Minimization of the First Nonzero Eigenvalue Problem for Two-Phase Conductors with Neumann Boundary Conditions

2020 ◽  
Vol 80 (4) ◽  
pp. 1607-1628 ◽  
Author(s):  
Di Kang ◽  
Patrick Choi ◽  
Chiu-Yen Kao
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Two Phase ◽  
Nonzero Eigenvalue ◽  
Phase Conductors

scholarly journals On the spectrum of one dimensional p-Laplacian for an eigenvalue problem with Neumann boundary conditions

2013 ◽  
Vol 33 (1) ◽  
pp. 9
Author(s):  
Ahmed Dakkak ◽  
Siham El Habib ◽  
Najib Tsouli
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
Strict Monotonicity ◽  
Indefinite Weight ◽  
One Dimensional ◽  
Weight One

This work deals with an indefinite weight one dimensional eigenvalue problem of the p-Laplacian operator subject to Neumann boundary conditions. We are interested in some properties of the spectrum like simplicity, monotonicity and strict monotonicity with respect to the weight. We also aim the study of zeros points of eigenfunctions.

scholarly journals Various Half-Eigenvalues of Scalarp-Laplacian with Indefinite Integrable Weights

2009 ◽  
Vol 2009 ◽  
pp. 1-27 ◽  
Author(s):  
Wei Li ◽  
Ping Yan
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Lebesgue Space ◽  
Weak Topology ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Fréchet Differentiable

Consider the half-eigenvalue problem(ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0a.e.t∈[0,1], where1<p<∞,ϕp(x)=|x|p−2x,x±(⋅)=max⁡{±x(⋅),0}forx∈&#x1D49E;0:=C([0,1],ℝ), anda(t)andb(t)are indefinite integrable weights in the Lebesgue spaceℒγ:=Lγ([0,1],ℝ),1≤γ≤∞. We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in(a,b)∈(ℒγ,wγ)2, wherewγdenotes the weak topology inℒγspace. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in(a,b)∈(ℒγ,‖⋅‖γ)2, where‖⋅‖γis theLγnorm ofℒγ.

scholarly journals THE BEST SOBOLEV TRACE CONSTANT IN PERIODIC MEDIA FOR CRITICAL AND SUBCRITICAL EXPONENTS

2009 ◽  
Vol 51 (3) ◽  
pp. 619-630
Author(s):  
JULIÁN FERNÁNDEZ BONDER ◽  
RAFAEL ORIVE ◽  
JULIO D. ROSSI
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Limit Problem ◽  
Neumann Boundary Conditions ◽  
Periodic Media ◽  
Best Constant ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Trace Embedding

AbstractIn this paper we study homogenisation problems for Sobolev trace embedding H1(Ω) ↪ Lq(∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, and we consider H1 and Lq spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of a homogenised limit problem, and the best trace constant converges to a homogenised best trace constant. Our results are in fact more general; we can also consider general operators of the form aɛ(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) ↪ Lq(∂Ω).

Invariant manifolds for metastable patterns in ut = ε2uxx—f(u)

1990 ◽  
Vol 116 (1-2) ◽  
pp. 133-160 ◽  
Author(s):  
Jack Carr ◽  
Robert Pego
Keyword(s):  
Energy Density ◽  
Initial Data ◽  
Boundary Conditions ◽  
Density Function ◽  
Invariant Manifolds ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Metastable States ◽  
Two Phase ◽  
Unstable Manifolds

SynopsisWe consider the above equation on the interval 0 ≦ x ≦ 1 subject to Neumann boundary conditions with f(u) = F′(u) where F is a double well energy density function with equal minima. Our previous work [3] proved the existence and persistence of very slowly evolving patterns (metastable states) in solutions with two-phase initial data. Here we characterise these metastable states in terms of the global unstable manifolds of equilibria, as conjectured by Fusco and Hale [6].

Asymptotic behaviour of solutions of elliptic equations with critical exponents and Neumann boundary conditions

1991 ◽  
Vol 117 (3-4) ◽  
pp. 225-250 ◽  
Author(s):  
C. Budd ◽  
M. C. Knaap ◽  
L. A. Peletier
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
Nonlinear Eigenvalue Problem ◽  
Neumann Boundary ◽  
Critical Sobolev Exponent ◽  
Neumann Boundary Conditions ◽  
Supremum Norm ◽  
Asymptotic Estimates ◽  
Sobolev Exponent

SynopsisAsymptotic estimates are established for nontrivial positive radial eigenfunctions of the nonlinear eigenvalue problem −Δu= λ(up−uq) in the unit ballBin ℝN(N> 2) with Neumann boundary conditions, as the supremum norm tends to infinity. Herepis the critical Sobolev exponent (N+ 2)/(N− 2) and 0 <q<p− 1 = 4/(N− 2).

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions

2020 ◽  
Vol 28 (2) ◽  
pp. 237-241
Author(s):  
Biljana M. Vojvodic ◽  
Vladimir M. Vladicic
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions

AbstractThis paper deals with non-self-adjoint differential operators with two constant delays generated by {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})}, where {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials {q_{j}} are real-valued functions, {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions {y(0)=y(\pi)=0} and the remaining two under boundary conditions {y(0)=y^{\prime}(\pi)=0}.

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