权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Spectral-type problems and nonlinear boundary value problems

谱型问题和非线性边值问题

基本信息

批准号：
EP/H030514/1
负责人：
Bryan Rynne
金额：
$ 32.57万
依托单位：
Heriot-Watt University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH030514%2F1
关键词：
Spectral type problems nonlinear boundary

项目摘要

A vast range of problems in applied mathematics can be modelled usingdifferential equations. Whilst simple models can be constructed using socalled `linear' differential equations, more realistic models usuallydemand the use of nonlinear equations. However, the nonlinearities insuch equations can range from being relatively simple to beinghopelessly complicated, so to make progress with any sort of theory ofsuch equations one must impose restrictions on the type of nonlinearityconsidered.A type of problem which is amenable to considerable analysis is one inwhich the nonlinearity is `asymptotically linear' - this means that whenthe unknown variable becomes large the problem essentially behaveslinearly, and we can then use the well understood linear theory toprovide information about the nonlinear problem. However, the assumptionof asymptotic linearity means that the nonlinear term behaves in thesame way when the unknown variable, say u, is both large and positive,and large and negative. Often, this is unrealistic, and in fact manyproblems have a certain type of (nearly) linear behaviour when u islarge and positive, and a different type of linear behaviour when u islarge and negative. Linearities with this type of behaviour are termed`jumping'. Jumping nonlinearities arise naturally in many applications,such as elasticity problems where the elastic constant involved isdifferent when the material is being stretched and compressed. Forexample, wires may be strong under tension, but have no resistance tocompression. Indeed, the theory of jumping nonlinearities has beenapplied to suspension bridges (a jumping nonlinearity model is relevanthere due to the cable supports for the road deck of the bridge), andleads to an explanation for the well-known Tacoma bridge collapse thatis different to the standardAlthough at first sight, jumping nonlinearities may seem to be only asimple generalisation of asymptotically linear problems, they are,surprisingly, much more complicated to deal with, and have been activelystudied since the late 70's. A generalisation of the usual idea of thespectrum of a linear operator to deal with jumping nonlinearities wasintroduced in about 1977. This was called the Fucik spectrum, and anextension of this, called the set of `half-eigenvalues' has been studiedmore recently. Each of these sets has been studied extensively, and leadto results that have no counterpart in linear or asymptotically linearproblems. In particular, for periodic problems the structure of theFucik spectrum or the set of half-eigenvalues is still not understood,although recent results show that this structure is much morecomplicated than in the linear case.Most work on these sets has dealt with the semilinear case, where astandard linear, second (or higher) order, elliptic differentialoperator has a jumping nonlinearity added to it. However, there is aquasilinear generalisation of such linear differential operators calledthe p-Laplacian. The p-Laplacian arises in many applications, such asnon-Newtonian fluid flows and percolation problems, and is currentlyunder intense study by mathematicians worldwide (as is shown bysearching for `p-Laplacian' on mathscinet). Many properties of the usualLaplacian extend to the p-Laplacian, but not all do so. In particular,since the p-Laplacian has a positive homogeneity property, it makessense to define a spectrum for it, and some of the spectral propertiesof the linear problem generalise to the p-Laplacian.It is our view that the interaction of the p-Laplacian and jumpingnon-linearities will provide fascinating and rich solution behaviour andwill provide a fundamental framework from which a better understandingof more complex applications can be obtained.

应用数学中的许多问题都可以用微分方程来模拟.虽然简单的模型可以用所谓的“线性”微分方程来构造，但更现实的模型通常需要使用非线性方程。然而，这种方程的非线性可以从相对简单到完全复杂，因此，要想在这种方程的任何理论上取得进展，就必须对所考虑的非线性类型加以限制。有一类问题可以进行大量的分析，其中的非线性是“渐近线性的”--这意味着当未知变量变大时，问题基本上是线性的，然后我们就可以用我们熟知的线性理论来提供关于非线性问题的信息。然而，渐近线性的解释意味着当未知变量，比如u，既大又正，又大又负时，非线性项的行为是相同的。通常，这是不切实际的，事实上，当u为大且正值时，许多问题都有某种类型的（接近）线性行为，而当u为大且负值时，则有另一种类型的线性行为。具有这种行为的线性称为“跳跃”。跳跃非线性在许多应用中自然出现，例如弹性问题，当材料被拉伸和压缩时，所涉及的弹性常数是不同的。例如，金属丝在张力下可能很强，但对压缩没有抵抗力。实际上，跳跃非线性理论已经应用于悬索桥（由于桥面的缆索支撑，跳跃非线性模型是相关的），并导致对著名的塔科马桥倒塌的解释，这与标准不同。尽管乍一看，跳跃非线性似乎只是渐近线性问题的简单概括，但令人惊讶的是，处理起来要复杂得多，而且从70年代后期开始就一直在积极研究。一个概括的通常想法的频谱的线性算子处理跳跃非线性是在1977年左右。这就是所谓的Fucik谱，和一个扩展，这被称为“半本征值”的一套最近已经研究。每一个这些集已被广泛研究，并导致结果，没有对应的线性或渐近linearproblems。特别是对于周期问题，Fucik谱或半特征值集的结构仍然不清楚，尽管最近的结果表明，这种结构比线性情况复杂得多。关于这些集的大多数工作都是处理半线性情况，其中标准线性，二阶线性，二阶线性和三阶线性。（或更高）阶，椭圆微分算子有一个跳跃的非线性添加到它。然而，有拟线性推广的线性微分算子称为p-Laplacian。p-Laplacian在许多应用中出现，例如非牛顿流体流动和渗流问题，目前正受到全世界数学家的密切研究（如在mathscinet上搜索“p-Laplacian”所示）。通常拉普拉斯算子的许多性质都可以扩展到p-Laplacian算子，但并非所有的都可以。特别是，由于p-Laplacian有一个积极的齐次属性，它makesense定义一个频谱，和一些谱性质的线性问题推广到p-Laplacian.It是我们的观点，相互作用的p-Laplacian和跳跃非线性将提供迷人的和丰富的解决方案的行为，并将提供一个基本框架，从中可以得到更好的理解更复杂的应用。