On the first eigenvalue of the normalized $p$-Laplacian
HTML articles powered by AMS MathViewer
- by Graziano Crasta, Ilaria Fragalà and Bernd Kawohl
- Proc. Amer. Math. Soc. 148 (2020), 577-590
- DOI: https://doi.org/10.1090/proc/14823
- Published electronically: November 6, 2019
- PDF | Request permission
Abstract:
We prove that if $\Omega$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty )$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (nonoptimal) lower bound for the eigenvalue in terms of the measure of $\Omega$, and we address the open problem of proving a Faber–Krahn-type inequality with balls as optimal domains.References
- Amal Attouchi and Eero Ruosteenoja, Remarks on regularity for $p$-Laplacian type equations in non-divergence form, J. Differential Equations 265 (2018), no. 5, 1922–1961. MR 3800106, DOI 10.1016/j.jde.2018.04.017
- Amal Attouchi, Mikko Parviainen, and Eero Ruosteenoja, $C^{1,\alpha }$ regularity for the normalized $p$-Poisson problem, J. Math. Pures Appl. (9) 108 (2017), no. 4, 553–591 (English, with English and French summaries). MR 3698169, DOI 10.1016/j.matpur.2017.05.003
- Agnid Banerjee and Bernd Kawohl, Overdetermined problems for the normalized $p$-Laplacian, Proc. Amer. Math. Soc. Ser. B 5 (2018), 18–24. MR 3797009, DOI 10.1090/bproc/33
- M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel) 73 (1999), 276–285., DOI 10.1007/s000130050399
- H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), no. 1, 47–92. MR 1258192, DOI 10.1002/cpa.3160470105
- Isabeau Birindelli and Françoise Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differential Equations 11 (2006), no. 1, 91–119. MR 2192416
- I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations 249 (2010), 1089–1110., DOI 10.1016/j.jde.2010.03.015
- P. Blanc, A lower bound for the principal eigenvalue of fully nonlinear elliptic operators, with arXiv:1709.02455 (2017).
- K. K. Brustad, Superposition in the $p$-Laplace equation, Nonlinear Anal. 158 (2017), 23–31., DOI 10.1016/j.na.2017.04.004
- Xavier Cabré, Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: a survey, Chinese Ann. Math. Ser. B 38 (2017), no. 1, 201–214. MR 3592160, DOI 10.1007/s11401-016-1067-0
- P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control, Progress in Nonlinear Differential Equations and their Applications, vol. 58, Birkhäuser, Boston, 2004., DOI 10.1007/b138356
- F. Charro, G. De Philippis, A. Di Castro, and D. Máximo, On the Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian, Calc. Var. Partial Differential Equations 48 (2013), 667–693., DOI 10.1007/s00526-012-0567-3
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- G. Crasta and I. Fragalà, On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: regularity and geometric results, Arch. Ration. Mech. Anal. 218 (2015), 1577–1607., DOI 10.1007/s00205-015-0888-4
- Graziano Crasta and Ilaria Fragalà, A $C^1$ regularity result for the inhomogeneous normalized infinity Laplacian, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2547–2558. MR 3477071, DOI 10.1090/proc/12916
- G. Crasta and I. Fragalà, Characterization of stadium-like domains via boundary value problems for the infinity Laplacian, Nonlinear Anal. 133 (2016), 228–249., DOI 10.1016/j.na.2015.12.007
- Graziano Crasta and Ilaria Fragalà, Rigidity results for variational infinity ground states, Indiana Univ. Math. J. 68 (2019), no. 2, 353–367. MR 3951067, DOI 10.1512/iumj.2019.68.7617
- Gonzalo Dávila, Patricio Felmer, and Alexander Quaas, Harnack inequality for singular fully nonlinear operators and some existence results, Calc. Var. Partial Differential Equations 39 (2010), no. 3-4, 557–578. MR 2729313, DOI 10.1007/s00526-010-0325-3
