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A free boundary problem for \(\infty\)–Laplace equation

  • Juan Manfredi
  • Arshak Petrosyan
  • Henrik Shahgholian
Original article

Abstract.

We consider a free boundary problem for the p-Laplacian

\(\Delta_pu={\rm div} (\vert\nabla u\vert^{p-2}\nabla u),\)

describing nonlinear potential flow past a convex profile K with prescribed pressure \(|\nabla u(x)| =a(x)\) on the free stream line. The main purpose of this paper is to study the limit as \(p\to\infty\) of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the \(\infty\)-Laplacian

\(\Delta_\infty u=\nabla^2u\nabla u\cdot\nabla u,\)

in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case \(a(x)\equiv a_0>0\) the limit is given by the distance function.

Mathematics Subject Classification (2000): 35R35, 35J60. 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Juan Manfredi
    • 1
  • Arshak Petrosyan
    • 2
  • Henrik Shahgholian
    • 2
  1. 1.Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA (e-mail: manfredi@pitt.edu) US
  2. 2.Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden (e-mail: arshak@math.kth.se / henriks@math.kth.se) SE

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