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Proceedings of the American Mathematical Society

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Smoothness of certain degenerate elliptic equations


Author: John L. Lewis
Journal: Proc. Amer. Math. Soc. 80 (1980), 259-265
MSC: Primary 35J70
DOI: https://doi.org/10.1090/S0002-9939-1980-0577755-6
MathSciNet review: 577755
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Abstract: Given $ p > 1,p \ne 2$, let u be a solution to $ {\text{div}}(\vert{\text{grad}}\;u{\vert^{P - 2}}{\text{grad}}\;u) = 0$ on a domain D in Euclidean two space. We prove that if u is nonconstant and real analytic in D, then the gradient of u does not vanish in D. Some examples of Krol' are used to show this result and a related result of Ural'tseva are nearly best possible.


References [Enhancements On Off] (What's this?)

  • [1] I. Krol', On the behavior of the solutions of a quasilinear equation near null salient points of the boundary, Proc. Steklov Inst. Math. 125 (1973), 130-136. MR 0344671 (49:9410)
  • [2] J. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66 (1977), 201-224. MR 0477094 (57:16638)
  • [3] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977), 219-240. MR 0474389 (57:14031)
  • [4] N. Ural'tseva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 7 (1968), 184-222. MR 0244628 (39:5942)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0577755-6
Keywords: Real analyticity, degenerate elliptic equations of divergence type
Article copyright: © Copyright 1980 American Mathematical Society

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