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 denotes such an operator, the first property deals with the possibility of removing singularities of solutions to the equation








- 1. Alain Bensoussan and Jacques-Louis Lions, Applications of variational inequalities in stochastic control, Studies in Mathematics and its Applications, vol. 12, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 653144
- 2. Lipman Bers, Isolated singularities of minimal surfaces, Ann. of Math. (2) 53 (1951), 364–386. MR 0043335, https://doi.org/10.2307/1969547
- 3. Isabeau Birindelli and Françoise Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 2, 261–287 (English, with English and French summaries). MR 2126744
- 4. 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
- 5. 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
- 6. Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, https://doi.org/10.1002/cpa.3160420304
- 7. Wen Xiong Chen and Congming Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622. MR 1121147, https://doi.org/10.1215/S0012-7094-91-06325-8
- 8. C. B. Clemons and C. K. R. T. Jones, A geometric proof of the Kwong-McLeod uniqueness result, SIAM J. Math. Anal. 24 (1993), no. 2, 436–443. MR 1205535, https://doi.org/10.1137/0524027
- 9. Alessandra Cutrì and Fabiana Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 2, 219–245 (English, with English and French summaries). MR 1753094, https://doi.org/10.1016/S0294-1449(00)00109-8
- 10. 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
- 11. Patricio L. Felmer and Alexander Quaas, On critical exponents for the Pucci’s extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 5, 843–865 (English, with English and French summaries). MR 1995504, https://doi.org/10.1016/S0294-1449(03)00011-8
- 12. Patricio L. Felmer and Alexander Quaas, Critical exponents for uniformly elliptic extremal operators, Indiana Univ. Math. J. 55 (2006), no. 2, 593–629. MR 2225447, https://doi.org/10.1512/iumj.2006.55.2864
- 13. Basilis 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) Lecture Notes in Pure and Appl. Math., vol. 54, Dekker, New York, 1980, pp. 255–273. MR 577096
- 14. B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, https://doi.org/10.1002/cpa.3160340406
- 15. 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
- 16. H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1990), no. 1, 26–78. MR 1031377, https://doi.org/10.1016/0022-0396(90)90068-Z
- 17. Robert Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), no. 1, 1–27. MR 920674, https://doi.org/10.1007/BF00281780
- 18. 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
- 19. 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
- 20. Carlo Pucci, Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc. 17 (1966), 788–795. MR 0199576, https://doi.org/10.1090/S0002-9939-1966-0199576-1
- 21. Carlo Pucci, Operatori ellittici estremanti, Ann. Mat. Pura Appl. (4) 72 (1966), 141–170 (Italian, with English summary). MR 0208150, https://doi.org/10.1007/BF02414332
- 22. 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
- 23. 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
- 24. James Serrin and Henghui Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), no. 1, 79–142. MR 1946918, https://doi.org/10.1007/BF02392645
- 25. Laurent Véron, Singularities of solutions of second order quasilinear equations, Pitman Research Notes in Mathematics Series, vol. 353, Longman, Harlow, 1996. MR 1424468
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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