Singular left-definite Sturm-Liouville problems

We study singular left-definite Sturm-Liouville problems with an indefinite weight function. The existence of eigenvalues is established based on the existence of eigenvalues of corresponding right-definite problems. Furthermore, for each singular left-definite problem with limit-circle non-oscillatory endpoints we construct a regular left-definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions. Inequalities among eigenvalues recently established for regular left-definite problems are extended to the singular case.     

On the existence of transmission eigenvalues in an inhomogeneous medium

We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both the scalar problem and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [Transmission eigenvalues, SIAM J. Math. Anal. 40, (2008) pp. 783–753] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that the contrasts are large enough.     

Eigenvalues in Non-Linear Potential Theory

In this work, we study the existence of eigenvalues for a type of non-linear equations and we give some conditions to get positive eigenfunctions. Particularly, we show that under some appropriate conditions, the set of eigenvalues is an interval.     

Eigenvalues for Radially Symmetric Fully Nonlinear Operators

In this paper we present an elementary theory about the existence of eigenvalues for fully nonlinear radially symmetric 1-homogeneous operators. A general theory for first eigenvalues and eigenfunctions of 1-homogeneous fully nonlinear operators exists in the framework of viscosity solutions. Here we want to show that for the radially symmetric operators or in the one dimensional case a much simpler theory, based on ode and degree theory arguments, can be established. We obtain the complete set of eigenvalues and eigenfunctions characterized by the number of zeroes.     

Exact boundary observability for a kind of second-order quasilinear hyperbolic system

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global classical solution, we establish the local exact boundary observability for a kind of second-order quasilinear hyperbolic system in which the number of positive eigenvalues and the number of negative ones are not equal. As an application, we obtain the one-sided local exact boundary observability and two-sided local exact boundary observability with fewer observed values for first-order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative ones are decoupled.     

Exact boundary observability for a kind of second order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C² solution, we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary observability for a kind of first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled.     

Spectral Problems for the Dirac System with Spectral Parameter in Local Boundary Conditions

We consider a spectral boundary value problem in a 3-dimensional bounded domain for the Dirac system that describes the behavior of a relativistic particle in an electromagnetic field. The spectral parameter is contained in a local boundary condition. We prove that the eigenvalues of the problem have finite multiplicities and two points of accumulation, zero and infinity and indicate the asymptotic behavior of the corresponding series of eigenvalues. We also show the existence of an orthonormal basis on the boundary consisting of two-dimensional parts of the four-dimensional eigenfunctions.     

Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails

We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of ‘symmetry’ in the spatial configuration of branching sources. The presented results are based on Green’s function representation of transition probabilities of an underlying random walk and cover not only the case of the finite variance of jumps but also a less studied case of infinite variance of jumps.     

Adiabatic Evolution Generated by a Schrödinger Operator with Discrete and Continuous Spectra

In the paper, we consider the one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. It is assumed that the corresponding stationary operator depending on time as a parameter has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. A solution close at some moment to an eigenfunction of the stationary operator is studied. We describe its asymptotic behavior in the case where the eigenvalues of the stationary operator move to the edge of the continuous spectrum and, having reached it, disappear one after another.     

Positive and Nontrivial Solutions for the Urysohn Integral Equation

We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.     

On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions

In the paper, we study the problem on the number of zeros of eigenfunctions of the fourth-order boundary-value problem with spectral and physical parameters in the boundary conditions. We show that the number of zeros of the eigenfunctions corresponding to eigenvalues of positive type behaves in a usual way (it is equal to the serial number of an eigenvalue increased by 1), but, however, the number of zeros of the eigenfunction corresponding to an eigenvalue of negative type can be arbitrary. In the case of a sufficient smoothness of coefficients of the differential expression, we write the asymptotics in the physical parameter for such a number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 41–52, 2006.     

Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators

We study uniformly elliptic fully nonlinear equations of the type F(D²u,Du,u,x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations.     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

Local Perturbations of the Schrödinger Operator on the Axis

We present necessary and sufficient conditions for the existence of eigenvalues of the Schrödinger operator on an axis under small local perturbations. In the case where the eigenvalues exist, we construct their asymptotic approximations.     

Mean curvature flow with obstacles

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.     

Higher order non‐resonance for differential equations with singularities

In this paper we prove an existence result of positive periodic solutions to second order differential equations with certain strong repulsive singularities near the origin and with some semilinear growth near infinity. Different from the nonsingular case, the result in this paper shows that both of the periodic and the antiperiodic eigenvalues play the same role in such an existence result. Copyright © 2003 John Wiley & Sons, Ltd.     

Decaying modes for the wave equation in the exterior of an obstacle

In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λ_k} of Δ on the geometry of the obstacle ??. In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle ??, both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres—for which the eigenvalues {λ_k} can be determined as roots of special functions—upper and lower bounds for the density of the real {λ_k}, and upper and lower bounds for λ₁, the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on ??. Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono-energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.     

Existence results for bellman equations and maximum principles in unbounded domains

The purpose of this paper is to extend known existence results for Hamilton-Jacobi-Bellman equations. The classical results give existence, uniqueness and Holder regularity when all elliptic operators involved have nonpositive zero-order term. We want to handle here the more general case where they have principal eigenvalues bounded below by a positive constant. As a motivation for this work, we give an application to the study of the Maximum Principle in infinite cylinders, following a work by Berestycki, Caffarelli and Nirenberg [2]. This is used to extend the cylindrical symmetry result in [2] to a more general class of operators.     

Banach空间非线性Sturm-Liouville边值问题的正解

通过对非紧性测度的精细计算, 结合相应的线性方程的特征值理论, 运用凝聚映射的不动点指数理论, 分别在超线性与次线性情形下, 讨论Banach空间Sturm-Liouville边值问题正解的存在性.     

On the eigenvalue set structure for higher-order equations of elliptic type with discontinuous nonlinearities

We study the existence of nonzero solutions of the Dirichlet problem for a higherorder equation of elliptic type with a discontinuous nonlinearity. By the variational method, we prove the existence of a ray of positive eigenvalues and prove an upper bound for the bifurcation parameter in the problem.