Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

Filippo Gazzola; Gianmarco Sperone; Filippo Gazzola; Gianmarco Sperone

doi:10.3934/mine.2022040

Mathematics in Engineering

2022, Volume 4, Issue 5: 1-24. doi: 10.3934/mine.2022040

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Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

Filippo Gazzola ¹,
Gianmarco Sperone ^{2
,
,}

1.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
2.
Department of Mathematical Analysis, Charles University in Prague, Sokolovská 83,186 75 Prague, Czech Republic

Received: 15 July 2021 Accepted: 30 September 2021 Published: 12 October 2021

Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in conformal dimensions, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.
- polyharmonic operators,
- Green function,
- radial symmetry,
- conformal dimensions
Citation: Filippo Gazzola, Gianmarco Sperone. Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations[J]. Mathematics in Engineering, 2022, 4(5): 1-24. doi: 10.3934/mine.2022040

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Abstract

Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in conformal dimensions, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.

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