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Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients
1. | Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", P.le A. Moro 5, I-00185 Roma, Italy |
2. | Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro 5, I-00185 Roma |
References:
[1] |
C. Bandle, V. Moroz and W. Reichel, "Boundary blowup'' type sub-solutions to semilinear elliptic equations with Hardy potential, J. London Math. Soc., 77 (2008), 503-523.
doi: 10.1112/jlms/jdm104. |
[2] |
R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'' Springer-Verlag, Berlin, 1992. |
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[4] |
M. Bertsch, R. Dal Passo and R. Van Deer Hout, Nonuniqueness for the heat flow of harmonic maps on the disk, Arch. Rat. Mech. Anal., 161 (2002), 93-112.
doi: 10.1007/s002050100171. |
[5] |
S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'' Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin, 2001. |
[6] |
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[7] |
S. D. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asympt. Anal., 22 (2000), 349-358. |
[8] |
G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, 5 (1956), 1-30. |
[9] |
A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[10] |
A. Friedman, "Stochastic Differential Equations and Applications,'' I, II, Probability and Mathematical Statistics, 28, Academic Press, New York - London, 1976. |
[11] |
I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'' MIR Publishers, Moscow, 1980. |
[12] |
A. Grigoryan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36 (1999), 135-249.
doi: 10.1090/S0273-0979-99-00776-4. |
[13] |
A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type, Russian Math. Surveys, 17 (1962), 1-144.
doi: 10.1070/RM1962v017n03ABEH004115. |
[14] |
H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkc. Ekv., 38 (1995), 101-120. |
[15] |
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.
doi: 10.1016/0022-0396(90)90068-Z. |
[16] |
S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279-298. |
[17] |
S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 33 (1981), 831-852.
doi: 10.1002/cpa.3160340605. |
[18] |
R. Z. Khas'minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain, Th. Prob. Appl., 4 (1958), 400-419.
doi: 10.1137/1103033. |
[19] |
L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'' Pure and Applied Mathematics, 283, Chapman & Hall/CRC, Boca Raton, FL, 2007. |
[20] |
A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$, Studia Math., 128 (1998), 171-198. |
[21] |
O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'' Amer. Math. Soc., Plenum Press, New York - London, 1973. |
[22] |
M. A. Pozio, F. Punzo and A. Tesei, Criteria for well-posedness of degenerate elliptic and parabolic problems, J. Math. Pures Appl., 90 (2008), 353-386.
doi: 10.1016/j.matpur.2008.06.001. |
[23] |
M. A. Pozio and A. Tesei, On the uniqueness of bounded solutions to singular parabolic problems, Discr. Cont. Dyn. Syst., 13 (2005), 117-137.
doi: 10.3934/dcds.2005.13.117. |
[24] |
F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 26 (2009), 2001-2024. |
[25] |
F. Punzo and A. Tesei, On the refined maximum principle for degenerate elliptic and parabolic problems, Nonlinear Anal., 70 (2009), 3047-3055.
doi: 10.1016/j.na.2008.12.032. |
[26] |
F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Rend. Lincei Mat. Appl., 21 (2010), 1-38. |
[27] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, New York, 1984. |
[28] |
D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associate diffusions, Comm. Pure Appl. Math., 25 (1972), 651-713.
doi: 10.1002/cpa.3160250603. |
[29] |
K. Taira, "Diffusion Processes and Partial Differential Equations,'' Academic Press, Inc., Boston, MA, 1988. |
show all references
References:
[1] |
C. Bandle, V. Moroz and W. Reichel, "Boundary blowup'' type sub-solutions to semilinear elliptic equations with Hardy potential, J. London Math. Soc., 77 (2008), 503-523.
doi: 10.1112/jlms/jdm104. |
[2] |
R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'' Springer-Verlag, Berlin, 1992. |
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[4] |
M. Bertsch, R. Dal Passo and R. Van Deer Hout, Nonuniqueness for the heat flow of harmonic maps on the disk, Arch. Rat. Mech. Anal., 161 (2002), 93-112.
doi: 10.1007/s002050100171. |
[5] |
S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'' Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin, 2001. |
[6] |
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[7] |
S. D. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asympt. Anal., 22 (2000), 349-358. |
[8] |
G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, 5 (1956), 1-30. |
[9] |
A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[10] |
A. Friedman, "Stochastic Differential Equations and Applications,'' I, II, Probability and Mathematical Statistics, 28, Academic Press, New York - London, 1976. |
[11] |
I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'' MIR Publishers, Moscow, 1980. |
[12] |
A. Grigoryan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36 (1999), 135-249.
doi: 10.1090/S0273-0979-99-00776-4. |
[13] |
A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type, Russian Math. Surveys, 17 (1962), 1-144.
doi: 10.1070/RM1962v017n03ABEH004115. |
[14] |
H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkc. Ekv., 38 (1995), 101-120. |
[15] |
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.
doi: 10.1016/0022-0396(90)90068-Z. |
[16] |
S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279-298. |
[17] |
S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 33 (1981), 831-852.
doi: 10.1002/cpa.3160340605. |
[18] |
R. Z. Khas'minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain, Th. Prob. Appl., 4 (1958), 400-419.
doi: 10.1137/1103033. |
[19] |
L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'' Pure and Applied Mathematics, 283, Chapman & Hall/CRC, Boca Raton, FL, 2007. |
[20] |
A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$, Studia Math., 128 (1998), 171-198. |
[21] |
O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'' Amer. Math. Soc., Plenum Press, New York - London, 1973. |
[22] |
M. A. Pozio, F. Punzo and A. Tesei, Criteria for well-posedness of degenerate elliptic and parabolic problems, J. Math. Pures Appl., 90 (2008), 353-386.
doi: 10.1016/j.matpur.2008.06.001. |
[23] |
M. A. Pozio and A. Tesei, On the uniqueness of bounded solutions to singular parabolic problems, Discr. Cont. Dyn. Syst., 13 (2005), 117-137.
doi: 10.3934/dcds.2005.13.117. |
[24] |
F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 26 (2009), 2001-2024. |
[25] |
F. Punzo and A. Tesei, On the refined maximum principle for degenerate elliptic and parabolic problems, Nonlinear Anal., 70 (2009), 3047-3055.
doi: 10.1016/j.na.2008.12.032. |
[26] |
F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Rend. Lincei Mat. Appl., 21 (2010), 1-38. |
[27] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, New York, 1984. |
[28] |
D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associate diffusions, Comm. Pure Appl. Math., 25 (1972), 651-713.
doi: 10.1002/cpa.3160250603. |
[29] |
K. Taira, "Diffusion Processes and Partial Differential Equations,'' Academic Press, Inc., Boston, MA, 1988. |
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