Skip to main content
1 of 5
liding
  • 153
  • 9

Is there a solution of a first order nonlinear PDE?

Let $a_{ij}, b_{i}, c,f$ are smooth function. Suppose $\Lambda I\geq (a^{i,j})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the following equation $$ \sum_{i,j}^{n}a^{ij}u_{i}u_{j}+\sum_{i}b_{i}u_{i}+cu=f$$

on a domain $\Omega\subset\mathbb{R}^{n}$ or a closed $n-$dimension manifold, where $u_{i}=\frac{\partial u}{x_{i}}$?

liding
  • 153
  • 9