Alternative proof for the existence of Green's function

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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

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

Abstract
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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.
[2]	D. Aronson, Non-negative solutions of linear parabolic equations,, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607.
[3]	P. Auscher, Regularity theorems and heat kernel for elliptic operators,, J. London Math. Soc., 54 (1996), 284. doi: 10.1112/jlms/54.2.284.
[4]	P. Bauman, Equivalence of the Green's functions for diffusion operators in $R^n$: a counterexample,, Proc. Amer. Math. Soc., 91 (1984), 64. doi: 10.1090/S0002-9939-1984-0735565-4.
[5]	P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints,, Ark. Mat., 22 (1984), 153. doi: 10.1007/BF02384378.
[6]	S. Cho, Two-sided global estimates of the Green's function of parabolic equations,, Potential Analysis, 25 (2006), 387. doi: doi:10.1007/s11118-006-9026-0.
[7]	R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. II.,, reprint of the 1962 original, (1962).
[8]	E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Univ. Press, (1989). doi: 10.1017/CBO9780511566158.
[9]	S. Èĭdel'man, "Parabolicheskie sistemy,", Izdat., (1964).
[10]	L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form,, Comm. Partial Differential Equations, 25 (2000), 821. doi: 10.1080/03605300008821533.
[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.
[12]	A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice Hall, (1964).
[13]	A. Il'in, A. Kalašnikov and O. Oleĭnik, Second-order linear equations of parabolic type,, Uspehi Mat. Nauk, 17 (1962), 3. doi: 10.1070/RM1962v017n03ABEH004115.
[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. doi: 10.1007/BF02844669.
[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, (1967).
[16]	E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali.,, Rend. Circ. Mat. Palermo, 24 (1907), 275.
[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) \textbf{16} (1909) 3-113, 16 (1909), 3.
[18]	G. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996).
[19]	V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations,, J. London Math. Soc., 62 (2000), 521. doi: 10.1112/S0024610700001332.
[20]	E. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series \textbf{31}, 31 (2005).
[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. doi: 10.1070/RM1984v039n03ABEH003164.
[22]	L. Saloff-Coste, "Aspects of Sobolev-type Inequalities,", London Mathematical Society Lecture Note Series \textbf{289}, 289 (2002).
[23]	P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory,, Ark. Mat., 11 (1973), 153. doi: 10.1007/BF02388513.
[24]	W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen,, Math. Z., 21 (1924), 286. doi: 10.1007/BF01187471.
[25]	Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians,, Journal of Differential Equations, 182 (2002), 416. doi: 10.1006/jdeq.2001.4112.

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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.
[2]	D. Aronson, Non-negative solutions of linear parabolic equations,, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607.
[3]	P. Auscher, Regularity theorems and heat kernel for elliptic operators,, J. London Math. Soc., 54 (1996), 284. doi: 10.1112/jlms/54.2.284.
[4]	P. Bauman, Equivalence of the Green's functions for diffusion operators in $R^n$: a counterexample,, Proc. Amer. Math. Soc., 91 (1984), 64. doi: 10.1090/S0002-9939-1984-0735565-4.
[5]	P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints,, Ark. Mat., 22 (1984), 153. doi: 10.1007/BF02384378.
[6]	S. Cho, Two-sided global estimates of the Green's function of parabolic equations,, Potential Analysis, 25 (2006), 387. doi: doi:10.1007/s11118-006-9026-0.
[7]	R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. II.,, reprint of the 1962 original, (1962).
[8]	E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Univ. Press, (1989). doi: 10.1017/CBO9780511566158.
[9]	S. Èĭdel'man, "Parabolicheskie sistemy,", Izdat., (1964).
[10]	L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form,, Comm. Partial Differential Equations, 25 (2000), 821. doi: 10.1080/03605300008821533.
[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.
[12]	A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice Hall, (1964).
[13]	A. Il'in, A. Kalašnikov and O. Oleĭnik, Second-order linear equations of parabolic type,, Uspehi Mat. Nauk, 17 (1962), 3. doi: 10.1070/RM1962v017n03ABEH004115.
[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. doi: 10.1007/BF02844669.
[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, (1967).
[16]	E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali.,, Rend. Circ. Mat. Palermo, 24 (1907), 275.
[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) \textbf{16} (1909) 3-113, 16 (1909), 3.
[18]	G. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996).
[19]	V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations,, J. London Math. Soc., 62 (2000), 521. doi: 10.1112/S0024610700001332.
[20]	E. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series \textbf{31}, 31 (2005).
[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. doi: 10.1070/RM1984v039n03ABEH003164.
[22]	L. Saloff-Coste, "Aspects of Sobolev-type Inequalities,", London Mathematical Society Lecture Note Series \textbf{289}, 289 (2002).
[23]	P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory,, Ark. Mat., 11 (1973), 153. doi: 10.1007/BF02388513.
[24]	W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen,, Math. Z., 21 (1924), 286. doi: 10.1007/BF01187471.
[25]	Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians,, Journal of Differential Equations, 182 (2002), 416. doi: 10.1006/jdeq.2001.4112.

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