Transversally Lipschitz harmonic functions are Lipschitz

Abstract

Let Ω ⊂ ℝ ⁿ be a bounded domain with C ^∞ boundary. We show that a harmonic function in Ω that is Lipschitz along a family of curves transversal to bΩ is Lipschitz in Ω. The space of Lipschitz functions which we consider is defined using the notion of a majorant which is a certain generalization of the power functions t ^α, 0 < α < 1

Keywords:

• harmonic functions
• transversally Lipschitz
• majorant

AMS Subject Classfications:

• 35B65

Acknowledgements

This work is part of my PhD dissertation 7 at The Ohio State University. I am deeply indebted to Jeffery McNeal, my thesis advisor, for his inspiration, motivation, and guidance over the years. I would like to thank Kenneth Koenig for his insightful feedback on this work. The exposition here, especially the organization of the proof of Theorem 4.3, has significantly benefited from his input. I would also like to thank the referees for several useful suggestions.

Notes

Note

1. For more on the distance to the boundary function, see Gilbarg-Trudinger 9, pp. 354–357] and Herbig-McNeal 8.