Generalized Regularly Discontinuous Solutions of the Einstein Equations

Regularly Varying Solutions of Generalized Thomas-fermi Equations

Some Remarks on Generalized Solutions of Discontinuous Differential Equations

In this paper we consider several concepts of generalized solutions for ordinary differential equations with a discontinuous right hand side. By means of some nontrivial examples, we show that Sentis solutions, Forward-Euler solutions and Carathéodory solutions are independent notions. AMS Subj. Classification: 34A38

Rough Solutions of the Einstein Constraint Equations

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in Hs loc with s > 3 2 . The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreover, every rough, maximal, asymptotically Euclidean solution can be approximated in an a...

Constructing Solutions of the Einstein Constraint Equations

The first step in the building of a spacetime solution of Einstein’s gravitational field equations via the initial value formulation is finding a solution of the Einstein constraint equations. We recall the conformal method for constructing solutions of the constraints and we recall what it tells us about the parameterization of the space of such solutions. One would like to know how to constru...

Viscosity solutions of discontinuous Hamilton–Jacobi equations

We define viscosity solutions for the Hamilton–Jacobi equation φt = v(x, t)H(∇φ) in RN × (0,∞) where v is positive and bounded measurable and H is non-negative and Lipschitz continuous. Under certain assumptions, we establish the existence and uniqueness of Lipschitz continuous viscosity solutions. The uniqueness result holds in particular for those v which are independent of t and piecewise co...