On the principal eigenvalue of degenerate quasilinear elliptic systems

We study the properties of the positive principal eigenvalue of a degenerate quasilinear elliptic system. We prove that this eigenvalue is simple, unique up to positive eigenfunctions and isolated. Under certain restrictions on the given data, the regularity of the corresponding eigenfunctions is established. The extension of the main result in the case of an unbounded domain is also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles

Let L be a uniformly elliptic second order differential operator with nice coefficients, defined on a smooth, bounded domain in ℝ ^d , d ≥ 2, with either the Dirichlet or an oblique-derivative boundary condition. In this work we study the asymptotics for the principal eigenvalue of L under hard and soft obstacle perturbations. The hard obstacle perturbation of L is obtained by making a finite number of holes with the Dirichlet boundary condition on their boundaries. The main result gives the asymptotic shift of the principal eigenvalue as the holes shrink to points. The rates are expressed in terms of the Newtonian capacity of the holes and the principal eigenfunctions for the unperturbed operator and its formal adjoint. The soft obstacle corresponds to a finite number of compactly supported finite potential wells. Here we only consider the oblique-derivative Laplacian. The main difference from the hard obstacle problem is that phase transitions occur, due to the various scaling possibilities. Our results generalize known results on similar perturbations for selfadjoint operators. Our approach is probabilistic.     

On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications

Dedicated to Professor Shmuel Agmon     

On coercivity and regularity for linear elliptic systems

We introduce a nonlinear method to study a ??universal?? strong coercivity problem for monotone linear elliptic systems by compositions of finitely many constant coefficient tensors satisfying the Legendre?CHadamard strong ellipticity condition. We give conditions and counterexamples for universal coercivity. In the case of non-coercive systems we give examples to show that the corresponding variational integral may have infinitely many nowhere C ¹ minimizers on their supports. For some universally coercive systems we also present examples with affine boundary values which have nowhere C ¹ solutions.     

Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture

Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D⊆R^d with oblique reflection at ∂D if D≠R^d. For each x∈D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as ε→0 of the expected hitting time of the ε-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at ∂D and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities.     

On a quasilinear elliptic eigenvalue problem with constraint

Via construction of pseudo gradient vector field and descending flow argument, we prove the existence of one positive, one negative and one sign-changing solutions for a quasilinear elliptic eigenvalue problem with constraint.     

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→+∞.     

The behavior of the principal eigenvalue of a mixed elliptic problem with respect to a parameter

In this paper we determine the exact asymptotic behavior of the principal eigenvalue of a mixed elliptic eigenvalue problem which depends on a positive parameter λ when λ→∞. We analyze the case in which the problem is considered in a smooth bounded domain Ω of RN, and also the case of planar domains which are smooth except for a finite number of corner points.     

On the principal minors of a matrix with a multiple eigenvalue

The property of a Hermitian n × n matrix A that all its principal minors of order n − 1 vanish is shown to be a purely algebraic implication of the fact that the lowest two coefficients of its characteristic polynomial are zero. To prove this assertion, no information on the rank or eigenvalues of A is required. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 47–49.     

The principal eigenvalue for periodic nonlocal dispersal systems with time delay

The theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time-periodic nonlocal dispersal cooperative system with time delay. Then we apply it to a Nicholson's blowflies population model and obtain a threshold type result on its global dynamics.     

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)