Alternative proof for the existence of Green's function

Previous Article
Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces
CPAA Home
This Issue
Next Article

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
Related Papers

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

[1]	Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407
[2]	Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[3]	Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations & Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61
[4]	Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[5]	Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[6]	Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335
[7]	Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92
[8]	Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001
[9]	Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745
[10]	Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[11]	Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[12]	Jiann-Sheng Jiang, Kung-Hwang Kuo, Chi-Kun Lin. Homogenization of second order equation with spatial dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 303-313. doi: 10.3934/dcds.2005.12.303
[13]	Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[14]	Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543
[15]	Florian Schneider. Second-order mixed-moment model with differentiable ansatz function in slab geometry. Kinetic & Related Models, 2018, 11 (5) : 1255-1276. doi: 10.3934/krm.2018049
[16]	Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010
[17]	P. Álvarez-Caudevilla, J. D. Evans, V. A. Galaktionov. The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 807-827. doi: 10.3934/dcds.2015.35.807
[18]	Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[19]	Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617
[20]	Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences