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arXiv · 2504.13559

Euler-Lagrange equations for variable-growth total variation

Wojciech Górny·Michał Łasica·Alexandros Matsoukas
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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.

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Wojciech Górny, Michał Łasica, Alexandros Matsoukas. 2025-04-18. Euler-Lagrange equations for variable-growth total variation. https://arxiv.org/abs/2504.13559

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