On the principal eigenvalue of a periodic-parabolic operator

1984 ◽  
Vol 9 (9) ◽  
pp. 919-941 ◽  
Author(s):  
Alex Beltramo ◽  
Peter Hess
Keyword(s):  
Principal Eigenvalue ◽  
Parabolic Operator

Protection Zones in Periodic-Parabolic Problems

2020 ◽  
Vol 20 (2) ◽  
pp. 253-276
Author(s):  
Julián López-Gómez
Keyword(s):  
Mixed Type ◽  
Principal Eigenvalue ◽  
Boundary Operator ◽  
General Boundary ◽  
Parabolic Problems ◽  
Parabolic Operator ◽  
Protection Zones ◽  
Competitive Environments

AbstractThis paper characterizes whether or not\Sigma_{\infty}\equiv\lim_{\lambda\uparrow\infty}\sigma[\mathcal{P}+\lambda m(% x,t),\mathfrak{B},Q_{T}]is finite, where {m\gneq 0} is T-periodic and {\sigma[\mathcal{P}+\lambda m(x,t),\mathfrak{B},Q_{T}]} stands for the principal eigenvalue of the parabolic operator {\mathcal{P}+\lambda m(x,t)} in {Q_{T}\equiv\Omega\times[0,T]} subject to a general boundary operator of mixed type, {\mathfrak{B}}, on {\partial\Omega\times[0,T]}. Then this result is applied to discuss the nature of the territorial refuges in periodic competitive environments.

scholarly journals Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator

2019 ◽  
Vol 147 (12) ◽  
pp. 5291-5302 ◽  
Author(s):  
Shuang Liu ◽  
Yuan Lou ◽  
Rui Peng ◽  
Maolin Zhou
Keyword(s):  
Linear Time ◽  
Principal Eigenvalue ◽  
Parabolic Operator ◽  
Time Periodic

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

Rearrangements and minimization of the principal eigenvalue of a nonlinear Steklov problem

2011 ◽  
Vol 74 (16) ◽  
pp. 5697-5704 ◽  
Author(s):  
Behrouz Emamizadeh ◽  
Mohsen Zivari-Rezapour
Keyword(s):  
Principal Eigenvalue ◽  
Steklov Problem

scholarly journals Eigenvalue problems for a quasilinear elliptic equation onℝN

2005 ◽  
Vol 2005 (18) ◽  
pp. 2871-2882 ◽  
Author(s):  
Marilena N. Poulou ◽  
Nikolaos M. Stavrakakis
Keyword(s):  
Elliptic Equation ◽  
Weight Function ◽  
Quasilinear Elliptic Equation ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Laplacian Operator ◽  
Quasilinear Elliptic

We prove the existence of a simple, isolated, positive principal eigenvalue for the quasilinear elliptic equation−Δpu=λg(x)|u|p−2u,x∈ℝN,lim|x|→+∞u(x)=0, whereΔpu=div(|∇u|p−2∇u)is thep-Laplacian operator and the weight functiong(x), being bounded, changes sign and is negative and away from zero at infinity.

scholarly journals Multiple solutions for Robin (p, q)-equations plus an indefinite potential and a reaction concave near the origin

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato
Keyword(s):  
Multiple Solutions ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Potential Term ◽  
Indefinite Potential

AbstractWe consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When $$q=2$$ q = 2 (a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems.

scholarly journals ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH SPATIALLY DEPENDENT RANDOM JUMPS

2011 ◽  
Vol 13 (06) ◽  
pp. 1077-1093
Author(s):  
NITAY ARCUSIN ◽  
ROSS G. PINSKY
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Large Time ◽  
Dirichlet Boundary ◽  
Principal Eigenvalue ◽  
Probability Measures ◽  
Rate Of Decay ◽  
Non Local ◽  
Random Jumps ◽  
Explicit Constant

Let D ⊂ Rd be a bounded domain and let [Formula: see text] denote the space of probability measures on D. Consider a Brownian motion in D which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity γV to a new point, according to a distribution [Formula: see text]. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator -Lγ,μ, defined by [Formula: see text] with the Dirichlet boundary condition, where Cμ is the "μ-centering" operator defined by [Formula: see text] The principal eigenvalue, λ0(γ, μ), of Lγ, μ governs the exponential rate of decay of the probability of not exiting D for large time. We study the asymptotic behavior of λ0(γ, μ) as γ → ∞. In particular, if μ possesses a density in a neighborhood of the boundary, which we call μ, then [Formula: see text] If μ and all its derivatives up to order k - 1 vanish on the boundary, but the kth derivative does not vanish identically on the boundary, then λ0(γ, μ) behaves asymptotically like [Formula: see text], for an explicit constant ck.

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