Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Removable singularities for degenerate elliptic equations without conditions on the growth of the solution


Author: Antonio Vitolo
Journal: Trans. Amer. Math. Soc. 370 (2018), 2679-2705
MSC (2010): Primary 35J67, 35J70, 35J60, 35D40
DOI: https://doi.org/10.1090/tran/7095
Published electronically: November 14, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of the paper is to state removable singularities results for solutions of fully nonlinear degenerate elliptic equations without any knowledge of the behaviour of the solution approaching the singular set and to obtain unconditional results of Brezis-Veron type for operators defined as the partial sum of the eigenvalues of the Hessian matrix.


References [Enhancements On Off] (What's this?)

  • [1] Luigi Ambrosio and Carlo Mantegazza, Curvature and distance function from a manifold, Dedicated to the memory of Fred Almgren, J. Geom. Anal. 8 (1998), no. 5, 723-748. MR 1731060, https://doi.org/10.1007/BF02922668
  • [2] Luigi Ambrosio and Halil Mete Soner, Level set approach to mean curvature flow in arbitrary codimension, J. Differential Geom. 43 (1996), no. 4, 693-737. MR 1412682
  • [3] M. E. Amendola, G. Galise, and A. Vitolo, Riesz capacity, maximum principle, and removable sets of fully nonlinear second-order elliptic operators, Differential Integral Equations 26 (2013), no. 7-8, 845-866. MR 3098990
  • [4] Scott N. Armstrong, Boyan Sirakov, and Charles K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math. 64 (2011), no. 6, 737-777. MR 2663711, https://doi.org/10.1002/cpa.20360
  • [5] Lipman Bers, Isolated singularities of minimal surfaces, Ann. of Math. (2) 53 (1951), 364-386. MR 0043335, https://doi.org/10.2307/1969547
  • [6] Maxime Bôcher, Singular points of functions which satisfy partial differential equations of the elliptic type, Bull. Amer. Math. Soc. 9 (1903), no. 9, 455-465. MR 1558016, https://doi.org/10.1090/S0002-9904-1903-01017-9
  • [7] B. Brandolini, F. Chiacchio, F. C. Cîrstea, and C. Trombetti, Local behaviour of singular solutions for nonlinear elliptic equations in divergence form, Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 367-393. MR 3116015, https://doi.org/10.1007/s00526-012-0554-8
  • [8] Haïm Brezis and Louis Nirenberg, Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal. 9 (1997), no. 2, 201-219. MR 1491843, https://doi.org/10.12775/TMNA.1997.009
  • [9] Haïm Brézis and Laurent Véron, Removable singularities for some nonlinear elliptic equations, Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 1-6. MR 592099, https://doi.org/10.1007/BF00284616
  • [10] Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007
  • [11] Luis Caffarelli, Yan Yan Li, and Louis Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. I, J. Fixed Point Theory Appl. 5 (2009), no. 2, 353-395. MR 2529505, https://doi.org/10.1007/s11784-009-0107-8
  • [12] Luis Caffarelli, Yan Yan Li, and Louis Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. II. Symmetry and monotonicity via moving planes, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 97-105. MR 3077250
  • [13] Luis Caffarelli, Yanyan Li, and Louis Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators, Comm. Pure Appl. Math. 66 (2013), no. 1, 109-143. MR 2994551, https://doi.org/10.1002/cpa.21412
  • [14] Italo Capuzzo Dolcetta, Fabiana Leoni, and Antonio Vitolo, Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.) 9 (2014), no. 2, 147-161. MR 3237064
  • [15] I. Capuzzo Dolcetta, F. Leoni, and A. Vitolo, On the inequality $ F(x,D^2u)\ge f(u)+g(u)\vert Du\vert^q$, Math. Ann. First Online (2015), 1-26. DOI: 10.1007/s00208-015-1280-2.
  • [16] Florica C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials, Mem. Amer. Math. Soc. 227 (2014), no. 1068, vi+85. MR 3135311
