Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1055-1092

EXISTENCE FOR EVOLUTIONARY NEUMANN PROBLEMS WITH LINEAR GROWTH BY STABILITY RESULTS

Leah Schätzler

Universität Erlangen–Nürnberg, Department Mathematik
Cauerstrasse 11, 91058 Erlangen, Germany; schaetzler 'at' math.fau.de

Abstract. We are concerned with the Neumann type boundary value problem to parabolic systems

∂_tu – div(D_ξ f(x,Du)) = –D_ug(x,u),

where u is vector-valued, f satisfies a linear growth condition and ξ ↦ f(x,ξ) is convex. We prove that variational solutions of such systems can be approximated by variational solutions to

∂_tu – div(D_ξ f^p(x,Du)) = –D_ug(x,u),

with p > 1. This can be interpreted both as a stability and existence result for general flows with linear growth.

2010 Mathematics Subject Classification: Primary 35K87, 35B35, 49J40, 49J45.

Key words: Total variation flow, flows with linear growth, stability, solutions obtained as limits of approximations.

Reference to this article: L. Schätzler: Existence for evolutionary Neumann problems with linear growth by stability results. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1055-1092.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4461