For the equation $\Delta_p u = 0 $ in $U$ ($U$ open and bounded), does a weak maximum principle hold? (The maximum and minimum occur on $\partial U$)? If yes, someone can indicate a book with the theorem?

Thanks in advance ( my english is horrible, sorry ... )


Yes. See Theorem 2.15 in Notes on the p-Laplace equation by Peter Lindqvist. It asserts more: the Comparison Principle holds for the p-Laplacian; the Maximum Principle amounts to comparison with a constant function.

If you wanted to prove it from scratch, you would argue that a $p$-harmonic function is the unique minimizer of $p$-energy for its boundary values. Since the truncation by $\sup_{\partial U} u$ does not increase the energy and does not change the boundary values, it has to keep the function the same.

  • $\begingroup$ from the theorem 2.15 can I obtain the weak maximum principle ? $\endgroup$ – math student Jul 23 '13 at 17:26
  • 1
    $\begingroup$ @LeandroTavares Yes, by letting $v$ be constant function. $\endgroup$ – 40 votes Jul 23 '13 at 17:29
  • $\begingroup$ Does it hold for $p=1$ ? $\endgroup$ – Hirak Mar 5 at 10:29

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