  • [17] Florica C. Cîrstea and Yihong Du, Isolated singularities for weighted quasilinear elliptic equations, J. Funct. Anal. 259 (2010), no. 1, 174-202. MR 2610383, https://doi.org/10.1016/j.jfa.2010.03.015
  • [18] Michael G. Crandall, Viscosity solutions: a primer, Viscosity solutions and applications (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997, pp. 1-43. MR 1462699, https://doi.org/10.1007/BFb0094294
  • [19] 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, https://doi.org/10.1090/S0273-0979-1992-00266-5
  • [20] Ennio De Giorgi and Guido Stampacchia, Sulle singolarità eliminabili delle ipersuperficie minimali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 38 (1965), 352-357 (Italian). MR 0187158
  • [21] Maria J. Esteban, Patricio L. Felmer, and Alexander Quaas, Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 1, 125-141. MR 2579683, https://doi.org/10.1017/S0013091507001393
  • [22] Patricio L. Felmer and Alexander Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5721-5736. MR 2529911, https://doi.org/10.1090/S0002-9947-09-04566-8
  • [23] G. Galise, S. Koike, O. Ley, and A. Vitolo, Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term, J. Math. Anal. Appl. 441 (2016), no. 1, 194-210. MR 3488054, https://doi.org/10.1016/j.jmaa.2016.03.083
  • [24] G. Galise and A. Vitolo, Viscosity solutions of uniformly elliptic equations without boundary and growth conditions at infinity, Int. J. Differ. Equ. (2011), Art. ID 453727, 18. MR 2854945
  • [25] Giulio Galise and Antonio Vitolo, Removable singularities for degenerate elliptic Pucci operators, Adv. Differential Equations 22 (2017), no. 1-2, 77-100. MR 3599512
  • [26] D. Gilbarg and James Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math. 4 (1955/56), 309-340. MR 0081416, https://doi.org/10.1007/BF02787726
  • [27] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [28] F. Reese Harvey and H. Blaine Lawson Jr., Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 (2009), no. 3, 396-443. MR 2487853, https://doi.org/10.1002/cpa.20265
  • [29] F. Reese Harvey and H. Blaine Lawson Jr., Plurisubharmonicity in a general geometric context, Geometry and analysis. No. 1, Adv. Lect. Math. (ALM), vol. 17, Int. Press, Somerville, MA, 2011, pp. 363-402. MR 2882430
  • [30] F. Reese Harvey and H. Blaine Lawson Jr., Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds, J. Differential Geom. 88 (2011), no. 3, 395-482. MR 2844439
  • [31] F. Reese Harvey and H. Blaine Lawson Jr., Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry, Surveys in differential geometry. Geometry and topology, Surv. Differ. Geom., vol. 18, Int. Press, Somerville, MA, 2013, pp. 103-156. MR 3087918, https://doi.org/10.4310/SDG.2013.v18.n1.a3
  • [32] F. Reese Harvey and H. Blaine Lawson Jr., Removable singularities for nonlinear subequations, Indiana Univ. Math. J. 63 (2014), no. 5, 1525-1552. MR 3283560, https://doi.org/10.1512/iumj.2014.63.5398
  • [33] F. Reese Harvey and H. Blaine Lawson Jr., The restriction theorem for fully nonlinear subequations, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 1, 217-265 (English, with English and French summaries). MR 3330548, https://doi.org/10.5802/aif.2846
  • [34] F. Reese Harvey and H. Blaine Lawson Jr., The Dirichlet problem with prescribed interior singularities, Adv. Math. 303 (2016), 1319-1357. MR 3552552, https://doi.org/10.1016/j.aim.2016.08.024
  • [35] W. K. Hayman and P. B. Kennedy, Subharmonic functions, Vol. 1, London Mathematical Society Monographs No 9, Academic Press, London, 1976.
  • [36] Hitoshi Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math. 42 (1989), no. 1, 15-45. MR 973743, https://doi.org/10.1002/cpa.3160420103
  • [37] Shigeaki Koike, A beginner's guide to the theory of viscosity solutions, MSJ Memoirs, vol. 13, Mathematical Society of Japan, Tokyo, 2004. MR 2084272
