In the present paper we explain the classification of oscillations and its relation to the loss of derivatives for a homogeneous hyperbolic operator of second order. In this way we answer the open question if the assumptions to get well posedness for weakly hyperbolic Cauchy problems or for strictly hyperbolic Cauchy problems with non-Lipschitz coefficients are optimal.
@article{ASNSP_2004_5_3_3_589_0,
author = {Hirosawa, Fumihiko and Reissig, Michael},
title = {Non-Lipschitz coefficients for strictly hyperbolic equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {589--608},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {3},
year = {2004},
mrnumber = {2099250},
zbl = {1170.35471},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_3_589_0/}
}
TY - JOUR AU - Hirosawa, Fumihiko AU - Reissig, Michael TI - Non-Lipschitz coefficients for strictly hyperbolic equations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2004 SP - 589 EP - 608 VL - 3 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2004_5_3_3_589_0/ LA - en ID - ASNSP_2004_5_3_3_589_0 ER -
%0 Journal Article %A Hirosawa, Fumihiko %A Reissig, Michael %T Non-Lipschitz coefficients for strictly hyperbolic equations %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2004 %P 589-608 %V 3 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2004_5_3_3_589_0/ %G en %F ASNSP_2004_5_3_3_589_0
Hirosawa, Fumihiko; Reissig, Michael. Non-Lipschitz coefficients for strictly hyperbolic equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 589-608. http://www.numdam.org/item/ASNSP_2004_5_3_3_589_0/
[1] - , Operators of -evolution with non regular coefficients in the time variable, J. Differential Equations 202 (2004), 143-157. | MR | Zbl
[2] , Coefficients with unbounded derivatives in hyperbolic equations, to appear in Math. Nachr. | MR | Zbl
[3] - - , Sur les equations hyperboliques avec des coefficients qui ne dependent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), 511-559. | EuDML | Numdam | MR | Zbl
[4] - - , Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1 (2002), 327-358. | EuDML | Numdam | MR | Zbl
[5] - -, On the optimal regularity of coefficients in hyperbolic Cauchy problems, Bull. Sci. Math. 127 (2003), 328-347. | MR | Zbl
[6] - , Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657-698. | MR | Zbl
[7] , On the Cauchy problem for second order strictly hyperbolic equations with non-regular coefficients, Math. Nachr. 256 (2003), 29-47. | MR | Zbl
[8] - , Well-posedness in Sobolev spaces for second order strictly hyperbolic equations with non-differentiable oscillating coefficients, Ann. Global Anal. Geom. 25 (2004), 99-119. | MR | Zbl
[9] , “Theory of the partial differential equations", Cambridge University Press, 1973. | MR | Zbl
[10] , Hyperbolic equations with non-Lipschitz coefficients, Rend. Sem. Mat. Univ. Politec. Torino 61 (2003), 135-182. | MR | Zbl
[11] - , estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients, Math. Nachr. 214 (2000), 71-104. | MR | Zbl
[12] , “The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics, Micro-Local Approach", Akademie-Verlag, Berlin, 1997. | MR | Zbl
[13] , On the well-posedness of some singular or degenerate partial differential equations of hyperbolic type, Comm. Partial Differential Equations 15 (1990), 1029-1078. | MR | Zbl
