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July  2011, 10(4): 1307-1314. doi: 10.3934/cpaa.2011.10.1307

Alternative proof for the existence of Green's function

Sungwon Cho 1,

1. 

Department of Mathematics Education, Gwangju National University of Education, 93 Pilmunlo Bugku, Gwangju 500-703, South Korea

Received  November 2009 Revised  September 2010 Published  April 2011

We present a new method for the existence of a Green's function of nod-divergence form parabolic operator with Hölder continuous coefficients. We also derive a Gaussian estimate. Main ideas involve only basic estimates and known results without a potential approach, which is used by E.E. Levi.
Keywords: second order parabolic equation, fundamental solution, Gaussian estimate., Green's function.
Mathematics Subject Classification: Primary: 35K10; Secondary: 31B1.
Citation: Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307
References:
[1]

A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble), 28 (1978), 169-213, x. See also for corrections: "Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien'' [Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 169-213; [MR 80d:31006], Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), Lecture Notes in Math., vol. 787, p. 28. (1980) Springer, Berlin.  Google Scholar

[2]

D. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.  Google Scholar

[3]

P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc., 54 (1996), 284-296. doi: 10.1112/jlms/54.2.284.  Google Scholar

[4]

P. Bauman, Equivalence of the Green's functions for diffusion operators in $R^n$: a counterexample, Proc. Amer. Math. Soc., 91 (1984), 64-68. doi: 10.1090/S0002-9939-1984-0735565-4.  Google Scholar

[5]

P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat., 22 (1984), 153-173. doi: 10.1007/BF02384378.  Google Scholar

[6]

S. Cho, Two-sided global estimates of the Green's function of parabolic equations, Potential Analysis, 25 (2006), 387-398. doi: doi:10.1007/s11118-006-9026-0.  Google Scholar

[7]

R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. II., reprint of the 1962 original, John Wiley & Sons Inc., New York, 1989.  Google Scholar

[8]

E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Univ. Press, Cambridge, UK 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[9]

S. Èĭdel'man, "Parabolicheskie sistemy," Izdat. "Nauka'', Moscow, 1964. Translation: "Parabolic systems," North-Holland Publishing Co., Amsterdam, 1964.  Google Scholar

[10]

L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form, Comm. Partial Differential Equations, 25 (2000), 821-845. doi: 10.1080/03605300008821533.  Google Scholar

[11]

E. Fabes, N. Garofalo and S. Salsa, A control on the set where a Green's function vanishes, Colloq. Math., 60/61 (1990), 637-647.  Google Scholar

[12]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, 1964.  Google Scholar

[13]

A. Il'in, A. Kalašnikov and O. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk, 17 (1962), 3-146. doi: 10.1070/RM1962v017n03ABEH004115.  Google Scholar

[14]

H. Kalf, On E. E. Levi's method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo, 41 (1992), 251-294. doi: 10.1007/BF02844669.  Google Scholar

[15]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, "Linear and Quasi-linear Equations of Parabolic Type," Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967.  Google Scholar

[16]

E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali., Rend. Circ. Mat. Palermo, 24 (1907), 275-317. Google Scholar

[17]

E. Levi, I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali., Memorie Mat. Fis. Soc. Ital. Scienze (detta dei XL) 16 (1909) 3-113 Google Scholar

[18]

G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996.  Google Scholar

[19]

V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations, J. London Math. Soc., 62 (2000), 521-543. doi: 10.1112/S0024610700001332.  Google Scholar

[20]

E. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[21]

F. Porper and S. Èĭdel'man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them, Uspekhi Math. Nauk, 39 (1984), 107-156. doi: 10.1070/RM1984v039n03ABEH003164.  Google Scholar

[22]

L. Saloff-Coste, "Aspects of Sobolev-type Inequalities," London Mathematical Society Lecture Note Series 289, Cambridge University Press, 2002.  Google Scholar

[23]

