Monotonicity and symmetry results for $p$-Laplace equations and applications

By Lucio Damascelli, Filomena Pacella. Advances in Differential Equations, 2000

62 citations recorded by the source corpus.

In this paper we prove monotonicity and symmetry properties of positive solutions of the equation−div(|Du|p−2Du) = f(u), 1 0, c ≥ 0, q ≥ p− 1. As a consequence we get an extension to the p–laplacian case of a symmetry theorem of Serrin for an overdetermined problem in bounded domains. Finally we apply the results obtained to the problem of finding the best constants for the classical isoperimetric inequality and for some Sobolev embeddings.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d8f240f2d8cce9a2f655872c0b938b6659acc142

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Citations in the corpus

A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative (2002, methodology): “in the case of the p- Laplacian for p �= 2) and if f is independent of x , then to prove the symmetry of the solution, the moving plane technique works only under additional geometric assumptions on the solutions (compare [8], [9], [10], [ 11 ], [12], [23]).”
Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations (2004, methodology): “1187 in [8] works as it is in our context and shows that for any direction n and l0 in the interval ðaðnÞ; l1ðnÞ there exist neighborhoods I of @O and J of l0 such that we have uðxÞpuðxlÞ for any xAOl-I ; lAJ: Of course this is true only up to l1ðnÞ (which can be strictly lower than l2ðnÞ), since th”
Overdetermined problems with possibly degenerate ellipticity, a geometric approach (2006, background): “Finally let us mention the paper by Damascelli-Pacella [13], where (2) is proved when A(t) = tp−2 and p ∈ (1, 2).”
A symmetry problem in the calculus of variations (2006, methodology): “Ω ( 1 p |Du| − u ) dx, u ∈ W 1,p 0 (Ω), 1 < p < ∞, using the analog of Serrin’s result for the p-Laplace operator (see [3, 13, 12]).”