Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence results for viscosity solutions of weakly coupled
  Hamilton-Jacobi systems   Assume $M$ is a closed, connected and smooth Riemannian manifold. We consider
the following weakly coupled system of Hamilton-Jacobi equations
\begin{equation*}
  H_i(x,Du_i(x),u_i(x),u_j(x))=0,\quad x\in M,\ i\neq j\in\{1,2\}.
\end{equation*} The classical monotone condition is not satisfied, so the
comparison principle and the Perron's method are not applicable. A new
existence result for the viscosity solutions of the above system is obtained in
this paper. The functions $H_i(x,p,u_i,u_j)$ are assumed to be continuous,
convex and superlinear in $p$, uniformly Lipschitz in $u_i$ and $u_j$. Each
$H_i(x,p,u_i,u_j)$ is either strictly decreasing or strictly decreasing in
$u_i$. For linear coupling case, a vanishing discount method is used to get the
solvability of the weakly coupled system in a certain critical case. At last,
several remaining problems are discussed.

25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg