Isoperimetric conditions, lower semicontinuity, and existence results for perimeter functionals with measure data

• T. Schmidt

2025, Zeitschriftenaufsätze: Mathematische Annalen, Vol. 391, 4, 5729-5807, 79 S.

• DOI: 10.1007/s00208-024-03025-1

• Preprint CVGMT
• Preprint arXiv

Zusammenfassung

We establish lower semicontinuity results for perimeter functionals with measure data on Rⁿ and deduce the existence of minimizers to these functionals with Dirichlet boundary conditions, obstacles, or volume-constraints. In other words, we lay foundations of a perimeter-based variational approach to mean curvature measures on Rⁿ capable of proving existence in various prescribed-mean-curvature problems with measure data. As crucial and essentially optimal assumption on the measure data we identify a new condition, called small-volume isoperimetric condition, which sharply captures cancellation effects and comes with surprisingly many properties and reformulations in itself. In particular, we show that the small-volume isoperimetric condition is satisfied for a wide class of (n−1)-dimensional measures, which are thus admissible in our theory. Our analysis includes infinite measures and semicontinuity results on very general domains.