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On the regularity of domains satisfying a uniform hour–glass condition and a sharp version of the Hopf–Oleinik boundary point principle

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We prove that an open, proper, nonempty subset of \({\mathbb{R}}^n\) is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The limiting cases are as follows: Lipschitz domains may be characterized by a uniform double cone condition, and domains of class may be characterized by a uniform two-sided ball condition. We discuss a sharp generalization of the Hopf–Oleinik boundary point principle for domains satisfying an interior pseudoball condition, for semi-elliptic operators with singular drift and obtain a sharp version of the Hopf strong maximum principle for second order, nondivergence form differential operators with singular drift. Bibliography: 66 titles. Illustrations: 7 figures.

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Authors and Affiliations

  1. University of Missouri at Columbia, Columbia, MO, 65211, USA

    R. Alvarado, D. Brigham, M. Mitrea & E. Ziadé

  2. University of Liverpool, Liverpool, L69 3BX, UK

    V. Maz’ya

  3. Linköping University, Linköping, SE-581 83, Sweden

    V. Maz’ya

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  1. R. Alvarado
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  2. D. Brigham
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  3. V. Maz’ya
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  4. M. Mitrea
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Correspondence to M. Mitrea.

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Translated from Problems in Mathematical Analysis 57, May 2011, pp. 3–68.

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Alvarado, R., Brigham, D., Maz’ya, V. et al. On the regularity of domains satisfying a uniform hour–glass condition and a sharp version of the Hopf–Oleinik boundary point principle. J Math Sci 176, 281–360 (2011). https://doi.org/10.1007/s10958-011-0398-3

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