , where is any given subset of . Moreover the amplitude of the standing wave is of the level . This extends the result of Byeon and Wang (Arch. Rational Mech. Anal. 165 (2002) 295) and is in striking contrast with the non-critical frequency case , which has been studied extensively in the past 20 years.     

Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities

In this paper we consider the following Schrödinger equation:

     

The Schrödinger-Poisson equation under the effect of a nonlinear local term

In this paper we study the problem

     

Non-differentiable symmetric duality for multiobjective programming with cone constraints

Two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597] to the non-differentiable multiobjective symmetric dual problem.     

Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems

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 Euler-Lagrange 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.     

A variational multiscale method based on bubble functions for convection-dominated convection-diffusion equation

This work presents a variational multiscale method based on polynomial bubble functions as subgrid scale and a numerical implementation based on two local Gauss integrations. This method can be implemented easily and efficiently for the convection-dominated problem. Static condensation of the bubbles suggests the stability of the method and we establish its global convergence. Representative numerical tests are presented.     

Projected subgradient methods with non-Euclidean distances for non-differentiable convex minimization and variational inequalities

We study subgradient projection type methods for solving non-differentiable convex minimization problems and monotone variational inequalities. The methods can be viewed as a natural extension of subgradient projection type algorithms, and are based on using non-Euclidean projection-like maps, which generate interior trajectories. The resulting algorithms are easy to implement and rely on a single projection per iteration. We prove several convergence results and establish rate of convergence estimates under various and mild assumptions on the problem’s data and the corresponding step-sizes. We dedicate this paper to Boris Polyak on the occasion of his 70th birthday.     

Liouville Theorems for Generalized Harmonic Functions

Each nonzero solution of the stationary Schrödinger equation u(x)–c(r)u(x)=0 in R ⁿ with a nonnegative radial potential c(r) must have certain minimal growth at infinity. If r ² c(r)=O(1), r, then a solution having power growth at infinity, is a generalized harmonic polynomial.     

Existence of gap solitons in periodic discrete nonlinear Schrödinger equations

In this paper, we discuss how to use the critical point theory to study the existence of gap solitons for periodic discrete nonlinear Schrödinger equations. An open problem proposed by Professor Alexander Pankov is solved.     

Ground state solutions for some indefinite variational problems

We consider the nonlinear stationary Schrödinger equation −Δu+V(x)u=f(x,u) in . Here f is a superlinear, subcritical nonlinearity, and we mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −Δ+V. Inspired by previous work of Li et al. (2006) [11] and Pankov (2005) [13], we develop an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy. The approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value. Our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed. For odd f, we obtain infinitely many geometrically distinct solutions. The approach also yields new existence and multiplicity results for the Dirichlet problem for the same type of equations in a bounded domain.     

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.