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p-harmonic functions by way of intrinsic mean value properties

  • Ángel Arroyo ORCID logo EMAIL logo and José G. Llorente ORCID logo
Published/Copyright: April 30, 2021
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 16 Issue 1

Abstract

Let Ω be a bounded domain in ℝ n . Under appropriate conditions on Ω, we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to certain nonlinear mean value properties in Ω with respect to balls of variable radius. We also show that, when properly normalized, such functions converge to the p-harmonic solution of the Dirichlet problem in Ω for p ⩾ 2 . Existence is obtained via iteration, a fundamental tool being the construction of explicit universal barriers in Ω.

Keywords: Dirichlet problem; mean value properties; approximation of solutions
MSC 2010: 31B35; 31C05; 31C45; 35A35; 35B05; 35J92

Communicated by Juan Manfredi

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2017-85666-P

Funding source: Agència de Gestió d’Ajuts Universitaris i de Recerca

Award Identifier / Grant number: 2017-SGR-395

Funding statement: Partially supported by Ministerio de Economía y Competitividad grant MTM2017-85666-P and by Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017-SGR-395. Part of this work has been carried out at the Machine Learning Genoa (MaLGa) center, Universitá di Genova (IT). Á. Arroyo is supported by the UniGe starting grant “curiosity driven”.

A Auxiliary lemmas

Lemma A.1.

Let T ∈ [ 0 , π 2 ] and γ > 0 . Then

( cos ⁡ t + sin 2 ⁡ T - sin 2 ⁡ t ) γ ± ( cos ⁡ t - sin 2 ⁡ T - sin 2 ⁡ t ) γ ⩽ ( 1 + sin ⁡ T ) γ ± ( 1 - sin ⁡ T ) γ

whenever | t | ⩽ T .

Proof.

Let a ∈ [ 0 , 1 ] and define φ ± : [ a , 1 ] → ℝ by

φ ± ⁢ ( x ) = ( x + x 2 - a 2 ) γ ± ( x - x 2 - a 2 ) γ .

Direct computation shows that

φ ± ′ ⁢ ( x ) = γ x 2 - a 2 ⁢ φ ∓ ⁢ ( x ) ⩾ 0 .

Therefore, φ ± is positive and increasing in [ a , 1 ] . In particular, φ ± ⁢ ( x ) ⩽ φ ± ⁢ ( 1 ) for every x ∈ [ a , 1 ] . Then the result follows by letting a = cos ⁡ T and performing the change of variables x = cos ⁡ t . ∎

Lemma A.2.

Let γ ∈ ( 0 , 1 ) . Then

(A.1) x 2 ⁢ [ ( 1 + x ) γ - ( 1 - x ) γ ] 1 - 1 2 ⁢ [ ( 1 + x ) γ + ( 1 - x ) γ ] ⩽ 2 1 - γ

for all x ∈ ( 0 , 1 ] .

Proof.

Let us recall the Taylor series of f ⁢ ( x ) = ( 1 + x ) γ :

(A.2) ( 1 + x ) γ = 1 + ∑ k = 1 ∞ ( γ k ) ⁢ x k

for | x | ⩽ 1 , where

( γ k ) = ( - 1 ) k - 1 ⁢ γ ⁢ ( 1 - γ ) ⁢ ( 2 - γ ) ⁢ ⋯ ⁢ ( k - 1 - γ ) k !

for each k ∈ ℕ . Observe that, since γ ∈ ( 0 , 1 ) , we have that

( γ 2 ⁢ k - 1 ) > 0   and   ( γ 2 ⁢ k ) < 0    for each  ⁢ k ∈ ℕ .

We can rewrite the left-hand side in (A.1) by replacing (A.2):

x 2 ⁢ [ ( 1 + x ) γ - ( 1 - x ) γ ] 1 - 1 2 ⁢ [ ( 1 + x ) γ + ( 1 - x ) γ ] = ∑ k = 1 ∞ ( γ 2 ⁢ k - 1 ) ⁢ x 2 ⁢ k - ∑ k = 1 ∞ ( γ 2 ⁢ k ) ⁢ x 2 ⁢ k .

Hence, (A.1) follows from the fact that

∑ k = 1 ∞ [ ( γ 2 ⁢ k - 1 ) + 2 1 - γ ⁢ ( γ 2 ⁢ k ) ] ⁢ x 2 ⁢ k ⩽ 0

for every x ∈ ( 0 , 1 ) . In fact, every coefficient in the above series is nonpositive, that is,

( γ 2 ⁢ k - 1 ) + 2 1 - γ ⁢ ( γ 2 ⁢ k ) = ( γ 2 ⁢ k - 1 ) ⁢ [ 1 + 2 1 - γ ⋅ γ - 2 ⁢ k + 1 2 ⁢ k ] ⩽ 0

for every k ∈ ℕ . ∎

Acknowledgements

We wish to thank F. del Teso, J. J. Manfredi and M. Parviainen for bringing to our attention their preprint [8], which motivated part of this work. The first author also wishes to thank the Department of Mathematics and the Machine Learning Genoa (MaLGa) center of the University of Genoa for the support during the elaboration of this article.