- Kerstin Does, An evolution equation involving the normalized $p$-Laplacian, Commun. Pure Appl. Anal. 10 (2011), no. 1, 361–396. MR 2746543, DOI 10.3934/cpaa.2011.10.361
- L. Esposito, B. Kawohl, C. Nitsch, and C. Trombetti, The Neumann eigenvalue problem for the $\infty$-Laplacian, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 2, 119–134. MR 3341101, DOI 10.4171/RLM/697
- Mariano Giaquinta, Giuseppe Modica, and Jiří Souček, Cartesian currents in the calculus of variations. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 37, Springer-Verlag, Berlin, 1998. Cartesian currents. MR 1645086, DOI 10.1007/978-3-662-06218-0
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
- Petri Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations 236 (2007), no. 2, 532–550. MR 2322023, DOI 10.1016/j.jde.2007.01.020
- Petri Juutinen and Bernd Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann. 335 (2006), no. 4, 819–851. MR 2232018, DOI 10.1007/s00208-006-0766-3
- B. Kawohl, Variational versus PDE-based approaches in mathematical image processing, Singularities in PDE and the calculus of variations, CRM Proc. Lecture Notes, vol. 44, Amer. Math. Soc., Providence, RI, 2008, pp. 113–126. MR 2528737, DOI 10.1090/crmp/044/08
- Bernd Kawohl, Variations on the $p$-Laplacian, Nonlinear elliptic partial differential equations, Contemp. Math., vol. 540, Amer. Math. Soc., Providence, RI, 2011, pp. 35–46. MR 2807407, DOI 10.1090/conm/540/10657
- B. Kawohl and J. Horák, On the geometry of the $p$-Laplacian operator, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), 799–813.
- Bernd Kawohl, Stefan Krömer, and Jannis Kurtz, Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differential Integral Equations 27 (2014), no. 7-8, 659–670. MR 3200758
- Michael Kühn, Power- and log-concavity of viscosity solutions to some elliptic Dirichlet problems, Commun. Pure Appl. Anal. 17 (2018), no. 6, 2773–2788. MR 3814398, DOI 10.3934/cpaa.2018131
- G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, Geometric methods in PDE’s, 2008, pp. 207–222.
- Pedro J. Martínez-Aparicio, Mayte Pérez-Llanos, and Julio D. Rossi, The limit as $p\rightarrow \infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian, Rev. Mat. Complut. 27 (2014), no. 1, 241–258. MR 3149186, DOI 10.1007/s13163-013-0124-4
- P.J. Martínez-Aparicio, M. Pérez-Llanos, and J.D. Rossi, The sublinear problem for the 1-homogeneous $p$-Laplacian, Proc. Amer. Math. Soc. 142 (2014), 2641–2648., DOI 10.1090/S0002-9939-2014-12108-3
- Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. MR 2449057, DOI 10.1090/S0894-0347-08-00606-1
- Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J. 145 (2008), no. 1, 91–120. MR 2451291, DOI 10.1215/00127094-2008-048
- Shigeru Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 3, 403–421 (1988). MR 951227
Bibliographic Information
- Graziano Crasta
- Affiliation: Dipartimento di Matematica “G. Castelnuovo”, Univ. di Roma I Piazzale Aldo Moro 2 – 00185 Roma, Italy
- MR Author ID: 355300
- ORCID: 0000-0003-3673-6549
- Email: crasta@mat.uniroma1.it
- Ilaria Fragalà
- Affiliation: Dipartimento di Matematica, Politecnico Piazza Leonardo da Vinci, 32 –20133 Milano, Italy
- MR Author ID: 629098
- Email: ilaria.fragala@polimi.it
- Bernd Kawohl
- Affiliation: Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany
- MR Author ID: 99465
- Email: kawohl@math.uni-koeln.de
- Received by editor(s): January 16, 2019
- Published electronically: November 6, 2019
- Additional Notes: The first and second authors were supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
- Communicated by: Joachim Krieger
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 577-590
- MSC (2010): Primary 49K20, 35J60, 47J10
- DOI: https://doi.org/10.1090/proc/14823
- MathSciNet review: 4052196