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With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation (as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum).
In any physical theory, it is important to understand when solutions to the fundamental field equation exist, and answering this question has been the central theme of York's scientific work, culminating in the achievement, with Yvonne Choquet-Bruhat, of formulating the Einstein field equation as a well-posed system in the sense of the theory of partial differential equations.
In the mathematical study of partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions.
Further, it led to the theory of microfunctions, interest in microlocal aspects of linear partial differential equations and Fourier theory such as wave fronts, and ultimately to the current developments in D-module theory.
He also substantially contributed to the development of the theory of capacities, nonlinear potential theory, the asymptotic and qualitative theory of arbitrary order elliptic equations, the theory of ill-posed problems, the theory of boundary value problems in domains with piecewise smooth boundary.
Calabi transformed the Calabi conjecture into a non–linear partial differential equation of complex Monge–Ampere type, and showed that this equation has at most one solution, thus establishing the uniqueness of the required Kähler metric.
Eikonal equation, a non-linear partial differential equation encountered in problems of wave propagation.
for example, the work of Eric J. Heller and Emmanuel David Tannenbaum using a partial differential equation gradient descent approach.
Kaup–Kupershmidt equation, the nonlinear fifth-order partial differential equation