Global Nonlinear Brascamp–Lieb Inequalities

We prove global versions of certain known nonlinear Brascamp–Lieb inequalities under a natural homogeneity assumption. We also establish a conditional theorem allowing one to generally pass from local to global nonlinear Brascamp–Lieb estimates under such a homogeneity assumption.     

Nonlinear Potentials for Hamilton–Jacobi–Bellman Equations. II

A formal method of constructing the viscosity solutions for abstract nonlinear equations of Hamilton–Jacobi–Bellman (HJB) type was developed in the previous work of the author. A new advantage of this method (which was called an nonlinear potentials method) is that it gives a possibility to choose at the first step an expected regularity of the solution and then – to construct this solution. This makes the whole procedure more simple because an analysis of regularity of viscosity solutions is usually the most complicated step.Nonlinear potentials method is a generalization of Krylov's approach to study HJB equations.In this article nonlinear potentials method is applied to elliptic degenerate HJB equations in R^d with variable coefficients.     

Logarithmic Asymptotics of the Nonlinear Cauchy–Riemann–Beltrami Equation

Ukrainian Mathematical Journal - We study regular solutions of the nonlinear Cauchy–Riemann–Beltrami equation for the logarithmic asymptotics in terms of the lower limits and solve an...     

Nonlinear self-adjointness of the Krichever–Novikov equation

It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie–Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever–Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever–Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.     

Convergence to Stationary Measures in Nonlinear Fokker–Planck–Kolmogorov Equations

The convergence of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary solutions is studied. Broad sufficient conditions for convergence in variation with an exponential bound are obtained.     

A Nonlinear Extension of the Coale–Lopez Theorem

For a sequence (T _n)_nof nonlinear positive operators on a real Banach space, certain monotonicity conditions are specified under which asymptotic equality holds for the inhomogeneous iterates T _n T _n-1 T ₁ T ₀. As a consequence, weak ergodicity or even path stability holds in the norm-topology. Finally, a nonlinear extension of the Coale–Lopez theorem in demography is shown.     

Nonlinear Calderón–Zygmund inequalities for maps

Being motivated by the problem of deducing \(\mathsf {L}^{p}\)-bounds on the second fundamental form of an isometric immersion from \(\mathsf {L}^{p}\)-bounds on its mean curvature vector field, we prove a nonlinear Calderón–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.     

A Levenberg–Marquardt Scheme for Nonlinear Image Registration

This paper presents a Levenberg—Marquardt scheme to obtain a displacement vector field u(x)=(u ₁(x),u ₂(x)) ^t , which matches two images recorded with the same imaging machinery. The displacement vector should transform the image location x=(x ₁,x ₂) ^t of an image T, such that the grey level are equal to another image R. The so-called mono-modal image registration problem leads to minimize the nonlinear least squares functional D(u(x))=R(x)–T(x–u(x))².To apply the Levenberg—Marquardt method, we replace the nonlinear functional D by its linearization around a current approximation. The resulting quadratic minimization problem is ill-posed, due to the fact that determining the unknown components of the displacements merely from the images is an underdetermined problem. We use an auxiliary Lagrange term borrowed from linear elasticity theory, which incorporates smoothness constraints to the displacement field. Finally, numerical experiments demonstrate the robustness and effectiveness of the proposed approach.This revised version was published online in October 2005 with corrections to the Cover Date.     

Superbranes and Super Born–Infeld Theories as Nonlinear Realizations

We use a few instructive examples to outline the characteristic features of the approach to superbranes and super Born–Infeld theories based on the concept of partial spontaneous breaking of global supersymmetry (PBGS). The examples include the N=1, D=4 supermembrane and the space-filling D2- and D3-branes. In addition to a short account of the available results for these systems, we present some new developments. For the supermembrane, we prove that the equation of motion following from the off-shell Goldstone superfield action is equivalent to the one derived directly from the nonlinear-realization formalism. We use a universal procedure inspired by the relation between linear and nonlinear realizations of the PBGS to obtain a new derivation of the off-shell Goldstone superfield actions for these systems.     

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

Nonlinear discrete Sturm–Liouville problems with global boundary conditions

This paper is devoted to the study of nonlinear difference equations subject to global nonlinear boundary conditions. We provide sufficient conditions for the existence of solutions based on properties of the nonlinearities and the eigenvalues of an associated linear Sturm–Liouville problem.     

Variational Methods of Micro–Macro Transitions in Nonlinear Elasticity

Continuum micromechanics deals with idealized materials where the macroscopic material response is modelled in an averaged or homogenized sense based on the information of the heterogeneous microstructure. In general, an efficient treatment of multiscale systems requires the application of equivalent structural problems where the constituents are governed by overall properties. The key contribution of this paper is the computational exploitation of variational methods for a numerical upper and lower bound estimation of the effective material response. We present aspects for the formulation of an appropriate minimizing principle yielding the displacement fluctuations on the microstructure and the local effective constitutive variables of the macrostructure depending on the choice whether we apply linear displacement, traction or periodic boundary conditions to the displacement fluctuations on the boundary of the microstructure. The proposed concept will be demonstrated in the scope of some representative model problems.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

The Convergence of a Levenberg–Marquardt Method for Nonlinear Inequalities

In this paper, we consider the least l ₂-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l ₂-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions.     

On a Nonlinear Heat Equation Associated with Dirichlet–Robin Conditions

This article is devoted to the study a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo–Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as t → + ∞. Finally, we give numerical results.     

Nonlinear dynamics of a symmetric Davydov–Fröhlich dimer

An analysis of a dimer, modeling two interacting equal monomer units in which complex monomer excitations can arise, is performed. The analyzed classical dynamical system corresponds to the basic unit of a novel microscopic model offered by a unified theory of the well-known Davydov and Fröhlich models of energy transport in proteins. The transition between regular and chaotic dynamics, which depends on the energy pumping and the monomer–monomer interaction parameters, is analyzed. There is a region of values of the relevant parameters when the system is either in an ordered and stable state or goes through a succession of disordered and unstable states. This could have important biological applications.