P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory, Ark. Mat., 11 (1973), 153-165. doi: 10.1007/BF02388513.  Google Scholar

[24]

W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen, Math. Z., 21 (1924), 286-311. doi: 10.1007/BF01187471.  Google Scholar

[25]

Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians, Journal of Differential Equations, 182 (2002), 416-430. doi: 10.1006/jdeq.2001.4112.  Google Scholar

show all references

References:
[1]

A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble), 28 (1978), 169-213, x. See also for corrections: "Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien'' [Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 169-213; [MR 80d:31006], Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), Lecture Notes in Math., vol. 787, p. 28. (1980) Springer, Berlin.  Google Scholar

[2]

D. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.  Google Scholar

[3]

P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc., 54 (1996), 284-296. doi: 10.1112/jlms/54.2.284.  Google Scholar

[4]

P. Bauman, Equivalence of the Green's functions for diffusion operators in $R^n$: a counterexample, Proc. Amer. Math. Soc., 91 (1984), 64-68. doi: 10.1090/S0002-9939-1984-0735565-4.  Google Scholar

[5]

P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat., 22 (1984), 153-173. doi: 10.1007/BF02384378.  Google Scholar

[6]

S. Cho, Two-sided global estimates of the Green's function of parabolic equations, Potential Analysis, 25 (2006), 387-398. doi: doi:10.1007/s11118-006-9026-0.  Google Scholar

[7]

R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. II., reprint of the 1962 original, John Wiley & Sons Inc., New York, 1989.  Google Scholar

[8]

E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Univ. Press, Cambridge, UK 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[9]

S. Èĭdel'man, "Parabolicheskie sistemy," Izdat. "Nauka'', Moscow, 1964. Translation: "Parabolic systems," North-Holland Publishing Co., Amsterdam, 1964.  Google Scholar

[10]

L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form, Comm. Partial Differential Equations, 25 (2000), 821-845. doi: 10.1080/03605300008821533.  Google Scholar

[11]

E. Fabes, N. Garofalo and S. Salsa, A control on the set where a Green's function vanishes, Colloq. Math., 60/61 (1990), 637-647.  Google Scholar

[12]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, 1964.  Google Scholar

[13]

A. Il'in, A. Kalašnikov and O. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk, 17 (1962), 3-146. doi: 10.1070/RM1962v017n03ABEH004115.  Google Scholar

[14]

H. Kalf, On E. E. Levi's method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo, 41 (1992), 251-294. doi: 10.1007/BF02844669.  Google Scholar

[15]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, "Linear and Quasi-linear Equations of Parabolic Type," Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967.  Google Scholar

[16]

E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali., Rend. Circ. Mat. Palermo, 24 (1907), 275-317. Google Scholar

[17]

E. Levi, I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali., Memorie Mat. Fis. Soc. Ital. Scienze (detta dei XL) 16 (1909) 3-113 Google Scholar

[18]

G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996.  Google Scholar

[19]

V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations, J. London Math. Soc., 62 (2000), 521-543. doi: 10.1112/S0024610700001332.  Google Scholar

[20]

E. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[21]

F. Porper and S. Èĭdel'man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them, Uspekhi Math. Nauk, 39 (1984), 107-156. doi: 10.1070/RM1984v039n03ABEH003164.  Google Scholar

[22]

L. Saloff-Coste, "Aspects of Sobolev-type Inequalities," London Mathematical Society Lecture Note Series 289, Cambridge University Press, 2002.  Google Scholar

[23]

P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory, Ark. Mat., 11 (1973), 153-165. doi: 10.1007/BF02388513.  Google Scholar

[24]

W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen, Math. Z., 21 (1924), 286-311. doi: 10.1007/BF01187471.  Google Scholar

[25]

Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians, Journal of Differential Equations, 182 (2002), 416-430. doi: 10.1006/jdeq.2001.4112.  Google Scholar

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