# Existence theory for perimeter functionals with measure data under isoperimetric conditions

## Abstract

The talk is concerned with minimization problems for parametric perimeter functionals with measure data where - formally - the Euler-Lagrange equation is the geometric prescribed-mean-curvature equation with a signed measure on the right-hand side. This approach extends the classical variational theory based on Massari's functional and yields semicontinuity and existence results under the small-volume isoperimetric condition, a new and optimal assumption on the measures. The latter assumption, in fact, can be illustrated with several examples, admits a wide class of (n-1)-dimensional measures, and will be shown to be govern cancellation-compensation effects between perimeter and measure terms. Related developments for non-parametric functionals may also be discussed.

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