Green's functions for parabolic systems of second order in time-varying domains

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July 2014, 13(4): 1407-1433. doi: 10.3934/cpaa.2014.13.1407

Green's functions for parabolic systems of second order in time-varying domains

1.	Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912
2.	Department of Mathematics, Yonsei University, Seoul 120-749, South Korea

Received March 2013 Revised January 2013 Published February 2014

Abstract
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We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and $1/2$-Hölder continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior Hölder continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder continuity estimate. In particular, our results apply to complex perturbations of a single real equation.

Keywords: Green's function, time-varying domain., parabolic system, Green's matrix.

Mathematics Subject Classification: Primary: 35A08, 35K40; Secondary: 35B4.

Citation: 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

References:

[1]	D. G. Aronson, Bounds for the fundamental solution of a parabolic equation,, Bull. Amer. Math. Soc., 73 (1967), 890. Google Scholar
[2]	P. Auscher, Regularity theorems and heat kernel for elliptic operators,, J. London Math. Soc., 54 (1996), 284. Google Scholar
[3]	R. M. Brown, W. Hu, G. M. Lieberman, Weak solutions of parabolic equations in non-cylindrical domains,, Proc. Amer. Math. Soc., 125 (1997), 1785. Google Scholar
[4]	S. Cho, H. Dong, S. Kim, On the Green's matrices of strongly parabolic systems of second order,, Indiana Univ. Math. J., 57 (2008), 1633. Google Scholar
[5]	S. Cho, H. Dong, S. Kim, Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains,, Potential Anal., 36 (2012), 339. Google Scholar
[6]	E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels,, Amer. J. Math., 109 (1987), 319. Google Scholar
[7]	G. Dolzmann, S. Müller, Estimates for Green's matrices of elliptic systems by $L^ p$ theory,, Manuscripta Math., 88 (1995), 261. Google Scholar
[8]	H. Dong, S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains,, Trans. Amer. Math. Soc., 361 (2009), 3303. Google Scholar
[9]	E. B. Fabes, D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash,, Arch. Rational Mech. Anal., 96 (1986), 327. Google Scholar
[10]	M. Fuchs, The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition,, Manuscripta Math., 46 (1984), 97. Google Scholar
[11]	M. Fuchs, The Green matrix for strongly elliptic systems of second order with continuous coefficients,, Z. Anal. Anwendungen, 5 (1986), 507. Google Scholar
[12]	M. Grüter, K.-O. Widman, The Green function for uniformly elliptic equations,, Manuscripta Math., 37 (1982), 303. doi: 10.1007/BF01166225. Google Scholar
[13]	S. Hofmann, S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems,, Publ. Mat., 48 (2004), 481. doi: 10.5565/PUBLMAT_48204_10. Google Scholar
[14]	S. Hofmann, S. Kim, The Green function estimates for strongly elliptic systems of second order,, Manuscripta Math., 124 (2007), 139. doi: 10.1007/s00229-007-0107-1. Google Scholar
[15]	S. Hofmann, J. L. Lewis, $L^2$ solvability and representation by caloric layer potentials in time-varying domains,, Ann. of Math., 144 (1996), 349. doi: 10.2307/2118595. Google Scholar
[16]	S. Hofmann, K. Nyström, Dirichlet problems for a nonstationary linearized system of Navier-Stokes equations in non-cylindrical domains,, Methods Appl. Anal., 9 (2002), 13. Google Scholar
[17]	M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems,, Princeton University Press: Princeton, (1983). Google Scholar
[18]	S. Kim, Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients,, Trans. Amer. Math. Soc., 360 (2008), 6031. doi: 10.1090/S0002-9947-08-04485-1. Google Scholar
[19]	N. V. Krylov, Parabolic and elliptic equations with VMO coefficients,, Comm. Partial Differential Equations, 32 (2007), 453. doi: 10.1080/03605300600781626. Google Scholar
[20]	O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society: Providence, (1967). Google Scholar
[21]	J. L. Lewis, M. A. M. Murray, The method of layer potentials for the heat equation in time-varying domains,, Mem. Amer. Math. Soc., 114 (1995). doi: 10.1090/memo/0545. Google Scholar
[22]	G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). Google Scholar
[23]	W. Littman, G. Stampacchia, H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients,, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 43. Google Scholar
[24]	J. Moser, A Harnack inequality for parabolic differential equations,, Comm. Pure Appl. Math., 17 (1964), 101. Google Scholar
[25]	J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. Google Scholar
[26]	K. Nyström, The Dirichlet problem for second order parabolic operators,, Indiana Univ. Math. J., 46 (1997), 183. doi: 10.1512/iumj.1997.46.1277. Google Scholar
[27]	K. Nyström, On area integral estimates for solutions to parabolic systems in time-varying and non-smooth cylinders,, Trans. Amer. Math. Soc., 360 (2008), 2987. doi: 10.1090/S0002-9947-07-04328-0. Google Scholar
[28]	F. O. Porper, S. D. Eidel'man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them,, (Russian) Uspekhi Mat. Nauk, 39 (1984), 107. Google Scholar
[29]	J. Rivera-Noriega, Absolute continuity of parabolic measure and area integral estimates in non-cylindrical domains,, Indiana Univ. Math. J., 52 (2003), 477. doi: 10.1512/iumj.2003.52.2210. Google Scholar
[30]	M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems,, Manuscripta Math., 35 (1981), 125. doi: 10.1007/BF01168452. Google Scholar

