Spectral-type problems and nonlinear boundary value problems

应用数学中的许多问题都可以用微分方程来建模。虽然简单的模型可以使用所谓的“线性”微分方程来构建，但更现实的模型通常需要使用非线性方程。然而，这些方程中的非线性可以是相对简单的，也可以是极其复杂的，因此，为了在这些方程的任何一种理论中取得进展，人们必须对所考虑的非线性类型施加限制。有一类问题是可以进行大量分析的，其中非线性是“渐近线性”的——这意味着当未知变量变大时，问题本质上表现为线性，然后我们可以使用很好理解的线性理论来提供有关非线性问题的信息。然而，渐近线性的假设意味着，当未知变量，比如u，既大又正，又大又负时，非线性项的表现是一样的。通常，这是不现实的，事实上，当u较大且为正时，许多问题具有某种类型的（接近）线性行为，而当u较大且为负时，则具有不同类型的线性行为。具有这种行为的线性被称为“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，电线在拉力下可能很结实，但对压缩没有抵抗力。事实上，跳跃非线性理论已经应用于悬索桥（跳跃非线性模型是相关的，因为桥的道路甲板上有电缆支撑），并导致了对著名的塔科马大桥倒塌的解释，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，它们处理起来要复杂得多。自70年代末以来一直在积极研究。关于处理跳跃非线性的线性算子的谱的一般思想的推广是在1977年左右提出的。这被称为福西克谱，它的一个扩展，被称为“半特征值集”，最近被研究了。这些集合中的每一个都得到了广泛的研究，并导致了在线性或渐近线性问题中没有对应的结果。特别地，对于周期问题，尽管最近的结果表明这种结构比线性情况要复杂得多，但fucik谱或半特征值集的结构仍然不为人所知。大多数关于这些集合的工作都是处理半线性情况，其中标准线性二阶（或更高）阶椭圆微分算子具有跳跃非线性。然而，这种线性微分算子有一种近似线性的推广，称为p-拉普拉斯算子。p-Laplacian出现在许多应用中，例如非牛顿流体流动和渗透问题，并且目前正受到全世界数学家的密切研究（如在数学科学上搜索“p-Laplacian”所示）。通常拉普拉斯算子的许多性质可以推广到p-拉普拉斯算子，但不是所有的都可以。特别地，由于p-拉普拉斯算子具有正的同质性，为它定义一个谱是有意义的，并且线性问题的一些谱性质可以推广到p-拉普拉斯算子。我们认为，p-拉普拉斯非线性和跳跃非线性的相互作用将提供令人着迷和丰富的解行为，并将提供一个基本框架，从中可以更好地理解更复杂的应用。

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.