I have the following Dirichlet problem \begin{equation} \begin{cases} a(x)Du\cdot Du-b(x)\cdot Du=0 \ \ \ \ \text{in} \ \Omega, \\u(x) = g(x) \ \ \ \ \text{on} \ \partial\Omega. \end{cases} \end{equation} Here $\Omega$ is an open bounded subset of $\mathbb{R}^N$, $a \in C(\bar\Omega, \mathbb{R}^{N \times N})$, $b \in C(\bar\Omega, \mathbb{R}^N)$ and $g \in C(\partial\Omega)$. I make the following assumptions:
1) $a(x)\xi \cdot \xi \geq |\xi|^2$ for $x \in \Omega$ and $\xi \in \mathbb{R}^N$.
2) There is a function $\psi \in C^1(\bar\Omega)$ such that $b(x) \cdot D\psi(x) \geq 1$ for $x \in \Omega$.
Now, I define $H(p,x) = a(x)p \cdot p- b(x) \cdot p$
How can I prove (if it is possible) that $H$ satisfies \begin{equation} |H(x,p)-H(y,p)| \leq \omega(|x-y|(1+|p|)) \ \ \ \text{for} \ x,y \in \Omega \ \text{and} \ p \in \mathbb{R}^N? \end{equation} Here $\omega$ is a modulus, i.e. a function $\omega\colon[0,+\infty[\to[0,+\infty[$ continuous, nondecreasing, and such that $\omega(0) = 0$.
My attempt: \begin{align} |H(x,p)-H(y,p)|&=|a(x)p \cdot p- b(x) \cdot p - a(y)p \cdot p + b(y) \cdot p|\\ &=|(a(x)-a(y))p \cdot p - (b(x)-b(y)) \cdot p| \end{align} but I don't know how to proceed from here. Any help would be appreciated.
Thanks in advance.