  • [38] Denis A. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Ration. Mech. Anal. 155 (2000), no. 3, 201-214. MR 1808368, https://doi.org/10.1007/s002050000108
  • [39] Denis A. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations 177 (2001), no. 1, 49-76. MR 1867613, https://doi.org/10.1006/jdeq.2001.3998
  • [40] Denis A. Labutin, Singularities of viscosity solutions of fully nonlinear elliptic equations, Viscosity solutions of differential equations and related topics (Japanese) (Kyoto, 2001), Sūrikaisekikenkyūsho Kōkyūroku 1287 (2002), 45-57. MR 1959709
  • [41] Denis A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J. 111 (2002), no. 1, 1-49. MR 1876440, https://doi.org/10.1215/S0012-7094-02-11111-9
  • [42] N. S. Landkof, Foundations of modern potential theory, Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. MR 0350027
  • [43] Yan Yan Li, Conformally invariant fully nonlinear elliptic equations and isolated singularities, J. Funct. Anal. 233 (2006), no. 2, 380-425. MR 2214582, https://doi.org/10.1016/j.jfa.2005.08.009
  • [44] Charles Loewner and Louis Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245-272. MR 0358078
  • [45] Juan J. Manfredi, Isolated singularities of $ p$-harmonic functions in the plane, SIAM J. Math. Anal. 22 (1991), no. 2, 424-439. MR 1084966, https://doi.org/10.1137/0522028
  • [46] S. Salsa, Lezioni su Equazioni Ellittiche, Corso di Dottorato, 2009.
  • [47] Ovidiu Savin, Changyou Wang, and Yifeng Yu, Asymptotic behavior of infinity harmonic functions near an isolated singularity, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnm163, 23. MR 2427455, https://doi.org/10.1093/imrn/rnm163
  • [48] James Serrin, Removable singularities of solutions of elliptic equations, Arch. Rational Mech. Anal. 17 (1964), 67-78. MR 0170095, https://doi.org/10.1007/BF00283867
  • [49] James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302. MR 0170096, https://doi.org/10.1007/BF02391014
  • [50] James Serrin, Removable singularities of solutions of elliptic equations. II, Arch. Rational Mech. Anal. 20 (1965), 163-169. MR 0186919, https://doi.org/10.1007/BF00276442
  • [51] James Serrin, Isolated singularities of solutions of quasi-linear equations, Acta Math. 113 (1965), 219-240. MR 0176219, https://doi.org/10.1007/BF02391778
  • [52] James Serrin, Singularities of solutions of nonlinear equations, Proc. Sympos. Appl. Math., Vol. XVII, Amer. Math. Soc., Providence, R.I., 1965, pp. 68-88. MR 0186903
  • [53] Ji-Ping Sha, $ p$-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437-447. MR 827362, https://doi.org/10.1007/BF01394417
  • [54] Ji-Ping Sha, Handlebodies and $ p$-convexity, J. Differential Geom. 25 (1987), no. 3, 353-361. MR 882828
  • [55] K. Takimoto, Isolated singularities for some types of curvature equations, J. Differential Equations 197 (2004), no. 2, 275-292. MR 2034161, https://doi.org/10.1016/j.jde.2003.10.010
  • [56] H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525-548. MR 905609, https://doi.org/10.1512/iumj.1987.36.36029

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35J67, 35J70, 35J60, 35D40

Retrieve articles in all journals with MSC (2010): 35J67, 35J70, 35J60, 35D40


Additional Information

Antonio Vitolo
Affiliation: Department of Mathematics, University of Salerno, Italy
Email: vitolo@unisa.it

DOI: https://doi.org/10.1090/tran/7095
Keywords: Elliptic equations, removable singularities
Received by editor(s): June 28, 2014
Received by editor(s) in revised form: January 4, 2016, January 18, 2016, and October 12, 2016
Published electronically: November 14, 2017
Additional Notes: The author wishes to thank GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) for partial support.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society