On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations

We show in this paper that maximal $L^q$-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic $\gamma$-growth in the gradient holds in the full range $q>(N+2)\frac{\gamma -1}{\gamma }$. Our approach is based on new $\frac{\gamma -2}{\gamma -1}$-Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.