Comparison Principles for the Finsler Infinity Laplacian with Applications to Minimal Lipschitz Extensions

This paper proves comparison principles for elliptic PDE involving the Finsler infinity Laplacian, a second-order differential operator with discontinuities in the gradient variable arising in $L^{\mathrm{\infty }}$-variational problems and tug-of-war games. The core of the paper consists in proving generalized cone comparison principles. Among other consequences, these results imply that, for any Finsler norm $\phi$ in ${\mathbb{R}}^d$, a function $u$ is a $\phi$-absolutely minimizing Lipschitz extension if and only if it is a viscosity solution of the $\phi$-infinity Laplace equation, settling a longstanding question in the $L^{\mathrm{\infty }}$-calculus of variations. The proofs combine new geometric constructions with classical notions from convex analysis.