Regularity Results for Eikonal-Type Equations with Nonsmooth Coefficients
This provides theoretical regularity results for PDEs with nonsmooth coefficients, which is incremental for the field of viscosity solutions.
The paper proves that solutions of certain Hamilton-Jacobi equations with Hölder continuous coefficients are locally semiconcave with a power-like modulus, establishing regularity results for eikonal-type equations with nonsmooth coefficients.
Solutions of the Hamilton-Jacobi equation $H(x,-Du(x))=1$, with $H(\cdot,p)$ Hölder continuous and $H(x,\cdot)$ convex and positively homogeneous of degree 1, are shown to be locally semiconcave with a power-like modulus. An essential step of the proof is the ${\mathcal C}^{1,α}$-regularity of the extremal trajectories associated with the multifunction generated by $D_pH$.