scholarly journals Nodal Sets of Steklov Eigenfunctions near the Boundary: Inner Radius Estimates

2021 ◽  
Author(s):  
Stefano Decio
Keyword(s):  
Lipschitz Domain ◽  
Nodal Domain ◽  
Nodal Sets ◽  
Bounded Lipschitz Domain ◽  
Inner Radius ◽  
Steklov Eigenfunctions

Abstract We show that Steklov eigenfunctions in a bounded Lipschitz domain have wavelength dense nodal sets near the boundary, in contrast to what can happen deep inside the domain. Conversely, in a 2D Lipschitz domain $\Omega $, we prove that any nodal domain of a Steklov eigenfunction contains a half-ball centered at $\partial \Omega $ of radius $c_{\Omega }/{\lambda }$.

On the Inner Radius of a Nodal Domain

2008 ◽  
Vol 51 (2) ◽  
pp. 249-260 ◽  
Author(s):  
Dan Mangoubi
Keyword(s):  
Riemannian Manifold ◽  
Lower Bounds ◽  
Type Inequality ◽  
Large Eigenvalue ◽  
Upper And Lower Bounds ◽  
Local Behavior ◽  
Nodal Domain ◽  
Closed Riemannian Manifold ◽  
Inner Radius

AbstractLet M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λα(log λ)β. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

On Korn's first inequality with non-constant coefficients

2002 ◽  
Vol 132 (1) ◽  
pp. 221-243 ◽  
Author(s):  
Patrizio Neff
Keyword(s):  
Lipschitz Domain ◽  
Large Deformations ◽  
Type Inequality ◽  
Two Dimensional ◽  
Smooth Part ◽  
Bounded Lipschitz Domain ◽  
Elasto Plasticity ◽  
Constant Coefficients ◽  
Image Position ◽  
Dimensional Lebesgue Measure

In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let Ω ⊂ R3 be a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define and let be given with det Fp(x) ≥ μ+ > 0. Moreover, suppose that Rot . Then Clearly, this result generalizes the classical Korn's first inequality which is just our result with Fp = 11. With slight modifications, we are also able to treat forms of the type

scholarly journals Analysis of the controllability from the exterior of strong damping nonlocal wave equations

2020 ◽  
Vol 26 ◽  
pp. 42 ◽  
Author(s):  
Mahamadi Warma ◽  
Sebastián Zamorano
Keyword(s):  
Wave Equation ◽  
Lipschitz Domain ◽  
Wave Equations ◽  
Control Function ◽  
Complete Analysis ◽  
Strong Damping ◽  
Open Set ◽  
Bounded Lipschitz Domain ◽  
Approximately Controllable ◽  
Fractional Laplace Operator

We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation associated with the fractional Laplace operator subject to the non-homogeneous Dirichlet type exterior condition. In the first part, we show that if 0 <s< 1, Ω ⊂ ℝN(N≥ 1) is a bounded Lipschitz domain and the parameterδ> 0, then there is no control functiongsuch that the following system\begin{align} u_{1,n}+ u_{0,n}\widetilde{\lambda}_{n}^++ \delta u_{0,n}\lambda_{n}=\int_0^{T}\int_{\Omc}(g(x,t)+\delta g_t(x,t))e^{-\widetilde{\lambda}_{n}^+ t}\mathcal{N}_{s}\varphi_{n}(x)\d x\d t,\label{39}\\ u_{1,n}+ u_{0,n}\widetilde{\lambda}_{n}^- +\delta u_{0,n}\lambda_{n}=\int_0^{T}\int_{\Omc}(g(x,t)+\delta g_t(x,t))e^{-\widetilde{\lambda}_{n}^- t}\mathcal{N}_{s}\varphi_{n}(x)\d x\d t,\label{40} \end{align}is exact or null controllable at timeT> 0. In the second part, we prove that for everyδ≥ 0 and 0 <s< 1, the system is indeed approximately controllable for anyT> 0 andg∈D(O× (0,T)), whereO⊂ ℝN\ Ω is any non-empty open set.

