On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two‐sided estimates for this term in a variety of situations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem

We writef=(g) iff(x)cg(x) with some positive constantc for allx from the domain of functionsf andg. We show that at least (n ² /r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank belowr. This improves the previously known bound (n ²/r ²). If we additionally know that the changes are bounded above in absolute value by some numbern/r, then the number of these entries is bounded below by (n ³/(r ² )), which improves upon the previously known bound (n ² / ² ).Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 535–540, April, 1998.The research of the first author was supported by the Russian Foundation for Basic Research under grants No. 96-01-00094 and No. 96-15-96102 and by the INTAS Foundation under grant No. 93-1376. The research of the second author was supported by the Russian Foundation for Basic Research under grants No. 96-01-01222 and No. 96-15-96090.     

On a nonlinear eigenvalue problem

In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R ¹. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.     

Nonclassical eigenvalue asymptotics

The leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {$\text{(x,y)∥x¦}{}^{\mathrm{\mu }}\text{¦y¦? 1}$} and for $\text{?Δ + ¦x¦}{}^{\mathrm{\alpha }}\text{¦ y¦}{}^{\mathrm{\beta }}$ all of which are non-Weyl because of infinite phase space volumes are computed. Along the way, a general inequality on quantum partition functions computed in a kind of Born-Oppenheimer approximation is proved.     

On the asymptotics of the maximal eigenvalue for a family of branching processes

We establish an asymptotic representation of the maximal eigenvalue ρ_ε for a family of branching processes with arbitrary number of types of particles which are close to critical ones. L’viv University, L’viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1112–1117, August, 1999.     

On the condition numbers of a multiple eigenvalue of a generalized eigenvalue problem

For standard eigenvalue problems, closed-form expressions for the condition numbers of a multiple eigenvalue are known. In particular, they are uniformly 1 in the Hermitian case and generally take different values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers. Our main result is that a multiple eigenvalue generally has multiple condition numbers, even in the Hermitian definite case. The condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors.     

On a nonlinear elliptic eigenvalue problem

Let N(λ) be the number of the solutions of the equation: , where Ω is a bounded domain in with smooth boundary. Under suitable conditions on f, we proved that N(λ)→+∞ as λ→+∞.     

On a recursive inverse eigenvalue problem

Let s ₁, ..., s _n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s _i is an eigenvalue of the leading principal submatrix A _i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s ₁, ..., s _n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A _i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A _i and A _{i + 1}.     

On solving a nonlinear matrix eigenvalue problem

Journal of Mathematical Sciences -     

On a nonlinear eigenvalue problem in ODE

In this paper we shall study the following variant of the logistic equation with diffusion:

−du^″(x)=g(x)u(x)−u²(x)     

On the asymptotic solutions of a generalized eigenvalue problem

Summary The asymptotic values of the eigenvalues and the corresponding eigenfunctions of the kernel (1/2)Ln|r(s)–r(s)| are obtained, where |r(s)–r(s)| is the distance between two pointsr(s) andr(s) on a closed smooth curveC.

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Résumé Soit |r(s)–r(s)| la distance entre deux points d'une courbe fermée plane; on détermine le comportement asymptotique des valeurs propres et fonctions propres du noyau (1/2)Ln|r(s)–r(s)|.

     

On the eigenvalue problem of 't Hooft

We determine the asymptotic distribution of the eigenvalues in t Hoofts eigenvalue problem in two-dimensional quantum chromodynamics. We formulate the problem as an eigenvalue problem for a singular pseudodifferential operator and use systematically its basic invariance properties.     

On the Robin problem in fractal domains

We study the solution to the Robin boundary problem for theLaplacian in a Euclidean domain. We present some families offractal domains where the infimum of the solution to the mixedDirichlet–Robin boundary problem is greater than 0, andsome other families of domains where it is equal to 0. We alsogive a new result on ‘trap domains’ defined in Burdzy,Chen and Marshall (Math. Z.), that is, domains where reflectingBrownian motion takes a long time to reach the center of thedomain.     

On the second solution to a critical growth Robin problem

We investigate the existence of the second mountain-pass solution to a Robin problem, where the equation is at critical growth and depends on a positive parameter λ. More precisely, we determine existence and nonexistence regions for this type of solutions, depending both on λ and on the parameter in the boundary conditions.