# On a specific non-linear partial differential equation

Given an $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$ and the functions $h_i: \mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\dots,l$, we would like to find a solution of the following equation:

$$\mathbf{x} + \sum_{i=1}^{l} exp\{-h_i^2(\mathbf{x})\} \frac{\partial}{\partial\mathbf{x}}h_i(\mathbf{x}) + \mathbf{c} = \mathbf{0},$$

where

$$h_i(\mathbf{x}) = -\frac{\mathbf{x}^T\mathbf{a}_i+b-1}{\sqrt{2\mathbf{x}^T\Sigma_i\mathbf{x}}},$$

$\mathbf{c}=(c_1,\cdots,c_n)^T, \mathbf{a}_i=(a_{i,1},\cdots,a_{i,n})^T \in \mathbb{R}^n$, $b\in\mathbb{R}$, and $\Sigma_i$ is an $n\times n$ positive definite symmetric matrix, i.e., $\Sigma_i \in \mathbb{S}_{++}^{n}$.

Thus, unless I have some mistake, it holds that

$$\frac{\partial}{\partial\mathbf{x}}h_i(\mathbf{x}) = \frac{(\mathbf{x}^T\mathbf{a}_i+b-1)2\mathbf{x}^T\Sigma_i - (2\mathbf{x}^T\Sigma_i\mathbf{x})\mathbf{a}_i^T}{2\mathbf{x}^T\Sigma_i\mathbf{x}\sqrt{2\mathbf{x}^T\Sigma_i\mathbf{x}}}$$

Is that a feasible equation? Could I find a $\mathbf{x}\in\mathbb{R}^n$ that satisfies the above equation?

Thanks in advance!

• I made some necessary edits. I hope it's now more clear... – nullgeppetto Dec 1 '13 at 22:22