Differentiable Peak-Interpolation on Bounded Domains with Smooth Boundary

1985 ◽  
Vol 17 (2) ◽  
pp. 134-136 ◽  
Author(s):  
Alan Noell
Keyword(s):  
Smooth Boundary ◽  
Bounded Domains ◽  
Peak Interpolation

scholarly journals On the cubic NLS on 3D compact domains

2013 ◽  
Vol 13 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Fabrice Planchon
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Schrodinger Equation ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Dirichlet Boundary Conditions ◽  
Well Posedness ◽  
Bounded Domains ◽  
Bilinear Estimates

AbstractWe prove bilinear estimates for the Schrödinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the ${ \mathbb{R} }^{3} $ case, while on bounded domains they match the generic boundaryless manifold case. We obtain, as an application, global well-posedness for the defocusing cubic NLS for data in ${ H}_{0}^{s} (\Omega )$, $1\lt s\leq 3$, with $\Omega $ any bounded domain with smooth boundary.

On a Kirchhoff Equation in Bounded Domains

2018 ◽  
Vol 18 (3) ◽  
pp. 613-648
Author(s):  
Yisheng Huang ◽  
Yuanze Wu
Keyword(s):  
Variational Method ◽  
Smooth Boundary ◽  
Uniqueness Result ◽  
Kirchhoff Equation ◽  
Bounded Domains ◽  
Multiplicity Results ◽  
Nodal Domains ◽  
Existence And Multiplicity ◽  
Sobolev Exponent ◽  
Sign Changing Solutions

AbstractIn this paper, we consider the following Kirchhoff equation:\left\{\begin{aligned} &\displaystyle{-}\bigg{(}a+b\int_{\Omega}\lvert\nabla u% |^{2}\,dx\bigg{)}\Delta u=\lambda u+|u|^{p-2}u&&\displaystyle\text{in }\Omega,% \\ &\displaystyle u=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right.where{\Omega\subset\mathbb{R}^{N}}({N\geq 3}) is a bounded domain with smooth boundary{\partial\Omega},{2<p<2^{*}=\frac{2N}{N-2}}is the Sobolev exponent anda,b, λ are positive parameters. By the variational method, we obtain some existence and multiplicity results of the sign-changing solutions (including the radial sign-changing solution in the case of{\Omega=\mathbb{B}_{R}}) for this problem. Some further properties of these sign-changing solutions, such as the numbers of the nodal domains, the concentration behaviors as{b\to 0^{+}}, the estimates of the energy values and so on, are also obtained. Our results generalize and improve some known results in the literature. Moreover, we also obtain a uniqueness result of the radial positive solution.

scholarly journals The geometric invariants for the spectrum of the Stokes operator

2021 ◽  
Author(s):  
Genqian Liu
Keyword(s):  
Asymptotic Expansion ◽  
Surface Area ◽  
Smooth Boundary ◽  
Stokes Problem ◽  
Stokes Operator ◽  
Bounded Domains ◽  
Spectral Invariants ◽  
Precise Information ◽  
Geometric Invariants ◽  
Dimensional Ball

AbstractFor a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion for the integral of the trace of the Stokes semigroup $$e^{-t S}$$ e - t S as $$t\rightarrow 0^+$$ t → 0 + . These coefficients (i.e., spectral invariants) provide precise information for the volume of the domain $$\Omega $$ Ω and the surface area of the boundary $$\partial \Omega $$ ∂ Ω by the spectrum of the Stokes problem. As an application, we show that an n-dimensional ball is uniquely determined by its Stokes spectrum among all Euclidean bounded domains with smooth boundary.

scholarly journals Interpolating sequences for weighted Bergman spaces on strongly pseudoconvex bounded domains

2021 ◽  
pp. 2150026
Author(s):  
Hamzeh Keshavarzi
Keyword(s):  
Unit Ball ◽  
Bounded Domain ◽  
Interpolation Problem ◽  
Smooth Boundary ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Bounded Domains ◽  
Interpolating Sequences ◽  
Conjugate Exponent

Let [Formula: see text], [Formula: see text], and [Formula: see text] be a strongly pseudoconvex bounded domain with a smooth boundary in [Formula: see text]. We will study the interpolation problem for weighted Bergman spaces [Formula: see text]. In the case, [Formula: see text], and [Formula: see text], where [Formula: see text] is the conjugate exponent of [Formula: see text] (let [Formula: see text], for [Formula: see text]), we show that a sequence in [Formula: see text], the unit ball in [Formula: see text], is interpolating for [Formula: see text] if and only if it is separated.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

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