Analysis of Divergence pp 441-462 | Cite as

# Surfaces minimizing integrals of divergent integrands

## Abstract

We consider parametric integrands which are unbounded above as a function of position. When one seeks a surface T minimizing the integral of such an integrand, subject to a boundary condition ∂T = R, the type of minimizing surface that can be found—even what is meant by “minimizing”—is dependent on various characteristics of the boundary R and the divergent behavior of the integrand. A notion of overtaking minimization is defined, and the existence of overtaking minimizers is proved under fairly weak hypotheses. As a class of examples which is hoped to be prototypical, we consider the area integrand in \({{\mathbb{R}}^{3}} \) multiplied by a function that diverges as the z-axis is approached. For such integrands, satisfying some additional technical conditions, overtaking minimizers are shown to be well-behaved near the z-axis.

## Keywords

Hyperbolic Space Principal Curvature Christoffel Symbol Integral Current Geometric Measure Theory## Preview

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