Parametric Variational System with a Smooth-Boundary Constraint Set

2010 ◽  
pp. 205-221 ◽  
Author(s):  
J.-C. Yao ◽  
N. D. Yen
Keyword(s):  
Smooth Boundary ◽  
Variational System ◽  
Boundary Constraint ◽  
Constraint Set

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.

scholarly journals Virtual Angle Boundary-Aware Particle Swarm Optimization to Maximize the Coverage of Directional Sensor Networks

Sensors ◽  
2021 ◽  
Vol 21 (8) ◽  
pp. 2868
Author(s):  
Gong Cheng ◽  
Huangfu Wei
Keyword(s):  
Particle Swarm Optimization ◽  
Sensor Networks ◽  
Particle Swarm ◽  
Pso Algorithm ◽  
The Internet ◽  
Directional Sensor Networks ◽  
Swarm Optimization ◽  
Coverage Optimization ◽  
Boundary Constraint ◽  
Directional Sensor

With the transition of the mobile communication networks, the network goal of the Internet of everything further promotes the development of the Internet of Things (IoT) and Wireless Sensor Networks (WSNs). Since the directional sensor has the performance advantage of long-term regional monitoring, how to realize coverage optimization of Directional Sensor Networks (DSNs) becomes more important. The coverage optimization of DSNs is usually solved for one of the variables such as sensor azimuth, sensing radius, and time schedule. To reduce the computational complexity, we propose an optimization coverage scheme with a boundary constraint of eliminating redundancy for DSNs. Combined with Particle Swarm Optimization (PSO) algorithm, a Virtual Angle Boundary-aware Particle Swarm Optimization (VAB-PSO) is designed to reduce the computational burden of optimization problems effectively. The VAB-PSO algorithm generates the boundary constraint position between the sensors according to the relationship among the angles of different sensors, thus obtaining the boundary of particle search and restricting the search space of the algorithm. Meanwhile, different particles search in complementary space to improve the overall efficiency. Experimental results show that the proposed algorithm with a boundary constraint can effectively improve the coverage and convergence speed of the algorithm.

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

2020 ◽  
Vol 20 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Solutions ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Nonlinear Transformations

AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

Steklov eigenvalue problem with a-harmonic solutions and variable exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Belhadj Karim ◽  
Abdellah Zerouali ◽  
Omar Chakrone
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Boundary ◽  
Normal Vector ◽  
Variable Exponents ◽  
Variational Technique ◽  
Unit Normal Vector ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Outward Unit ◽  
Outward Unit Normal Vector

AbstractUsing the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences:\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}}{(N\geq 2)} is a bounded domain of smooth boundary {\partial\Omega} and ν is the outward unit normal vector on {\partial\Omega}. The functions {m\in L^{\infty}(\partial\Omega)}, {p\colon\overline{\Omega}\mapsto\mathbb{R}} and {a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

Sign in / Sign up

Export Citation Format

Share Document