Euler-Lagrange equations for variable-growth total variation
Wojciech Górny, Michał Łasica, Alexandros Matsoukas
TL;DR
The paper addresses the Euler–Lagrange framework for variable-growth total variation by identifying the $L^2$-subdifferential of $TV_\varphi$ and providing a local characterisation via a variant Anzellotti product. A Fenchel duality approach connects the subdifferential to a pair $(\xi, Dv)$ with a divergence constraint and a duality equality involving $\varphi(|Dv|)$ and $\varphi^*(x,|\xi|)$. The main results deliver both an abstract and a local characterisation, enabling the derivation of a verifiable Euler–Lagrange condition for a variable-growth ROF model in image denoising. The framework encompasses double-phase and variable-exponent growth, supported by smooth-approximation results and gradient-flow interpretation, offering a flexible regularisation paradigm with potential reductions in staircasing.
Abstract
We consider a class of integral functionals with Musielak-Orlicz type variable growth, possibly linear in some regions of the domain. This includes $p(x)$ power-type integrands with $p(x)\ge 1$ as well as double-phase $p\!-\!q$ integrands with $p=1$. The main goal of this paper is to identify the $L^2$-subdifferential of the functional, including a local characterisation in terms of a variant of the Anzellotti product defined through the Young's inequality. As an application, we obtain the Euler-Lagrange equation for the variant of the Rudin-Osher-Fatemi image denoising problem with variable growth regularising term.