Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

By variational methods and Morse theory, we prove the existence of uncountably many \((\alpha ,\beta )\in \mathbb R ^2\) for which the equation \(-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}\) in \(\Omega \) , has a sign changing solution under the Neumann boundary condition, where a map \(A\) from \(\overline{\Omega }\times \mathbb R ^N\) to \(\mathbb R ^N\) satisfying certain regularity conditions. As a special case, the above equation contains the \(p\) -Laplace equation. However, the operator \(A\) is not supposed to be \((p-1)\) -homogeneous in the second variable. In particular, it is shown that generally the Fu?ík spectrum of the operator \(-\mathrm{div}\, A(x, \nabla u)\) on \(W^{1,p}(\Omega )\) contains some open unbounded subset of \(\mathbb R ^2\) .     

Discrete analogue of Fučik spectrum of the Laplacian

The discrete analog of the Fučik spectrum for elliptic equations, namely M-matrices, is shown to have properties analogous to the continuum. In particular, the Fučik spectrum of a M-matrix contains a continuous and decreasing curve which is symmetric with respect to the diagonal.     

On Fučik Spectra and Periodic Solutions

In this paper, we continue extending the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand side. To this end, we construct a new version of the method of shifts along trajectories. We compare the results obtained by the new approach and those obtained by the method of Fuik spectra.     

On the Fučik point spectrum for Schrödinger operators on {\mathbb{R}}^{N}

We investigate the Fučik point spectrum of the Schr?dinger operator when the potential V_λ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that S_λ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schr?dinger equations with jumping nonlinearity.      

On the Fu?ik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equation

utt−uss=au⁺−bu⁻+p(s,t,u),     

On absolute continuity of the spectrum of periodic Schrödinger operators

In this paper we find a new condition on a real periodic potential for which the self-adjoint Schrödinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted \(L^2\) spaces on the torus, and an oscillatory integral theorem.     

On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae

In this paper we give an unified approach to some questions arising in different fields of nonlinear analysis, namely: (a) the study of the structure of the Fuík spectrum and (b) possible variants and extensions of the monotonicity formula by Alt-Caffarelli-Friedman [1]. In the first part of the paper we present a class of optimal partition problems involving the first eigenvalue of the Laplace operator. Beside establishing the existence of the optimal partition, we develop a theory for the extremality conditions and the regularity of minimizers. As a first application of this approach, we give a new variational characterization of the first curve of the Fuík spectrum for the Laplacian, promptly adapted to more general operators. In the second part we prove a monotonicity formula in the case of many subharmonic components and we give an extension to solutions of a class of reaction-diffusion equation, providing some Liouville-type theorems.Received: 27 December 2003, Accepted: 29 January 2004, Published online: 2 April 2004Mathematics Subject Classification (2000): 35J65 (58E05)Work partially supported by MIUR project Metodi Variazionali ed Equazioni Differenziali Non Lineari     

On the absolutely continuous spectrum of magnetic Schrödinger operators

We prove that [0, ∞) is an essential support for the absolutely continuous part of the spectral measure associated with the magnetic Schr?dinger operator (i∇ + A)² in , given certain conditions on the decay of A. Bibliography: 8 titles. Illustrations: 1 figure. Translated from Problemy Matematicheskogo Analiza, No. 38, 2008, pp. 121–143.     

On the essential spectrum of Schrödinger operators on Riemannian manifolds

The main result of this paper is a lower bound for the essential spectrum of Schrödinger operators −Δ+V on Riemannian manifolds. In particular, we obtain conditions on V which imply the discreteness of the spectrum, or equivalently, the compactness of the resolvent.     

On approximating the spectrum of convolution-type operators, 1. Wiener—Hopf matricial integral operators

Letk be an inverse Fourier transform of a real valued bounded and summable functionK, and let {λ _j ^τ (τ > 0)} denote the eigenvalues of the Hermitian integral operator (W _k ^(τ) ?)(t) = ∫ ₀ ^τ k (t ?s)?(s)ds (? ∈L ₂(0,τ)). The well known Kac, Murdock and Szegö formula asserts that $$\mathop {\lim }\limits_{\tau \to \infty } \tau ^{ - 1} \sum\limits_{j = 1}^\infty {[\lambda _j^{(\tau )} ]^3 = (2\pi )} ^{ - 1} \int {_{ - \infty }^\infty [K(x)]^5 dx (s = 2,3, \cdot \cdot \cdot ,)} $$ . The main aim of the present paper is to extend this formula to the case of a complex-valued matrix functionK. We achieve this extension by developing an operator approach which is valid for a wide class of convolution type operators.