Uniqueness of Solutions to Degenerate Elliptic Problems With Unbounded Coefficients

Punzo, Fabio; Tesei, Alberto

doi:10.1016/j.anihpc.2009.04.005

Punzo, Fabio ; Tesei, Alberto

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 5, pp. 2001-2024.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2009.04.005

@article{AIHPC_2009__26_5_2001_0,
     author = {Punzo, Fabio and Tesei, Alberto},
     title = {Uniqueness of {Solutions} to {Degenerate} {Elliptic} {Problems} {With} {Unbounded} {Coefficients}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {2001--2024},
     publisher = {Elsevier},
     volume = {26},
     number = {5},
     year = {2009},
     doi = {10.1016/j.anihpc.2009.04.005},
     mrnumber = {2566719},
     zbl = {1176.35087},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.04.005/}
}

TY  - JOUR
AU  - Punzo, Fabio
AU  - Tesei, Alberto
TI  - Uniqueness of Solutions to Degenerate Elliptic Problems With Unbounded Coefficients
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2009
SP  - 2001
EP  - 2024
VL  - 26
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.04.005/
DO  - 10.1016/j.anihpc.2009.04.005
LA  - en
ID  - AIHPC_2009__26_5_2001_0
ER  -

%0 Journal Article
%A Punzo, Fabio
%A Tesei, Alberto
%T Uniqueness of Solutions to Degenerate Elliptic Problems With Unbounded Coefficients
%J Annales de l'I.H.P. Analyse non linéaire
%D 2009
%P 2001-2024
%V 26
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.04.005/
%R 10.1016/j.anihpc.2009.04.005
%G en
%F AIHPC_2009__26_5_2001_0

Punzo, Fabio; Tesei, Alberto. Uniqueness of Solutions to Degenerate Elliptic Problems With Unbounded Coefficients. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 5, pp. 2001-2024. doi : 10.1016/j.anihpc.2009.04.005. http://www.numdam.org/articles/10.1016/j.anihpc.2009.04.005/

Bibliographie
Cité par

[1] Amano K., Maximum Principles for Degenerate Elliptic-Parabolic Operators, Indiana Univ. Math. J. 28 (1979) 545-557. | MR | Zbl

[2] Bardi M., Mannucci P., On the Dirichlet Problem for Non-Totally Fully Nonlinear Degenerate Equations, Comm. Pure Appl. Anal. 5 (2006) 709-731. | MR | Zbl

[3] Berestycki H., Nirenberg L., Varadhan S. R.S., The Principal Eigenvalue and the Maximum Principle for Second-Order Elliptic Operators in General Domains, Comm. Pure Appl. Math. 47 (1994) 47-92. | MR | Zbl

[4] Bony J. M., Principe Du Maximum, Inégalité De Harnack Et Unicité Du Problème De Cauchy Pour Les Opérateurs Elliptiques Dégénérés, Ann. Inst. Fourier 19 (1969) 277-304. | Numdam | MR | Zbl

[5] Cerrai S., Second Order PDE's in Finite and Infinite Dimension. a Probabilistic Approach, Springer, 2001. | MR | Zbl

[6] Crandall M. G., Ishii H., Lions P. L., User's Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR | Zbl

[7] Denson Hill C., A Sharp Maximum Principle for Degenerate Elliptic-Parabolic Equations, Indiana Univ. Math. J. 20 (1970) 213-229. | MR | Zbl

[8] Eidelman S. D., Kamin S., Porper F., Uniqueness of Solutions of the Cauchy Problem for Parabolic Equations Degenerating at Infinity, Asympt. Anal. 22 (2000) 349-358. | MR | Zbl

[9] Fichera G., Sulle Equazioni Differenziali Lineari Ellittico-Paraboliche Del Secondo Ordine, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 5 (1956) 1-30. | MR | Zbl

[10] Fichera G., On a Unified Theory of Boundary Value Problem for Elliptic-Parabolic Equations of Second Order, in: Boundary Value Problems in Differential Equations, Univ. of Wisconsin Press, 1960, pp. 97-120. | MR | Zbl

