Everywhere differentiability of infinity harmonic functions

Abstract

We show that an infinity harmonic function, that is, a viscosity solution of the nonlinear PDE − ∞ u = −u x i u x j u x i x j = 0, is everywhere differentiable. Our new innovation is proving the uniqueness of appropriately rescaled blow-up limits around an arbitrary point. Mathematics Subject Classification (2000)Primary 49N60 · Secondary 35J20 · 35J65

On solutions to Dirichlet problems involving the infinity-Laplacian

“…It is not clear whether these solutions possess additional regularity. We direct the reader to a recent work [10] in the context of infinity-harmonic functions.…”

Optimal C1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition

“…This conjecture has been answered positively by Savin [21] in the plane. Over here, we would like to mention that although the conjecture is open, nevertheless it is well known that that solutions to ∆ ∞ v = 0 are locally of class C 1,α in the plane, for some exponent α depending only n for instance, [14] and quite recently, Evans and Smart [15] proved that infinity-harmonic functions are everywhere differentiable regardless the dimension. We remember the famous example of the infinity-harmonic function u(x, y) = x 4/3 − y 4/3 due to Aronsson from the late 1960s sets the ideal optimal regularity theory for such problem.…”

Adjoint Methods for the Infinity Laplacian Partial Differential Equation

“…We record some first bounds, uniform in ε, proved in our other paper [8]: (ii) We have u ε → u locally uniformly onŪ , (2.3) where u is the unique viscosity solution of the boundary value problem (1.1). 3) follows from standard linear elliptic PDE theory, and σ ε is smooth away from the singularity at x 0 .…”

“…The main new advances are a proof that u is everywhere differentiable and a rigorous interpretation of the infinity Laplace equation as a parabolic partial differential equation (PDE), at least generically. Our companion paper [8] provides a simpler proof of the everywhere differentiability, employing only the maximum principle. This alternative proof was inspired by the adjoint methods set forth here, which however provide much more detailed information, as we will see.…”