Surfaces minimizing integrals of divergent integrands

  • Harold R. Parks
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


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.


Hyperbolic Space Principal Curvature Christoffel Symbol Integral Current Geometric Measure Theory 
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© Birkhäuser Boston 1999

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  • Harold R. Parks

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