[11] Freidlin M., Functional Integration and Partial Differential Equations, Princeton Univ. Press, 1985. | MR | Zbl

[12] Friedman A., Stochastic Differential Equations and Applications, I, II, Academic Press, 1976. | Zbl

[13] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer, 1983. | MR | Zbl

[14] Grigoryan A., Heat Kernels on Weighted Manifolds and Applications, Contemp. Math. 398 (2006) 93-191. | MR | Zbl

[15] Guikhman I., Skorokhod A., Introduction À La Théorie Des Processus Aléatoires, Éditions MIR, Moscou, 1977. | Zbl

[16] Ishii H., On the Equivalence of Two Notions of Weak Solutions, Viscosity Solutions and Distribution Solutions, Funkcial. Ekvac. 38 (1995) 101-120. | MR | Zbl

[17] Ishii H., Lions P. L., Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Partial Differential Equations, J. Differential Equations 83 (1990) 26-78. | MR | Zbl

[18] Kamin S., Pozio M. A., Tesei A., Admissible Conditions for Parabolic Equations Degenerating at Infinity, St. Petersburg Math. J. 19 (2007) 105-121. | MR | Zbl

[19] Keldysh M. V., On Certain Cases of Degeneration of Equations of Elliptic Type on the Boundary of a Domain, Dokl. Akad. Nauk SSSR 77 (1951) 181-183, (in Russian). | MR

[20] Khas'Minskii R. Z., Diffusion Processes and Elliptic Differential Equations Degenerating at the Boundary of the Domain, Theory Probab. Appl. 4 (1958) 400-419. | Zbl

[21] Littman W., Generalized Subharmonic Functions: Monotonic Approximations and an Improved Maximum Principle, Ann. Sc. Norm. Super. Pisa 17 (1963) 207-222. | Numdam | MR | Zbl

[22] Littman W., A Strong Maximum Principle for Weakly L-Subharmonic Functions, J. Math. Mech. 8 (1959) 761-770. | MR | Zbl

[23] Lorenzi L., Bertoldi M., Analytical Methods for Markov Semigroups, CRC Press, 2006. | MR | Zbl

[24] Oleinik O. A., A Problem of Fichera, Dokl. Akad. Nauk SSSR 157 (1964) 1297-1300, (in Russian); English transl.:, Soviet Math. Dokl. 5 (1964) 1129-1133. | MR | Zbl

[25] Oleinik O. A., Radkevic E. V., Second Order Equations With Nonnegative Characteristic Form, Amer. Math. Soc., Plenum Press, 1973. | MR

[26] Picone M., Teoremi Di Unicità Nei Problemi Dei Valori Al Contorno Per Le Equazioni Ellittiche E Paraboliche, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. 22 (1913) 275-282. | JFM

[27] Pozio M. A., Punzo F., Tesei A., Criteria for Well-Posedness of Degenerate Elliptic and Parabolic Problems, J. Math. Pures Appl. 90 (2008) 353-386. | MR | Zbl

[28] Pozio M. A., Tesei A., On the Uniqueness of Bounded Solutions to Singular Parabolic Problems, Discrete Contin. Dyn. Syst. 13 (2005) 117-137. | MR | Zbl

[29] Protter M. H., Weinberger H. F., Maximum Principles in Differential Equations, Prentice-Hall, 1967. | MR | Zbl

[30] Redheffer R. M., The Sharp Maximum Principle for Nonlinear Inequalities, Indiana Univ. Math. J. 21 (1971) 227-248. | MR | Zbl

[31] Stroock D., Varadhan S. R.S., On Degenerate Elliptic-Parabolic Operators of Second Order and Their Associate Diffusions, Comm. Pure Appl. Math. 25 (1972) 651-713. | MR | Zbl

[32] Taira K., Diffusion Processes and Partial Differential Equations, Academic Press, 1998. | MR | Zbl

[33] Tesei A., On Uniqueness of the Positive Cauchy Problem for a Class of Parabolic Equations, in: Problemi Attuali Dell'Analisi E Della Fisica Matematica, Taormina, 1998, Aracne, 2000, pp. 145-160. | MR | Zbl

Cité par Sources :