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的理论相对较好理解，但混合椭圆-双曲型非线性方程的理论却欠发达。然而，混合椭圆-双曲型方程在几何中有着重要的应用，在流体力学、固体力学和弹性力学的数学研究等相关领域也有着广泛的应用。因此，加深对混合型方程的理解对许多领域的进展至关重要。我们现在通过给出两个问题的例子来说明这些方程的作用，这两个问题可以简化为混合椭圆-双曲型方程的研究，每个问题都包含一些开放问题，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等距浸入R-3。这要求我们解高斯-科达齐方程，它将第二种基本形式的系数与度规的系数联系起来。对于正高斯曲率曲面，这些可以表述为椭圆边值问题，而对于负高斯曲率曲面，我们有一个双曲初值或初值边值问题。因此，在高斯曲率改变符号的曲面上，我们必须解决混合椭圆-双曲型初边值问题，其可用结果比前面的情况弱得多。高维流形或伪黎曼度量的等效问题也可以进行类似的研究，并应用于相对论等物理领域。我们也给出了一个几何以外的例子，它是由跨声速势流的研究引起的。我们考虑一个平面激波正面撞击一个有角度的楔子，并希望研究由此产生的反射-衍射模式，该模式在数学上被建模为二维黎曼问题的双曲守恒律的全局熵解。方程在远场是双曲的，在楔顶点附近是椭圆的，所以我们又得到了一个混合型的非线性偏微分方程。然后我们感兴趣的是反射-衍射图样的潜在的相当复杂的结构，以及这些图样是如何依赖于楔形角和各种物理参数的。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，无论是在微分几何的浸入问题的背景下，还是在几何、力学或其他数学领域的相关领域中。该项目属于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.