Generalized principal eigenvalues for indefinite-weight elliptic problems

In this paper we prove a necessary and a sufficient conditions for the existence of a positive solution of the equation (P − μW)_u = 0 in Ω, where P is a critical, secondorder, linear elliptic operator which is defined on a subdomain Ω of a noncompact Riemannian manifold X. It is assumed that W ε Cδ (Ω) is a “weak” perturbation and μ < 0 is small enough.     

Partial symmetry and asymptotic behavior for some elliptic variational problems

A short elementary proof based on polarizations yields a useful (new) rearrangement inequality for symmetrically weighted Dirichlet type functionals. It is then used to answer some symmetry related open questions in the literature. The non symmetry of the Hénon equation ground states (previously proved in [19]) as well as their asymptotic behavior are analyzed more in depth. A special attention is also paid to the minimizers of the Caffarelli-Kohn-Nirenberg [8] inequalities.Received: 5 May 2002, Accepted: 3 September 2002, Published online: 17 December 2002Mathematics Subject Classification (2000): 35B40 - 35J20     

On the asymptotic behavior of eigenvalues of the radial p-laplacian

We study the asymptotic behavior and distribution of the eigenvalues of the singular radial p-laplacian. We prove a Weyl type asymptotic formula for the number of eigenvalues less than a given value.     

On the asymptotic behavior of some algorithms

A simple approach is presented to study the asymptotic behavior of some algorithms with an underlying tree structure. It is shown that some asymptotic oscillating behaviors can be precisely analyzed without resorting to complex analysis techniques as it is usually done in this context. A new explicit representation of periodic functions involved is obtained at the same time. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

On the asymptotic behaviour of elliptic problems with periodic data

We study the asymptotic behaviour of the solution of elliptic problems with periodic data when the size of the domain on which the problem is set becomes unbounded. To cite this article: M. Chipot, Y. Xie, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On the asymptotic behavior of solutions of quasilinear elliptic equations

The asymptotic behavior of solutions of second-order quasilinear elliptic and nonhyperbolic partial differential equations defined on unbounded domains inR ⁿ contained in\(\{ x_1 ,...,x_n :\left| {x_n } \right|< \lambda \sqrt {x_1^2 + ...x_{n - 1}^2 } \) for certain sublinear functions λ is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data has appropriate asymptotic behavior at infinity. We prove Phragmèn-Lindelöf theorems for large classes of nonhyperbolic operators, without «lower order terms”, including uniformly elliptic operators and operators with well-definedgenre, using special barrier functions which are constructed by considering an operator associated to our original operator. We also estimate the rate at which a solution converges to its limiting function at infinity in terms of properties of the top order coefficienta _nn of the operator and the rate at which the boundary values converge to their limiting function; these results are proven using appropriate barrier functions which depend on the behavior of the coefficients of the operator and the rate of convergence of boundary values.     

On the identification of coefficients of elliptic problems by asymptotic regularization

The problem of recovering coefficients of elliptic problems from measured data is considered. An algorithm is developed to identify the unknown coefficients without a minimization technique. The method is based on the construction of certain time-dependent problems which contain the original equation as asymptotic steady state. A Liapunovtype a-priori estimate is fundamental to prove that the solution of the time-dependent regularized equations approach a solution of the original problem as t →∞. A related behavior is proved for the solution of corresponding finite-dimensional Galerkin approximations. A stability result is proved for the Galerkin approximations.     

On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting

((without abstract.)) Received January 20, 1998     

On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting

((without abstract.)) Received January 20, 1998     

Spectral asymptotic behavior of elliptic systems

One investigates a first-order elliptic self-adjoint pseudodifferential operator A (x,D) acting in sections of a Hermitian vector bundle over a compactn-dimensional manifold x. It is assumed that the principal symbol A(x, ξ) of the operator is locally diagonalizable and that its eigenvaluesaj(x, ξ) have a variable multiplicity and that {a _i,a _k}≠0 whenevera _i=a _k. Under indicated conditions one constructs an expansion of the fundamental solution of the hyperbolic system \(i\frac{{\partial u}}{{\partial t}} = A(x,D)u\) and one investigates the asymptotic properties of the spectrum of the operator A (x,D). For the distribution functionN(λ) of the eigenvalues one establishes that . Under further assumptions on the properties of the bicharacteristic of the symbolsaj(x, ξ) one establishes a stronger estimate of Duistermaat-Guillemin type:N(λ)=Cλ ⁿ +C′λ ^n?i +0(γ ^n?1 )     

On the principal eigenvalue of some nonlocal diffusion problems

In this paper we analyze some properties of the principal eigenvalue λ₁(Ω) of the nonlocal Dirichlet problem (J∗u)(x)−u(x)=−λu(x) in Ω with u(x)=0 in RN?Ω. Here Ω is a smooth bounded domain of RN and the kernel J is assumed to be a C¹ compactly supported, even, nonnegative function with unit integral. Among other properties, we show that λ₁(Ω) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Ω. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Λ(γ)=λ₁(γΩ) when the dilatation parameter γ>0 tends to zero or to infinity.     

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions ofthe quasilinear elliptic problem –_pu = a u^p–1–b(x)u^q, u| = 0, as q p – 1 + 0 and as q , via a scale argument.Here _p is the p-Laplacian with 1 < p and q > p –1. If p = 2, such problems arise in population dynamics. Ourmain results generalize the results for p = 2, but some technicaldifficulties arising from the nonlinear degenerate operator–_p are successfully overcome. As a by-product, we cansolve a free boundary problem for a nonlinear p-Laplacian equation.     

On the index instability for some nonlocal elliptic problems

The index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at the conjugation points do not affect the index of the unbounded operator. In this paper, we construct examples showing that the index may change under nonlocal perturbations with coefficients not vanishing at the points of conjugation of boundary-value conditions. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 178–193, 2007.