# American Institute of Mathematical Sciences

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

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.
Citation: Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure and 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), 890-896. [2] P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc., 54 (1996), 284-296. [3] R. M. Brown, W. Hu, G. M. Lieberman, Weak solutions of parabolic equations in non-cylindrical domains, Proc. Amer. Math. Soc., 125 (1997), 1785-1792. [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), 1633-1677. [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), 339-372. [6] E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-333. [7] G. Dolzmann, S. Müller, Estimates for Green's matrices of elliptic systems by $L^ p$ theory, Manuscripta Math., 88 (1995), 261-273. [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), 3303-3323. [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), 327-338. [10] M. Fuchs, The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition, Manuscripta Math., 46 (1984), 97-115. [11] M. Fuchs, The Green matrix for strongly elliptic systems of second order with continuous coefficients, Z. Anal. Anwendungen, 5 (1986), 507-531. [12] M. Grüter, K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342. doi: 10.1007/BF01166225. [13] S. Hofmann, S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat., 48 (2004), 481-496. doi: 10.5565/PUBLMAT_48204_10. [14] S. Hofmann, S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172. doi: 10.1007/s00229-007-0107-1. [15] S. Hofmann, J. L. Lewis, $L^2$ solvability and representation by caloric layer potentials in time-varying domains, Ann. of Math., 144 (1996), 349-420. doi: 10.2307/2118595. [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), 13-98. [17] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton University Press: Princeton, NJ, 1983. [18] S. Kim, Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients, Trans. Amer. Math. Soc., 360 (2008), 6031-6043. doi: 10.1090/S0002-9947-08-04485-1. [19] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations, 32 (2007), 453-475. doi: 10.1080/03605300600781626. [20] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence, RI, 1967. [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), no. 545. doi: 10.1090/memo/0545. [22] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [23] W. Littman, G. Stampacchia, H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963) 43-77. [24] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964), 101-134. [25] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. [26] K. Nyström, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J., 46 (1997), 183-245. doi: 10.1512/iumj.1997.46.1277. [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), 2987-3017. doi: 10.1090/S0002-9947-07-04328-0. [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), 107-156; English translation: Russian Math. Surveys, 39 (1984), 119-179. [29] J. Rivera-Noriega, Absolute continuity of parabolic measure and area integral estimates in non-cylindrical domains, Indiana Univ. Math. J., 52 (2003), 477-525. doi: 10.1512/iumj.2003.52.2210. [30] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscripta Math., 35 (1981), 125-145. doi: 10.1007/BF01168452.

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##### References:
 [1] D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73 (1967), 890-896. [2] P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc., 54 (1996), 284-296. [3] R. M. Brown, W. Hu, G. M. Lieberman, Weak solutions of parabolic equations in non-cylindrical domains, Proc. Amer. Math. Soc., 125 (1997), 1785-1792. [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), 1633-1677. [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), 339-372. [6] E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-333. [7] G. Dolzmann, S. Müller, Estimates for Green's matrices of elliptic systems by $L^ p$ theory, Manuscripta Math., 88 (1995), 261-273. [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), 3303-3323. [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), 327-338. [10] M. Fuchs, The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition, Manuscripta Math., 46 (1984), 97-115. [11] M. Fuchs, The Green matrix for strongly elliptic systems of second order with continuous coefficients, Z. Anal. Anwendungen, 5 (1986), 507-531. [12] M. Grüter, K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342. doi: 10.1007/BF01166225. [13] S. Hofmann, S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat., 48 (2004), 481-496. doi: 10.5565/PUBLMAT_48204_10. [14] S. Hofmann, S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172. doi: 10.1007/s00229-007-0107-1. [15] S. Hofmann, J. L. Lewis, $L^2$ solvability and representation by caloric layer potentials in time-varying domains, Ann. of Math., 144 (1996), 349-420. doi: 10.2307/2118595. [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), 13-98. [17] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton University Press: Princeton, NJ, 1983. [18] S. Kim, Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients, Trans. Amer. Math. Soc., 360 (2008), 6031-6043. doi: 10.1090/S0002-9947-08-04485-1. [19] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations, 32 (2007), 453-475. doi: 10.1080/03605300600781626. [20] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence, RI, 1967. [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), no. 545. doi: 10.1090/memo/0545. [22] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [23] W. Littman, G. Stampacchia, H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963) 43-77. [24] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964), 101-134. [25] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. [26] K. Nyström, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J., 46 (1997), 183-245. doi: 10.1512/iumj.1997.46.1277. [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), 2987-3017. doi: 10.1090/S0002-9947-07-04328-0. [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), 107-156; English translation: Russian Math. Surveys, 39 (1984), 119-179. [29] J. Rivera-Noriega, Absolute continuity of parabolic measure and area integral estimates in non-cylindrical domains, Indiana Univ. Math. J., 52 (2003), 477-525. doi: 10.1512/iumj.2003.52.2210. [30] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscripta Math., 35 (1981), 125-145. doi: 10.1007/BF01168452.
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