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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 |
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. |
show all references
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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