Challenges in geometric partial differential equations

我打算研究几何分析中的三个中心问题。两个来自爱因斯坦的广义相对论方程，而第三个是一个逆问题，一个非常活跃的应用数学领域。* 爱因斯坦的方程被认为提供了引力的正确数学描述。它们形成了一个非线性波动方程组，就像许多数学物理方程一样。因此，在这一领域开发的强大的数学方法具有更广泛的适用性的潜力。我想研究的两个问题涉及动态的黑洞解（即它们随时间变化）。它们都源于该学科的一个中心预测（彭罗斯在20世纪70年代创造了“建立观点”），该预测涉及此类解决方案外部区域的大时间行为。该预测断言，引力波的发射应该对基本的度规具有稳定作用，使其能够（渐近地）稳定到非辐射状态。我们被告知，这样的状态必须是静止的（即与时间无关）。问题就变成了理解方程可能的定态解。** 我提出的第一个项目是验证最后两个断言：不发射引力辐射的解确实是静止的。而且，这种定态必须与克尔解一致，这一点已经在非常严格的实分析性环境中得到了验证（霍金、卡特和罗宾逊）。我想讨论的第二个挑战是彭罗斯再次提出的，也是以他的名字命名的著名的不平等。这就提出了从下面通过被捕获表面的面积来限制时空的质量含量。这个不等式最初是作为对“建制观点”的检验而提出的，而这种观点（在整体上）完全脱离了现代数学分析的方法。* 最后一个问题是一个逆问题：挑战是纯粹通过在其边界进行测量来获得关于物体内部的信息，即不侵入内部。该领域具有非常广泛的适用性，从医学成像到地震学和地球物理学。在边界收集的数据类型根据所考虑的应用而变化。我提出了一个挑战的灵感来自断层扫描，其中的测量对应于施加一个对象（人体）的边界上的电荷和测量的边界电流。其目的是检测组织的电导率，作为内部结构的测量。这种导电性是由一个椭圆形偏微分算子，它描述了静电势的分布（未知）系数建模。* 所有上述项目都将由一个研究小组进行，该小组与这些学科的英才中心密切相关。*** *** *** *** **

I propose to study three central questions in geometric analysis. Two come from Einstein's equations of the general theory of relativity, while the third is an inverse problem,  a very active area of applied mathematics. *** Einstein's equations are thought to provide the correct mathematical description of gravity. They form a system of nonlinear wave equations, as is the case for many equations of mathematical physics. Thus, robust mathematical methods developed in this area have the potential for broader applicability. The two questions I wish to study deal with black hole solutions which are dynamical (i.e. they change in time). They both stem from a central prediction in the subject  (coined the ``establishment view'' by Penrose in the 1970s), which deals with the large-time behaviour of the exterior regions of such solutions. The prediction asserts that the emission of gravitational waves should have a stabilizing effect on the underlying metric, enabling it to settle down (asymptotically) to a non-radiating state. Such a state must be stationary (i.e. time-independent), we are told. The question then becomes to understand the possible stationary solutions of the equations. *** ***The first project I propose is to verify the last two assertions: That solutions that do not emit gravitational radiation must indeed be stationary. And that such stationary states must agree with the Kerr solutions, as has been verified (by Hawking, Carter and Robinson) in the very restrictive  setting of real-analyticity. The second challenge I wish to address was again proposed by Penrose and is the celebrated inequality that bears his name. This proposes to bound the space-time's mass content from below by the area of trapped surfaces. The inequality was originally proposed as a test of the ``establishment view'', which (in its entirety) lies completely out of the methods of modern mathematical analysis. *** The final question is an inverse problem: The challenge is to derive information on the interior of an object's interior, purely by making measurements at its boundary, i.e with no intrusion into the interior. This field has very wide applicability ranging from medical imaging to seismology and geophysics. The type of data one collects at the boundary varies according to the application one has in mind. I propose a challenge inspired by tomography, where the measurements correspond to applying a charge on the boundary of an object (a human body) and measuring the boundary electric current. The objective is to detect the conductivity of tissue, as a gauge for the interior structure. This conductivity is modelled by an (unknown) coefficient of an elliptic partial differential operator which describes the distribution of the electrostatic potential. *** All the above projects are meant to be undertaken by a team of researchers, tightly connected with centres of excellence in these subjects. *** *** *** *** **