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)

 
 

 

Fundamental solutions and two properties of elliptic maximal and minimal operators


Authors: Patricio L. Felmer and Alexander Quaas
Journal: Trans. Amer. Math. Soc. 361 (2009), 5721-5736
MSC (2000): Primary 35J60; Secondary 35B05, 35B60
DOI: https://doi.org/10.1090/S0002-9947-09-04566-8
Published electronically: June 16, 2009
MathSciNet review: 2529911
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a large class of nonlinear second order elliptic differential operators, we define a concept of dimension, upon which we construct a fundamental solution. This allows us to prove two properties associated to these operators, which are generalizations of properties for the Laplacian and Pucci's operators. If $ {\mathcal M}$ denotes such an operator, the first property deals with the possibility of removing singularities of solutions to the equation

$\displaystyle {\mathcal M}(D^2 u)-u^p=0,$   in$\displaystyle \quad B\setminus \{0\}, $

where $ B$ is a ball in $ \mathbb{R}^N$. The second property has to do with existence or nonexistence of solutions in $ R^N$ to the inequality

$\displaystyle {\mathcal M}(D^2 u)+u^p\le 0,$   in$\displaystyle \quad \mathbb{R}^N. $

In both cases a common critical exponent defined upon the dimension number is obtained, which plays the role of $ N/(N-2)$ for the Laplacian.


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

  • 1. A. Bensoussan, J.L. Lions, Applications of variational inequalities in stochastic control. Translated from the French. Studies in Mathematics and its Applications, 12. North-Holland Publishing Co., Amsterdam-New York, 1982. MR 653144 (83e:49012)
  • 2. L. Bers. Isolated singularities of minimal surfaces. Ann. of Math. (2) 53, (1951), 364-386. MR 0043335 (13:244c)
  • 3. I. Birindelli, F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 2, 261-287. MR 2126744 (2005k:35105)
  • 4. H. Brézis, L.Véron. Removable singularities for some nonlinear elliptic equations, Arch. Rational Mech. Anal. 75, (1980/81), 1-6. MR 592099 (83i:35071)
  • 5. X. Cabré, L. Caffarelli. Fully Nonlinear Elliptic Equations, American Mathematical Society, Colloquium Publication, Vol. 43, 1995. MR 1351007 (96h:35046)
  • 6. L. Caffarelli, B. Gidas, J. Spruck. Asymptotic symmetry and local behavior of semi-linear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42, 3 (1989) 271-297. MR 982351 (90c:35075)
  • 7. W. Chen, C. Li. Classification of solutions of some nonlinear elliptic equations. Duke Math. Journal, Vol. 3, No. 3, (1991), 615-622. MR 1121147 (93e:35009)
  • 8. C. Clemons, C. Jones, A geometric proof of the Kwong-McLeod uniqueness result, SIAM J. Math. Anal. 24 (1993) 436-443. MR 1205535 (94f:35047)
  • 9. A. Cutri, F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Analyse Non Linéaire 17 (2) (2000), 219-245. MR 1753094 (2001h:35053)
  • 10. E. De Giorgi, G. Stampacchia. Sulle singolarità eliminabili delle ipersuperficie minimali. (Italian) Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 38 (1965) 352-357. MR 0187158 (32:4612)
  • 11. P. Felmer, A. Quaas. On critical exponents for the Pucci's extremal operators. Ann Inst. Henri Poincaré, Analyse Non Linéaire 20, no. 5 (2003), pp. 843-865. MR 1995504 (2004g:35083)
  • 12. P. Felmer, A. Quaas. Critical Exponents for Uniformly Elliptic Extremal Operators. Indiana Univ. Math. J. 55 (2006), no. 2, 593-629. MR 2225447 (2007e:35084)
  • 13. B. Gidas, Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations. Nonlinear partial differential equations in engineering and applied science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 1979), pp. 255-273, Lecture Notes in Pure and Appl. Math., 54, Dekker, New York, 1980. MR 577096 (83m:35059)
  • 14. B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34, (1981), 525-598. MR 615628 (83f:35045)
  • 15. D. Gilbarg, N. S. Trudinger. Elliptic partial differential equation of second order, 2nd ed., Springer-Verlag, 1983. MR 737190 (86c:35035)
  • 16. P.L. Lions, H. Ishi, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. Journal of Differential Equations 3, 26-78, (1990). MR 1031377 (90m:35015)
  • 17. R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second-order partial differential equations. Arch. Rat. Mech. Anal. 101, 1-27, (1988). MR 920674 (89a:35038)
  • 18. D. Labutin, Removable singularities for Fully Nonlinear Elliptic Equations, Arch. Rational Mech. Anal. 155 (2000) 201-214. MR 1808368 (2002a:35072)
  • 19. D. Labutin, Isolated singularities for Fully Nonlinear Elliptic Equations. Journal of Differential Equation 177 (2001), 49-76. MR 1867613 (2003a:35065)
  • 20. C. Pucci. Maximum and minimum first eigenvalues for a class of elliptic operators. Proc. Amer. Math. Soc. 17, (1966), 788-795. MR 0199576 (33:7719)
  • 21. C. Pucci. Operatori ellittici estremanti, Ann. Mat. Pure Appl. 72 (1966), 141-170. MR 0208150 (34:7960)
  • 22. J. Serrin. Removable singularities of solutions of elliptic equations. Arch. Rational Mech. Anal. 17 (1964) 67-78. MR 0170095 (30:336)
  • 23. J. Serrin. Removable singularities of solutions of elliptic equations. II. Arch. Rational Mech. Anal. 20 (1965) 163-169. MR 0186919 (32:4374)
  • 24. J. Serrin, H. Zou. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189 (2002), no. 1, 79-142. MR 1946918 (2003j:35107)
  • 25. L. Véron. Singularities of solutions of second order quasilinear equations. Pitman Research Notes in Mathematics Series, 353. Longman, Harlow, 1996. MR 1424468 (98b:35053)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J60, 35B05, 35B60

Retrieve articles in all journals with MSC (2000): 35J60, 35B05, 35B60


Additional Information

Patricio L. Felmer
Affiliation: Departamento de Ingeniería Matemática, and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Alexander Quaas
Affiliation: Departamento de Matemática, Universidad Santa María, Casilla: V-110, Avda. España 1680, Valparaíso, Chile

DOI: https://doi.org/10.1090/S0002-9947-09-04566-8
Keywords: Extremal operators, viscosity solutions, fundamental solutions, removability of singularities, Liouville-type theorems
Received by editor(s): January 4, 2006
Received by editor(s) in revised form: May 10, 2007
Published electronically: June 16, 2009
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society