scholarly journals A geometric condition for the uniform stability of linear magnetoelasticity

2021 ◽  
Author(s):  
Thomas Duyckaerts
Keyword(s):  
Boundary Condition ◽  
High Frequency ◽  
Smooth Boundary ◽  
Geometric Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Decay ◽  
Type Boundary ◽  
Necessary And Sufficient ◽  
Type Boundary Condition

We give a necessary and sufficient condition, of geometrical type, for the uniform decay of energy of solutions of the linear system of magnetoelasticity in a bounded domain with smooth boundary. A Dirichlet-type boundary condition is assumed. Our strategy is to use microlocal defect measures to show suitable observability inequalities on high-frequency solutions of the Lamé system.

scholarly journals A necessary and sufficient condition for global existence for a quasilinear reaction-diffusion system

2005 ◽  
Vol 2005 (11) ◽  
pp. 1809-1818 ◽  
Author(s):  
Alan V. Lair
Keyword(s):  
Global Solution ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Special Case

We show that the reaction-diffusion systemut=Δφ(u)+f(v),vt=Δψ(v)+g(u), with homogeneous Neumann boundary conditions, has a positive global solution onΩ×[0,∞)if and only if∫∞ds/f(F−1(G(s)))=∞(or, equivalently,∫∞ds/g(G−1(F(s)))=∞), whereF(s)=∫0sf(r)drandG(s)=∫0sg(r)dr. The domainΩ⊆ℝN(N≥1)is bounded with smooth boundary. The functionsφ,ψ,f, andgare nondecreasing, nonnegativeC([0,∞))functions satisfyingφ(s)ψ(s)f(s)g(s)>0fors>0andφ(0)=ψ(0)=0. Applied to the special casef(s)=spandg(s)=sq,p>0,q>0, our result proves that the system has a global solution if and only ifpq≤1.

An elliptic equation with an indefinite sublinear boundary condition

2016 ◽  
Vol 8 (1) ◽  
pp. 175-192
Author(s):  
Humberto Ramos Quoirin ◽  
Kenichiro Umezu
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Additional Condition ◽  
Smooth Domain ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Abstract We investigate the problem \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\lvert u\rvert^{p-% 2}u&&\displaystyle\phantom{}\text{in ${\Omega}$},\\ \displaystyle\frac{\partial u}{\partial\mathbf{n}}&\displaystyle=\lambda b(x)% \lvert u\rvert^{q-2}u&&\displaystyle\phantom{}\text{on ${\partial\Omega}$},% \end{aligned}\right. where Ω is a bounded and smooth domain of {\mathbb{R}^{N}} ( {N\geq 2} ), {1<q<2<p} , {\lambda>0} , and {b\in C^{1+\alpha}(\partial\Omega)} for some {\alpha\in(0,1)} . We show that {\int_{\partial\Omega}b<0} is a necessary and sufficient condition for the existence of nontrivial non-negative solutions of this problem. Under the additional condition {b^{+}\not\equiv 0} we show that for {\lambda>0} sufficiently small this problem has two nontrivial non-negative solutions which converge to zero in {C(\overline{\Omega})} as {\lambda\to 0} . When {p<2^{*}} we also provide the asymptotic profiles of these solutions.

A Metric Deformation to Obtain a Positive/Negative Gaussian Curvature on a Disk

2008 ◽  
Vol 15 (1) ◽  
pp. 87-92
Author(s):  
Hajime Kawakami
Keyword(s):  
Boundary Condition ◽  
Gaussian Curvature ◽  
Neumann Boundary Condition ◽  
Differential Inequality ◽  
Neumann Boundary ◽  
Partial Answer ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Negative Gaussian Curvature ◽  
Necessary And Sufficient

Abstract We conjecture that the Gauss–Bonnet formula gives a necessary and sufficient condition for the existence of a metric deformation to obtain a positive/negative Gaussian curvature on a disk. The conjecture is reduced to a problem of a partial differential inequality with Dirichlet–Neumann boundary condition. We give a partial answer and an example.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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