Energy–based localization and multiplicity of radially symmetric states for the stationary p–Laplace diffusion

ABSTRACT

It is investigated the role of the state–dependent source–term for the localization by means of the kinetic energy of radially symmetric states for the stationary p–Laplace diffusion. It is shown that the oscillatory behavior of the source–term, with respect to the state amplitude, yields multiple possible states, located in disjoint energy bands. The mathematical analysis makes use of critical point theory in conical shells and of a version of Pucci–Serrin three–critical point theorem for the intersection of a cone with a ball. A key ingredient is a Harnack type inequality in terms of the energetic norm.     

A regularity criterion for axially symmetric solutions to the Navier–Stokes equations

The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v _r ) of the velocity belongs to L _∞(0, T;L ₃(Ω₀)) or v _r /r belongs to L _∞(0, T;L _3/2(Ω₀)), where Ω₀ is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω _k , k = 1, . . . , N, such that W₀ ì è_{k = 1}^N W_k {Omega_0} subset bigcuplimits_{k = 1}^N {{Omega_k}} , and assume that there exist constants α ₁, α ₂ such that either || v_r ||_{L￥ ( 0,T;L3( Wk ) )} ￡ a₁ or  || fracv_rr ||_{L￥ ( 0,T;L3/2( Wk ) )} ￡ a₂ {left| {{v_r}} right|_{{L_infty }left( {0,T;{L_3}left( {{Omega_k}} right)} right)}} leq {alpha_1},or;{left| {frac{{{v_r}}}{r}} right|_{{L_infty }left( {0,T;{L_{3/2}}left( {{Omega_k}} right)} right)}} leq {alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W₂^2,1( W×( 0,T ) ),?p ? L₂( W×( 0,T ) ) ) left( {v in W_2^{2,1}left( {Omega times left( {0,T} right)} right),nabla p in {L_2}left( {Omega times left( {0,T} right)} right)} right) . Bibliography: 28 titles.     

On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator

The long-time asymptotics are analyzed for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $t\to \pm \infty$ to the set of ‘nonlinear eigenfunctions’ $\psi (x){\mathrm{e}}^{?\mathrm{i}\omega t}$. To cite this article: A.I. Komech, A.A. Komech, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Quantitative properties on the steady states to a Schrödinger–Poisson–Slater System

In this paper, we obtain the existence, uniqueness and asymptotic behavior of steady states to a class of Schrödinger-Poisson-Slater System.     

Expansion with respect to ħ for bound states of the Schrödinger equation

A WKB-complementing expansion for bound states of the radial Schrödinger equation is discussed. A recursive method for calculating the quantum corrections of any order to the energy of the classical motion is presented. The use of quantization conditions makes it possible to write down recursion relations in an equally simple form for the ground and radially excited states. The connection between the approach and the 1/N expansion is considered. It is shown that the method can also be used for analysis in thel, E) plane in the form of a expansion for Regge trajectories.Dnepropetrovsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 208–217, February, 1992.     

A linear,symmetric and energy-conservative scheme for the space-fractional Klein–Gordon–Schrödinger equations

In this paper, we propose an efficient numerical scheme for the space-fractional Klein–Gordon–Schrödinger (SFKGS) equations. Motivated by the “Invariant Energy Quadratization” (IEQ) approach, we introduce two auxiliary variables to transform the SFKGS system into a new equivalent system in which the time derivative is discretized by the Crank–Nicolson method, and the space discretization is based on the Fourier spectral method. Consequently, the numerical scheme shares two good features. The first feature is that the nonlinear terms are treated semi-explicitly and a linear symmetric system is solved at each time step. The second feature is the energy conservation at the discrete level. These two advantages are proved by the theoretical analysis and illustrated by a given numerical example.     

Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems

We consider the two-dimensional stationary Schrödinger and Dirac equations in the case of radial symmetry. A radially symmetric potential simulates the tip of a scanning tunneling microscope. We construct semiclassical asymptotic forms for generalized eigenfunctions and study the local density of states that corresponds to the microscope measurements. We show that in the case of the Dirac equation, the tip distorts the measured density of states for all energies.     

Existence of bound states for the coupled Schrödinger–KdV system with cubic nonlinearity

We prove in this Note the existence of an infinite family of smooth positive bound states for the coupled Schrödinger–Korteweg–de Vries system, which decays exponentially at infinity.     

A generalization of the Pólya–Szegö inequality for one-dimensional functionals

We consider the Pólya–Szegö type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner’s symmetrization.     

On the Pólya–Szégö Inequality for Functionals with Variable Exponent

Analogues of the Pólya–Szégö inequality with variable exponent in the integrand are considered. Necessary and sufficient conditions for the fulfillment of these inequalities are obtained.     

Hölder Estimates for the Regular Component of the Solution to a Singularly Perturbed Convection–Diffusion Equation

In a half-plane, a homogeneous Dirichlet boundary value problem for an inhomogeneous singularly perturbed convection–diffusion equation with constant coefficients and convection directed orthogonally away from the boundary of the half-plane is considered. Assuming that the right-hand side of the equation belongs to the space C^λ, 0 < λ < 1, and the solution is bounded at infinity, an unimprovable estimate of the solution is obtained in a corresponding Hölder norm (anisotropic with respect to a small parameter).     

On global solutions to the initial–boundary value problem for the nonlinear Schrödinger equations in exterior domains

We consider the initial–boundary value problem for the nonlinear Schr‐dinger equations in an exterior domain. Global existence theorem of smooth solutions is established by using a–priori decay estimates of solutions which are obtained by the pseudoconformal indentity     

Reverse Hölder inequality for a class of suitable weak solutions to the Navier–Stokes equations

In the paper, a special class of suitable weak solutions to the three-dimensional nonstationary Navier–Stokes equations is considered, and a reverse H?lder inequality for them is proved. An interesting feature of this class is that it contains solutions having majorants invariant to the Navier-Stokes scaling. Bibliography: titles. Dedicated to Vsevolod Alekseevich Solonnikov Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 325–336.     

Another Approach to Hölder Estimates for the ∂¯-Operator on Polycylinders - Some Comments to P.W. Darko's Paper

P.W. Darko [Hölder estimates for the ?¯-operator on polycylinders. Complex Variables, 48(10), 929-934.] uses Hölder estimates for the ?¯-operator in order to solve some problems for holomorphic functions in several complex variables such as the corona problem. The present short note proves a generalized version of these estimates.