Abstract
We consider a model case for a strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with smooth boundary $\partial\Omega\neq\emptyset$, and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.
-
@article{Berry,
author={Berry, M. V.},
TITLE={Waves and {T}hom's theorem},
JOURNAL={Adv. in Phys.},
VOLUME={25},
PAGES={1--26},
YEAR={1976},
DOI = {10.1080/00018737600101342},
} -
[bss08]
M. D. Blair, H. F. Smith, and C. D. Sogge, "On Strichartz estimates for Schrödinger operators in compact manifolds with boundary," Proc. Amer. Math. Soc., vol. 136, iss. 1, pp. 247-256, 2008.
@article {bss08, MRKEY = {2350410},
AUTHOR = {Blair, Matthew D. and Smith, Hart F. and Sogge, Christopher D.},
TITLE = {On {S}trichartz estimates for {S}chrödinger operators in compact manifolds with boundary},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {136},
YEAR = {2008},
NUMBER = {1},
PAGES = {247--256},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {35Q40 (35B65 35J10)},
MRNUMBER = {2350410},
MRREVIEWER = {Atanas G. Stefanov},
ZBLNUMBER = {1169.35012},
DOI = {10.1090/S0002-9939-07-09114-9},
} -
[blsmso08]
M. D. Blair, H. F. Smith, and C. D. Sogge, "Strichartz estimates for the wave equation on manifolds with boundary," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 26, iss. 5, pp. 1817-1829, 2009.
@article {blsmso08, MRKEY = {2566711},
AUTHOR = {Blair, Matthew D. and Smith, Hart F. and Sogge, Christopher D.},
TITLE = {Strichartz estimates for the wave equation on manifolds with boundary},
JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
VOLUME = {26},
YEAR = {2009},
NUMBER = {5},
PAGES = {1817--1829},
ISSN = {0294-1449},
MRCLASS = {35R01 (35B33 35B45 35L20 42B30)},
MRNUMBER = {2566711},
MRREVIEWER = {Vladimir Tulovsky},
ZBLNUMBER = {1198.58012},
DOI = {10.1016/j.anihpc.2008.12.004},
} -
[BLP]
N. Burq, G. Lebeau, and F. Planchon, "Global existence for energy critical waves in 3-D domains," J. Amer. Math. Soc., vol. 21, iss. 3, pp. 831-845, 2008.
@article {BLP, MRKEY = {2393429},
AUTHOR = {Burq, Nicolas and Lebeau, Gilles and Planchon, Fabrice},
TITLE = {Global existence for energy critical waves in 3-{D} domains},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {21},
YEAR = {2008},
NUMBER = {3},
PAGES = {831--845},
ISSN = {0894-0347},
MRCLASS = {35L70 (35L20)},
MRNUMBER = {2393429},
MRREVIEWER = {Enzo Vitillaro},
ZBLNUMBER = {1204.35119},
DOI = {10.1090/S0894-0347-08-00596-1},
} -
[esk77]
G. Eskin, "Parametrix and propagation of singularities for the interior mixed hyperbolic problem," J. Analyse Math., vol. 32, pp. 17-62, 1977.
@article {esk77, MRKEY = {0477491},
AUTHOR = {Eskin, Gregory},
TITLE = {Parametrix and propagation of singularities for the interior mixed hyperbolic problem},
JOURNAL = {J. Analyse Math.},
FJOURNAL = {Journal d'Analyse Mathématique},
VOLUME = {32},
YEAR = {1977},
PAGES = {17--62},
ISSN = {0021-7670},
CODEN = {JOAMAV},
MRCLASS = {35L35},
MRNUMBER = {0477491},
MRREVIEWER = {D. Ciumasu},
ZBLNUMBER = {0375.35037},
DOI = {10.1007/BF02803574},
} -
[Hormander] L. Hörmander, "The analysis of linear partial differential operators. I," in Distribution Theory and Fourier Analysis, New York: Springer-Verlag, 2003, p. x.
@incollection{Hormander, MRKEY = {1996773},
AUTHOR = {H{ö}rmander, Lars},
TITLE = {The analysis of linear partial differential operators. {I}},
SERIES = {Classics in Math.},
BOOKTITLE = {Distribution Theory and Fourier Analysis},
NOTE={Reprint of the second (1990) edition [Springer, New York; \mr{1065993},
\zbl{0712.35001}]},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2003},
PAGES = {x+440},
ISBN = {3-540-00662-1},
MRCLASS = {35-02},
MRNUMBER = {1996773},
ZBLNUMBER = {1028.35001},
} -
[doi]
O. Ivanovici, "Counterexamples to Strichartz estimates for the wave equation in domains," Math. Ann., vol. 347, iss. 3, pp. 627-673, 2010.
@article {doi, MRKEY = {2640046},
AUTHOR = {Ivanovici, Oana},
TITLE = {Counterexamples to {S}trichartz estimates for the wave equation in domains},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {347},
YEAR = {2010},
NUMBER = {3},
PAGES = {627--673},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {35L10},
MRNUMBER = {2640046},
MRREVIEWER = {Elena Cordero},
ZBLNUMBER = {1201.35060},
DOI = {10.1007/s00208-009-0454-1},
} -
[doi2]
O. Ivanovici, "Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary," J. Eur. Math. Soc. $($JEMS$)$, vol. 14, iss. 5, pp. 1357-1388, 2012.
