Nonlinear partial differential equations of mixed elliptic-hyperbolic type in geometry and related areas

虽然纯椭圆型或纯双曲型偏微分方程组的理论相对较好，但椭圆型和双曲型混合非线性方程的理论发展较少。然而，混合椭圆-双曲型方程在几何以及流体力学、固体力学和弹性力学的数学研究等广泛的相关领域都有重要的应用。因此，加深对混合型方程的理解对于在许多领域取得进展至关重要。我们通过给出两个问题的例子来说明这些方程的作用，这两个问题可以归结为研究椭圆-双曲型混合方程，每个例子都包含一些公开问题，并且是当前研究的主题。给定一个二维黎曼流形，微分几何中的一个重要问题是度量是否可以实现为R-3中的等距浸入。这需要我们解Gauss-Codazzi方程，它将第二基本形式的系数与度规的系数联系起来。对于正高斯曲率的曲面，这些问题可以表示为椭圆边值问题，而对于负高斯曲率的曲面，我们则有一个双曲型初边值问题。在高斯曲率变号的曲面上，我们不得不解一个椭圆-双曲型混合初边值问题，可得的结果比前面的情况弱得多。高维流形或伪黎曼度量的等价问题也可以被类似地研究，并应用于相对论等物理领域。我们还给出了一个几何以外的例子，源于跨音速势流的研究。我们考虑平面激波迎面撞击成角度的楔形，并希望研究由此产生的反射-绕射图案，其数学模型为双曲型守恒律的二维Riemann问题的整体熵解。方程在远场是双曲型的，在楔形顶点附近是椭圆型的，所以我们又有了一个混合型的非线性偏微分方程组。然后我们感兴趣的是反射-衍射图案的潜在的相当复杂的结构，以及这些图案如何依赖于楔角和各种物理参数。因此，研究的目的将是研究混合椭圆-双曲非线性偏微分方程组，无论是在微分几何的浸没问题的背景下，还是在它们产生的几何、力学或其他数学领域的相关领域中。本项目属于EPSRC数学分析研究领域。

Whilst the theory for purely elliptic or purely hyperbolic partial differential equations is relatively well understood, the theory for nonlinear equations of mixed elliptic-hyperbolic type is much less developed. However, equations of mixed elliptic-hyperbolic type have important applications in geometry, as well as a wide range of related areas such as the mathematical study of fluid mechanics, solid mechanics, and elasticity. Developing a greater understanding of equations of mixed type is therefore critical to progress in a number of fields. We illustrate the role of these equations by now giving two examples of problems which can be reduced to the study of equations of mixed elliptic-hyperbolic type, each of which contains a number of open problems and is the subject of current research.Given a two-dimensional Riemannian manifold, an important problem in differential geometry concerns whether the metric can be realised as an isometric immersion into R-3. This requires us to solve the Gauss-Codazzi equations, which relate the coefficients of the second fundamental form to those of the metric. For surfaces of positive Gauss curvature these can be formulated as an elliptic boundary value problem, whilst for surfaces of negative Gauss curvature we instead have a hyperbolic initial or initial-boundary value problem. In a surface for which the Gauss curvature changes signs, we therefore have to solve an initial-boundary value problem of mixed elliptic-hyperbolic type, for which the available results are much weaker than in the preceding cases. The equivalent problems for higher dimensional manifolds or pseudo-Riemannian metrics can also be studied similarly, and have applications to areas of physics such as relativity.We also give an example outside geometry, arising from the study of transonic potential flows. We consider a plane shock wave hitting an angled wedge head-on and wish to study the resulting reflection-diffraction pattern, modelled mathematically as a global entropy solution of the two-dimensional Riemann problem for hyperbolic conservation laws. The equations are hyperbolic in the far field and elliptic near the wedge vertex, so again we have a nonlinear PDE of mixed type. We are then interested in the potentially quite complicated structure of the reflection-diffraction patterns, and in how these patterns depend on the wedge angle as well as various physical parameters.The aim of the research will therefore be to study mixed elliptic-hyperbolic nonlinear PDEs, either in the context of the immersion problem from differential geometry or in a related area of geometry, mechanics or other areas of mathematics in which they arise.This project falls within the EPSRC Mathematical Analysis research area.