Determination of the modulus of continuity

Question

I'm trying to prove the uniqueness of the viscosity solution of an Hamilton-Jacobi-Bellman equation. Thanks to a classical result, I'm left to check if it exist a modulus of continuity $\omega_1$ --- i.e. a non-decreasing function such that $\omega_1(0)=0$ --- such that \begin{equation} \bigl| H(t,x,p) - H(t,y,p) \bigr| \leq \omega_1 \bigl( |x-y| (1+|p|) \bigr) \end{equation} where H denotes the hamiltonian of the optimal problem. Now, from the calculation it results that \begin{equation} \bigl| H(t,x,p) - H(t,y,p) \bigr| = \bigl| a p (x-y) - b (x^2 - y^2) + 2bc (x-y) \bigr| \end{equation} ($a,b,c, >0$), and I am not able to move forward. Anyone can help me? Thanks in advance

Answer 1

You can't get an unconditional estimate of the form $$ \bigl| a p (x-y) - b (x^2 - y^2) + 2bc (x-y) \bigr| \leq \omega_1 \bigl( |x-y| (1+|p|) \bigr) \tag{1} $$ because if $p$ is fixed and $x=y+1$, the right side of (1) is constant while the left side is an unbounded function of $y$, due to $x^2-y^2= 2y+1$.

But if you restrict your attention to $|x|,|y|\le M$ for some $M$, the estimate follows from $|x^2-y^2|\le 2M |x-y|$.