Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients

Fabio Punzo; Alberto Tesei

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 5, page 2001-2024
  • ISSN: 0294-1449

How to cite


Punzo, Fabio, and Tesei, Alberto. "Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients." Annales de l'I.H.P. Analyse non linéaire 26.5 (2009): 2001-2024. <http://eudml.org/doc/78922>.

author = {Punzo, Fabio, Tesei, Alberto},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {degenerate second-order elliptic equations; unbounded coefficients; well-posedness; regular/singular boundary; comparison methods},
language = {eng},
number = {5},
pages = {2001-2024},
publisher = {Elsevier},
title = {Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients},
url = {http://eudml.org/doc/78922},
volume = {26},
year = {2009},

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
PB - Elsevier
VL - 26
IS - 5
SP - 2001
EP - 2024
LA - eng
KW - degenerate second-order elliptic equations; unbounded coefficients; well-posedness; regular/singular boundary; comparison methods
UR - http://eudml.org/doc/78922
ER -


  1. [1] Amano K., Maximum principles for degenerate elliptic–parabolic operators, Indiana Univ. Math. J.28 (1979) 545-557. Zbl0423.35023MR542943
  2. [2] Bardi M., Mannucci P., On the Dirichlet problem for non-totally fully nonlinear degenerate equations, Comm. Pure Appl. Anal.5 (2006) 709-731. Zbl1142.35041MR2246004
  3. [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. Zbl0806.35129MR1258192
  4. [4] Bony J.M., Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier19 (1969) 277-304. Zbl0176.09703MR262881
  5. [5] Cerrai S., Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Springer, 2001. Zbl0983.60004MR1840644
  6. [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. Zbl0755.35015MR1118699
  7. [7] Denson Hill C., A sharp maximum principle for degenerate elliptic–parabolic equations, Indiana Univ. Math. J.20 (1970) 213-229. Zbl0205.10402MR287175
  8. [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. Zbl0954.35095MR1753769
  9. [9] Fichera G., Sulle equazioni differenziali lineari ellittico–paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I5 (1956) 1-30. Zbl0075.28102MR89348
  10. [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. Zbl0122.33504MR111931
  11. [11] Freidlin M., Functional Integration and Partial Differential Equations, Princeton Univ. Press, 1985. Zbl0568.60057MR833742
  12. [12] Friedman A., Stochastic Differential Equations and Applications, I, II, Academic Press, 1976. Zbl0323.60057
  13. [13] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer, 1983. Zbl0562.35001MR737190
  14. [14] Grigoryan A., Heat kernels on weighted manifolds and applications, Contemp. Math.398 (2006) 93-191. Zbl1106.58016MR2218016
  15. [15] Guikhman I., Skorokhod A., Introduction à la Théorie des Processus Aléatoires, Éditions MIR, Moscou, 1977. Zbl0573.60003
  16. [16] Ishii H., On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac.38 (1995) 101-120. Zbl0833.35053MR1341739
  17. [17] Ishii H., Lions P.L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations83 (1990) 26-78. Zbl0708.35031MR1031377
  18. [18] Kamin S., Pozio M.A., Tesei A., Admissible conditions for parabolic equations degenerating at infinity, St. Petersburg Math. J.19 (2007) 105-121. Zbl1152.35413MR2333899
  19. [19] Keldysh M.V., On certain cases of degeneration of equations of elliptic type on the boundary of a domain, Dokl. Akad. Nauk SSSR77 (1951) 181-183, (in Russian). MR42031
  20. [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. Zbl0116.36302
  21. [21] Littman W., Generalized subharmonic functions: Monotonic approximations and an improved maximum principle, Ann. Sc. Norm. Super. Pisa17 (1963) 207-222. Zbl0123.29104MR177186
  22. [22] Littman W., A strong maximum principle for weakly L-subharmonic functions, J. Math. Mech.8 (1959) 761-770. Zbl0090.08201MR107746
  23. [23] Lorenzi L., Bertoldi M., Analytical Methods for Markov Semigroups, CRC Press, 2006. Zbl1109.35005MR2313847
  24. [24] Oleinik O.A., A problem of Fichera, Dokl. Akad. Nauk SSSR157 (1964) 1297-1300, (in Russian); English transl.:, Soviet Math. Dokl.5 (1964) 1129-1133. Zbl0142.37103MR171061
  25. [25] Oleinik O.A., Radkevic E.V., Second Order Equations with Nonnegative Characteristic Form, Amer. Math. Soc., Plenum Press, 1973. MR457908
  26. [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. Zbl44.0437.01JFM44.0437.01
  27. [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. Zbl1171.35056MR2454711
  28. [28] Pozio M.A., Tesei A., On the uniqueness of bounded solutions to singular parabolic problems, Discrete Contin. Dyn. Syst.13 (2005) 117-137. Zbl1165.35403MR2128795
  29. [29] Protter M.H., Weinberger H.F., Maximum Principles in Differential Equations, Prentice–Hall, 1967. Zbl0153.13602MR219861
  30. [30] Redheffer R.M., The sharp maximum principle for nonlinear inequalities, Indiana Univ. Math. J.21 (1971) 227-248. Zbl0235.35007MR422864
  31. [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. Zbl0344.35041MR387812
  32. [32] Taira K., Diffusion Processes and Partial Differential Equations, Academic Press, 1998. Zbl0652.35003MR954835
  33. [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. Zbl0987.35088MR1809023

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