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

2008 ◽  
Vol 15 (1) ◽  
pp. 87-92
Author(s):  
Hajime Kawakami

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.

2005 ◽  
Vol 2005 (11) ◽  
pp. 1809-1818 ◽  
Author(s):  
Alan V. Lair

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.


1989 ◽  
Vol 39 (2) ◽  
pp. 161-165
Author(s):  
Jurang Yan

A necessary and sufficient condition is obtained for a first order linear delay differential inequality to be oscillatory. Our main result improves and extends some known results.


2016 ◽  
Vol 8 (1) ◽  
pp. 175-192
Author(s):  
Humberto Ramos Quoirin ◽  
Kenichiro Umezu

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.


Author(s):  
Thomas Duyckaerts

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.


2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas

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.


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