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While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two problems as follow.
The theorem 6.6 of this chapter is to prove the $C^{2,\alpha}$ regularity of ...
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Suppose $\Omega$ is a bounded smooth domain in $\mathbb{R}^d$.
How does one prove that weak solutions are viscosity solutions and vice versa for the problem
$$
\begin{cases}
-\Delta u = f(x) & \text{ in } \Omega\\
u=g & \text{ on } \partial \Omega
\end{cases}
$$
under suitable assumptions (to ...
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In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in optimal control (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems, as well as second-order equations such as...
Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a counterexample.
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To fix ideas, let us recall that General Relativity describes gravitational phenomena on a 4-dimensional pseudo-Riemannian manifold $(X,g_{ab})$ with field equations that relate the energy-momentum tensor $T_{ab}\,$ of the matter distribution to the geometry of spacetime via the so called Einstei...
In theoretical physics, Lovelock's theory of gravity (often referred to as Lovelock gravity) is a generalization of Einstein's theory of general relativity introduced by David Lovelock in 1971. It is the most general metric theory of gravity yielding conserved second order equations of motion in arbitrary number of spacetime dimensions D. In this sense, Lovelock's theory is the natural generalization of Einstein's General Relativity to higher dimensions. In three and four dimensions (D = 3, 4), Lovelock's theory coincides with Einstein's theory, but in higher dimensions the theories are different...
