Functions with almost equal gradients

Problem's Description

Fix $u \in C^1(\Omega)$ with $|\nabla u| >0$ in $\Omega$. Let $\tilde{u} \in C^1(\Omega)$ be another function such that $u|_{\partial \Omega}=\tilde{u}|_{\partial \Omega}$, and $$ \nabla u \cdot J=|J||\nabla u| \ \ \hbox{and}\ \ \nabla \tilde{u} \cdot \tilde{J}=|\tilde{J}||\nabla \tilde{u}| \ \ $$ for some $J,\tilde{J}\in L^{\infty}(\Omega)$. Suppose $\parallel u-\tilde{u}\parallel_{L^1} \leq C_1 \parallel |J| -|\tilde{J}| \parallel_{L^{\infty}}$ and $\parallel J-\tilde{J} \parallel \leq C_2 \parallel |J| -|\tilde{J}| \parallel_{L^{\infty}}$, for some $C_1, C_2>0$. Is $\parallel \nabla u- \nabla\tilde{u}\parallel_{L^1} \leq C \parallel |J| -|\tilde{J}| \parallel_{L^{\infty}}$, for some $C>0$ independent of $\tilde{u}$? if not, provide a counterexample. One may assume $J, \tilde{J}$ are divergence-free, if necessary.

This problem arises in the stability of minimizers of least gradient problems arising in conductivity imaging.

1. Article Uniqueness of minimizers of weighted least gradient problems arising in hybrid inverse problems

   • Amir Moradifam,
   • Adrian Nachman,
   • Alexandru Tamasan

   Calculus of Variations and Partial Differential Equations 57 (6), 2018fulltext