Glasnik Matematicki, Vol. 52, No. 1 (2017), 107-113.

A REMARK ON GLOBAL W1,P BOUNDS FOR HARMONIC FUNCTIONS WITH LIPSCHITZ BOUNDARY VALUES

Nikos Katzourakis

Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, United Kindgom
e-mail: n.katzourakis@reading.ac.uk

Abstract.   In this note we show that gradient of harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in Lp for all finite p≥ 1.

2010 Mathematics Subject Classification.   31B05, 31B20, 31B25, 35B99.

Key words and phrases.   Harmonic functions, Dirichlet problem, Schauder estimates.


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DOI: 10.3336/gm.52.1.08


References:

  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, New York, 2000.
    MathSciNet    

  2. L. C. Evans, Partial differential equations, American Mathematical Society, Providence, 1998.
    MathSciNet    

  3. L. C. Evans and R. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, 1992.
    MathSciNet    

  4. I. Fonseca and G. Leoni, Modern methods in the calculus of variations: Lp spaces, Springer, New York, 2007.
    MathSciNet    

  5. D. Gilbarg and L. Hörmander, Intermediate Schauder estimates, Arch. Rational Mech. Anal. 74 (1980), 297-318.
    MathSciNet     CrossRef

  6. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1983.
    MathSciNet     CrossRef

  7. G. H. Hardy and J. E. Littlewood, Theorems concerning mean values of analytic or harmonic functions, Quart. J. of Math., Oxford Ser. 12 (1941), 221-256.
    MathSciNet     CrossRef

  8. G. Hile and A. Stanoyevitch, Gradient bounds for harmonic functions Lipschitz on the boundary, Appl. Anal. 73 (1999), 101-113.
    MathSciNet     CrossRef

  9. N. Katzourakis, An introduction to viscosity solutions for fully nonlinear pde with applications to calculus of variations in L, Springer, Cham, 2015.
    MathSciNet     CrossRef

  10. N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, ArXiv preprint, http://arxiv.org/pdf/1501.06164.pdf.

  11. N. Katzourakis, A new characterisation of -harmonic and p-harmonic mappings via affine variations in L, ArXiv preprint, http://arxiv.org/pdf/1509.01811.pdf.

  12. N. Katzourakis, Existence of vectorial absolute minimisers in calculus of variations in L, manuscript in preparation.

  13. O. D. Kellogg, On the derivatives of harmonic functions on the boundary, Trans. Amer. Math. Soc. 33 (1931), 486-510.
    MathSciNet     CrossRef

  14. G. M. Troianiello, Estimates of the Caccioppoli-Schauder type in weighted function spaces, Trans. Amer. Math. Soc. 334 (1992), 551-573.
    MathSciNet     CrossRef

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