A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution

TL;DR

This paper introduces a class of nowhere differentiable functions with singular derivatives, providing explicit examples that challenge existing PDE theories and contribute to the calculus of variations in $L^ Infty$.

Contribution

The paper constructs explicit functions with nowhere differentiability and singular derivatives, advancing the understanding of solutions to $L^ Infty$ variational PDEs and introducing the Contact solutions framework.

Findings

Functions in $\\mathcal{K}$ are in $C^{0,\al}$ but not in any higher $C^{0,\be}$ class.

All functions in $\\mathcal{K}$ are nowhere differentiable with derivatives as singular distributions.

Constructs explicit singular solutions for Aronsson PDE and related systems.

Abstract

We present a class of functions $\mathcal{K}$ in $C^0(\R)$ which is variant of the Knopp class of nowhere differentiable functions. We derive estimates which establish $\mathcal{K} \sub C^{0,\al}(\R)$ for $0<\al<1$ but no $K \in \mathcal{K}$ is pointwise anywhere improvable to $C^{0,\be}$ for any $\be>\al$. In particular, all $K$'s are nowhere differentiable with derivatives singular distributions. $\mathcal{K}$ furnishes explicit realizations of the functional analytic result of Berezhnoi. Recently, the author and simulteously others laid the foundations of Vector-Valued Calculus of Variations in $L^\infty$ (Katzourakis), of $L^\infty$-Extremal Quasiconformal maps (Capogna and Raich, Katzourakis) and of Optimal Lipschitz Extensions of maps (Sheffield and Smart). The "Euler-Lagrange PDE" of Calculus of Variations in $L^\infty$ is the nonlinear nondivergence form Aronsson PDE with as special case the $\infty$-Laplacian. Using $\mathcal{K}$, we construct singular solutions for these PDEs. In the scalar case, we partially answered the open $C^1$ regularity problem of Viscosity Solutions to Aronsson's PDE (Katzourakis). In the vector case, the solutions can not be rigorously interpreted by existing PDE theories and justify our new theory of Contact solutions for fully nonlinear systems (Katzourakis). Validity of arguments of our new theory and failure of classical approaches both rely on the properties of $\mathcal{K}$.