scholarly journals Remarques sur la frontière de martin biharmonique et la représentation intégrale des fonctions biharmoniques

2005 ◽  
Vol 2005 (9) ◽  
pp. 1461-1472 ◽  
Author(s):  
Mohamed El Kadiri ◽  
Sabah Haddad
Keyword(s):  
Lipschitz Domain ◽  
Classical Case ◽  
Martin Boundary ◽  
Minimal Points ◽  
Bounded Lipschitz Domain ◽  
Nous Montrons ◽  
Biharmonic Space ◽  
Représentation Intégrale

Soit(Ω,ℋ)un espace biharmonique fort au sens de Smyrnelis dont les espaces harmoniques associés sont des espaces de Brelot qui vérifient l'axiome de proportionnalité. On montre que s'il existe un coupleℋ-harmonique>0surΩ, alors lénsemble des points minimaux de la frontière de Martin biharmonique deΩqui ne sont pas les pôles de couples biharmoniques minimaux est négiligeable dans un sens que l'on précisera. Dans le cas classique d'un domaine lipschitzien borné deℝn, nous montrons que cet ensemble est vide.Let(Ω,ℋ)be a strong biharmonic space of Smyrnelis such that the harmonic spaces associeted are Brelot spaces satisfying the axiom of proportionnality. We prove that if there exists a biharmonic pair greater than0onΩ, then the set of minimal points of the biharmonic Martin boundary ofΩ, that are not the poles of minimal biharmonic pairs, is negligible in some meaning that we will precise. For the classical case of a bounded Lipschitz domain ofℝn, we prove that this set is empty.

scholarly journals Lower bounds for interior nodal sets of Steklov eigenfunctions

2016 ◽  
Vol 144 (11) ◽  
pp. 4715-4722 ◽  
Author(s):  
Christopher D. Sogge ◽  
Xing Wang ◽  
Jiuyi Zhu
Keyword(s):  
Lower Bounds ◽  
Nodal Sets ◽  
Steklov Eigenfunctions

scholarly journals Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Xiaming Chen ◽  
Renjin Jiang ◽  
Dachun Yang
Keyword(s):  
Sobolev Spaces ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Lipschitz Domain ◽  
Lipschitz Domains ◽  
Regularity Estimates ◽  
Bounded Lipschitz Domain ◽  
Divergence Equation ◽  
Strongly Lipschitz Domain

AbstractLet Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable regularity estimates.

scholarly journals Eigenvalues for anisotropic p-Laplacian under a Steklov-like boundary condition

2021 ◽  
Vol 66 (1) ◽  
pp. 85-94
Author(s):  
Luminita Barbu
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Variational Methods ◽  
Lipschitz Domain ◽  
Bounded Lipschitz Domain

"The eigenvalue problem $$-\mbox{div}~\Big(\frac{1}{p}\nabla_\xi \big(F^p\big (\nabla u)\Big)=\lambda a(x) \mid u\mid ^{q-2}u,$$ with $q\in (1, \infty),~ p\in \Big(\frac{Nq}{N+q-1}, \infty\Big),~ p\neq q,$ subject to Steklov-like boundary condition, $$F^{p-1}(\nabla u)\nabla _\xi F (\nabla u)\cdot \nu=\lambda b(x) \mid u\mid ^{q-2}u$$ is investigated on a bounded Lipschitz domain $\Omega\subset \mathbb{R}^ N,~N\geq 2$. Here, $F$ stands for a $C^2(\mathbb{R}^N\setminus \{0\})$ norm and $a\in L^{\infty}(\Omega),~ b\in L^{\infty}(\partial\Omega)$ are given nonnegative functions satisfying \[ \int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0. \] Using appropriate variational methods, we are able to prove that the set of eigenvalues of this problem is the interval $[0, \infty)$."

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