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References:

[1]	D. G. Aronson, Bounds for the fundamental solution of a parabolic equation,, Bull. Amer. Math. Soc., 73 (1967), 890. Google Scholar
[2]	P. Auscher, Regularity theorems and heat kernel for elliptic operators,, J. London Math. Soc., 54 (1996), 284. Google Scholar
[3]	R. M. Brown, W. Hu, G. M. Lieberman, Weak solutions of parabolic equations in non-cylindrical domains,, Proc. Amer. Math. Soc., 125 (1997), 1785. Google Scholar
[4]	S. Cho, H. Dong, S. Kim, On the Green's matrices of strongly parabolic systems of second order,, Indiana Univ. Math. J., 57 (2008), 1633. Google Scholar
[5]	S. Cho, H. Dong, S. Kim, Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains,, Potential Anal., 36 (2012), 339. Google Scholar
[6]	E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels,, Amer. J. Math., 109 (1987), 319. Google Scholar
[7]	G. Dolzmann, S. Müller, Estimates for Green's matrices of elliptic systems by $L^ p$ theory,, Manuscripta Math., 88 (1995), 261. Google Scholar
[8]	H. Dong, S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains,, Trans. Amer. Math. Soc., 361 (2009), 3303. Google Scholar
[9]	E. B. Fabes, D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash,, Arch. Rational Mech. Anal., 96 (1986), 327. Google Scholar
[10]	M. Fuchs, The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition,, Manuscripta Math., 46 (1984), 97. Google Scholar
[11]	M. Fuchs, The Green matrix for strongly elliptic systems of second order with continuous coefficients,, Z. Anal. Anwendungen, 5 (1986), 507. Google Scholar
[12]	M. Grüter, K.-O. Widman, The Green function for uniformly elliptic equations,, Manuscripta Math., 37 (1982), 303. doi: 10.1007/BF01166225. Google Scholar
[13]	S. Hofmann, S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems,, Publ. Mat., 48 (2004), 481. doi: 10.5565/PUBLMAT_48204_10. Google Scholar
[14]	S. Hofmann, S. Kim, The Green function estimates for strongly elliptic systems of second order,, Manuscripta Math., 124 (2007), 139. doi: 10.1007/s00229-007-0107-1. Google Scholar
[15]	S. Hofmann, J. L. Lewis, $L^2$ solvability and representation by caloric layer potentials in time-varying domains,, Ann. of Math., 144 (1996), 349. doi: 10.2307/2118595. Google Scholar
[16]	S. Hofmann, K. Nyström, Dirichlet problems for a nonstationary linearized system of Navier-Stokes equations in non-cylindrical domains,, Methods Appl. Anal., 9 (2002), 13. Google Scholar
[17]	M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems,, Princeton University Press: Princeton, (1983). Google Scholar
[18]	S. Kim, Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients,, Trans. Amer. Math. Soc., 360 (2008), 6031. doi: 10.1090/S0002-9947-08-04485-1. Google Scholar
[19]	N. V. Krylov, Parabolic and elliptic equations with VMO coefficients,, Comm. Partial Differential Equations, 32 (2007), 453. doi: 10.1080/03605300600781626. Google Scholar
[20]	O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society: Providence, (1967). Google Scholar
[21]	J. L. Lewis, M. A. M. Murray, The method of layer potentials for the heat equation in time-varying domains,, Mem. Amer. Math. Soc., 114 (1995). doi: 10.1090/memo/0545. Google Scholar
[22]	G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). Google Scholar
[23]	W. Littman, G. Stampacchia, H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients,, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 43. Google Scholar
[24]	J. Moser, A Harnack inequality for parabolic differential equations,, Comm. Pure Appl. Math., 17 (1964), 101. Google Scholar
[25]	J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. Google Scholar
[26]	K. Nyström, The Dirichlet problem for second order parabolic operators,, Indiana Univ. Math. J., 46 (1997), 183. doi: 10.1512/iumj.1997.46.1277. Google Scholar
[27]	K. Nyström, On area integral estimates for solutions to parabolic systems in time-varying and non-smooth cylinders,, Trans. Amer. Math. Soc., 360 (2008), 2987. doi: 10.1090/S0002-9947-07-04328-0. Google Scholar
[28]	F. O. Porper, S. D. Eidel'man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them,, (Russian) Uspekhi Mat. Nauk, 39 (1984), 107. Google Scholar
[29]	J. Rivera-Noriega, Absolute continuity of parabolic measure and area integral estimates in non-cylindrical domains,, Indiana Univ. Math. J., 52 (2003), 477. doi: 10.1512/iumj.2003.52.2210. Google Scholar
[30]	M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems,, Manuscripta Math., 35 (1981), 125. doi: 10.1007/BF01168452. Google Scholar

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