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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.   Google Scholar

[2]

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

[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.  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.  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.  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.  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, (1962).   Google Scholar

[8]

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

[9]

S. Èĭdel'man, "Parabolicheskie sistemy,", Izdat., (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.  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.   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.  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.  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, (1967).   Google Scholar

[16]

E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali.,, Rend. Circ. Mat. Palermo, 24 (1907), 275.   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) \textbf{16} (1909) 3-113, 16 (1909), 3.   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.  doi: 10.1112/S0024610700001332.  Google Scholar

[20]

E. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series \textbf{31}, 31 (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.  doi: 10.1070/RM1984v039n03ABEH003164.  Google Scholar

[22]

L. Saloff-Coste, "Aspects of Sobolev-type Inequalities,", London Mathematical Society Lecture Note Series \textbf{289}, 289 (2002).   Google Scholar

[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.  Google Scholar

[24]

W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen,, Math. Z., 21 (1924), 286.  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.  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.   Google Scholar

[2]

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

[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.  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.  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.  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.  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, (1962).   Google Scholar

[8]

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

[9]

S. Èĭdel'man, "Parabolicheskie sistemy,", Izdat., (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.  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.   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.  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.  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, (1967).   Google Scholar

[16]

E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali.,, Rend. Circ. Mat. Palermo, 24 (1907), 275.   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) \textbf{16} (1909) 3-113, 16 (1909), 3.   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.  doi: 10.1112/S0024610700001332.  Google Scholar

[20]

E. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series \textbf{31}, 31 (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.  doi: 10.1070/RM1984v039n03ABEH003164.  Google Scholar

[22]

L. Saloff-Coste, "Aspects of Sobolev-type Inequalities,", London Mathematical Society Lecture Note Series \textbf{289}, 289 (2002).   Google Scholar

[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.  Google Scholar

[24]

W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen,, Math. Z., 21 (1924), 286.  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.  doi: 10.1006/jdeq.2001.4112.  Google Scholar

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