References

[1] Á. Arroyo, J. Heino and M. Parviainen, Tug-of-war games with varying probabilities and the normalized p ⁢ ( x ) -Laplacian, Commun. Pure Appl. Anal. 16 (2017), no. 3, 915–944. 10.3934/cpaa.2017044Search in Google Scholar

[2] Á. Arroyo and J. G. Llorente, On the asymptotic mean value property for planar p-harmonic functions, Proc. Amer. Math. Soc. 144 (2016), no. 9, 3859–3868. 10.1090/proc/13026Search in Google Scholar

[3] Á. Arroyo and J. G. Llorente, On the Dirichlet problem for solutions of a restricted nonlinear mean value property, Differential Integral Equations 29 (2016), no. 1–2, 151–166. 10.57262/die/1448323257Search in Google Scholar

[4] Á. Arroyo and J. G. Llorente, A priori Hölder and Lipschitz regularity for generalized p-harmonious functions in metric measure spaces, Nonlinear Anal. 168 (2018), 32–49. 10.1016/j.na.2017.11.007Search in Google Scholar

[5] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no. 3, 271–283. 10.1109/CDC.1990.204046Search in Google Scholar

[6] C. Caratheodory, On Dirichlet’s Problem, Amer. J. Math. 59 (1937), no. 4, 709–731. 10.2307/2371339Search in Google Scholar

[7] D. DeBlassie and R. G. Smits, The p-harmonic measure of a small spherical cap, Matematiche (Catania) 71 (2016), no. 1, 149–171. Search in Google Scholar

[8] F. del Teso, J. J. Manfredi and M. Parviainen, Convergence of dynamic programming principles for the p-Laplacian, Adv. Calc. Var. (2020), 10.1515/acv-2019-0043. 10.1515/acv-2019-0043Search in Google Scholar

[9] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics Appl. Math. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. 10.1137/1.9781611972030Search in Google Scholar

[10] W. K. Hayman and P. B. Kennedy, Subharmonic Functions. Vol. I, London Math. Soc. Monogr. 9, Academic Press, London, 1976. Search in Google Scholar

[11] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover, Mineola, 2006. Search in Google Scholar

[12] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 (2001), no. 3, 699–717. 10.1137/S0036141000372179Search in Google Scholar

[13] O. D. Kellogg, Converses of Gauss’s theorem on the arithmetic mean, Trans. Amer. Math. Soc. 36 (1934), no. 2, 227–242. 10.1090/S0002-9947-1934-1501739-0Search in Google Scholar

[14] E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal. 29 (1998), no. 1, 279–292. 10.1137/S0036141095294067Search in Google Scholar

[15] H. Lebesgue, Sur le problème de Dirichlet, Comptes Rendus (Paris) 154 (1912), 335–337. 10.1007/BF03015070Search in Google Scholar

[16] P. Lindqvist, Notes on the Stationary p-Laplace Equation, Springer Briefs Math., Springer, Cham, 2019. 10.1007/978-3-030-14501-9Search in Google Scholar

[17] P. Lindqvist and J. Manfredi, On the mean value property for the p-Laplace equation in the plane, Proc. Amer. Math. Soc. 144 (2016), no. 1, 143–149. 10.1090/proc/12675Search in Google Scholar

[18] H. Luiro, M. Parviainen and E. Saksman, On the existence and uniqueness of p-harmonious functions, Differential Integral Equations 27 (2014), no. 3–4, 201–216. 10.57262/die/1391091363Search in Google Scholar

[19] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc. 138 (2010), no. 3, 881–889. 10.1090/S0002-9939-09-10183-1Search in Google Scholar

[20] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 2, 215–241. 10.2422/2036-2145.201005_003Search in Google Scholar

[21] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. 10.1007/978-1-4419-9675-6_18Search in Google Scholar

[22] Y. Peres and S. Sheffield, Tug-of-war with noise: A game-theoretic view of the p-Laplacian, Duke Math. J. 145 (2008), no. 1, 91–120. 10.1215/00127094-2008-048Search in Google Scholar

[23] V. Volterra, Alcune osservazioni sopra propietà atte ad individuare una funzione, Atti Real. Acad. Lincei Roma 18 (1909), 263–266. Search in Google Scholar
Received: 2020-10-14
Revised: 2021-03-15
Accepted: 2021-03-24
Published Online: 2021-04-30
Published in Print: 2023-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
  3. Compactness of 𝑀-uniform domains and optimal thermal insulation problems
  4. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
  5. Pansu–Wulff shapes in ℍ1
  6. Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group
  7. p-harmonic functions by way of intrinsic mean value properties
  8. The sharp quantitative isocapacitary inequality (the case of p-capacity)
  9. A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy
  10. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
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https://doi.org/10.1515/acv-2020-0101
Keywords for this article
Dirichlet problem; mean value properties; approximation of solutions

Articles in the same Issue

  1. Frontmatter
  2. The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
  3. Compactness of 𝑀-uniform domains and optimal thermal insulation problems
  4. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
  5. Pansu–Wulff shapes in ℍ1
  6. Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group
  7. p-harmonic functions by way of intrinsic mean value properties
  8. The sharp quantitative isocapacitary inequality (the case of p-capacity)
  9. A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy
  10. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
  11. Remarks on the vanishing discount problem for infinite systems of Hamilton–Jacobi–Bellman equations
  12. Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms
  13. Rectifiability of entropy defect measures in a micromagnetics model
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