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Monotone Sobolev Functions in Planar Domains: Level Sets and Smooth Approximation

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Abstract

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded 1-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard’s theorem, which asserts that for a \(C^2\)-smooth function in a planar domain almost every value is a regular value. As an application, using the theory of p-harmonic functions, we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.

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Acknowledgements

The author would like to thank Tadeusz Iwaniec for a motivating discussion, Matthew Romney for his comments on the manuscript, and the anonymous referees for their useful and thoughtful comments and corrections.

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  1. Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY, 11794, USA

    Dimitrios Ntalampekos

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  1. Dimitrios Ntalampekos
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Correspondence to Dimitrios Ntalampekos.

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Communicated by A. Figalli

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The author was partially supported by NSF Grant DMS-2000096

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Ntalampekos, D. Monotone Sobolev Functions in Planar Domains: Level Sets and Smooth Approximation. Arch Rational Mech Anal 238, 1199–1230 (2020). https://doi.org/10.1007/s00205-020-01563-x

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