@article {doi2, MRKEY = {2966654},
AUTHOR = {Ivanovici, Oana},
TITLE = {Counterexamples to the {S}trichartz inequalities for the wave equation in general domains with boundary},
JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
VOLUME = {14},
YEAR = {2012},
NUMBER = {5},
PAGES = {1357--1388},
ISSN = {1435-9855},
MRCLASS = {35R01 (35B45 35L20 58J45)},
MRNUMBER = {2966654},
MRREVIEWER = {Atanas G. Stefanov},
ZBLNUMBER = {1254.35035},
DOI = {10.4171/JEMS/335},
} -
[ip08] O. Ivanovici and F. Planchon, Square function and heat flow estimates on domains, 2008.
@misc{ip08,
author={Ivanovici, Oana and Planchon, Fabrice},
TITLE = {Square function and heat flow estimates on domains},
ARXIV={0812.2733},
YEAR={2008},
} -
[ip12]
O. Ivanovici and F. Planchon, "On the energy critical Schrödinger equation in $3D$ non-trapping domains," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 27, iss. 5, pp. 1153-1177, 2010.
@article {ip12, MRKEY = {2683754},
AUTHOR = {Ivanovici, Oana and Planchon, Fabrice},
TITLE = {On the energy critical {S}chrödinger equation in {$3D$} non-trapping domains},
JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
VOLUME = {27},
YEAR = {2010},
NUMBER = {5},
PAGES = {1153--1177},
ISSN = {0294-1449},
MRCLASS = {35Q55},
MRNUMBER = {2683754},
MRREVIEWER = {Olivier J. Goubet},
ZBLNUMBER = {1200.35066},
DOI = {10.1016/j.anihpc.2010.04.001},
} -
@misc{gle06,
author={Lebeau, G.},
TITLE={Estimation de dispersion pour les ondes dans un convexe},
NOTE={in \emph{Journées} ``\emph{Équations aux Dérivées Partielles}'' (Evian, 2006)},
YEAR={2006},
URL={http://www.numdam.org/numdam-bin/fitem?id=JEDP\_2006\_\_\_\_A7\_0},
} -
[mesj78]
R. B. Melrose and J. Sjöstrand, "Singularities of boundary value problems. I," Comm. Pure Appl. Math., vol. 31, iss. 5, pp. 593-617, 1978.
@article {mesj78, MRKEY = {0492794},
AUTHOR = {Melrose, R. B. and Sj{ö}strand, J.},
TITLE = {Singularities of boundary value problems. {I}},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {31},
YEAR = {1978},
NUMBER = {5},
PAGES = {593--617},
ISSN = {0010-3640},
MRCLASS = {35G15 (58G15)},
MRNUMBER = {0492794},
MRREVIEWER = {Makhlouf Derridj},
ZBLNUMBER = {0368.35020},
DOI = {10.1002/cpa.3160310504},
} -
[meta85]
R. B. Melrose and M. E. Taylor, "The radiation pattern of a diffracted wave near the shadow boundary," Comm. Partial Differential Equations, vol. 11, iss. 6, pp. 599-672, 1986.
@article {meta85, MRKEY = {0837278},
AUTHOR = {Melrose, Richard B. and Taylor, Michael E.},
TITLE = {The radiation pattern of a diffracted wave near the shadow boundary},
JOURNAL = {Comm. Partial Differential Equations},
FJOURNAL = {Communications in Partial Differential Equations},
VOLUME = {11},
YEAR = {1986},
NUMBER = {6},
PAGES = {599--672},
ISSN = {0360-5302},
CODEN = {CPDIDZ},
MRCLASS = {35L05 (35A27 35L67)},
MRNUMBER = {0837278},
MRREVIEWER = {Alejandro Uribe},
ZBLNUMBER = {0632.35056},
DOI = {10.1080/03605308608820439},
} -
[Olver] F. W. J. Olver, Asymptotics and Special Functions, New York: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], 1974.
@book {Olver, MRKEY = {0435697},
AUTHOR = {Olver, F. W. J.},
TITLE = {Asymptotics and Special Functions},
SERIES = {Computer Sci. Applied Math.},
PUBLISHER = {Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers]},
ADDRESS={New York},
YEAR = {1974},
PAGES = {xvi+572},
MRCLASS = {41A60},
MRNUMBER = {0435697},
MRREVIEWER = {Norman Bleistein},
ZBLNUMBER = {0303.41035},
} -
[Shatah-Struwe-IMRN]
J. Shatah and M. Struwe, "Well-posedness in the energy space for semilinear wave equations with critical growth," Internat. Math. Res. Notices, vol. 1994, p. 7.
@article {Shatah-Struwe-IMRN, MRKEY = {1283026},
AUTHOR = {Shatah, Jalal and Struwe, Michael},
TITLE = {Well-posedness in the energy space for semilinear wave equations with critical growth},
JOURNAL = {Internat. Math. Res. Notices},
FJOURNAL = {International Mathematics Research Notices},
VOLUME = {1994},
SORTYEAR={1994},
PAGES = {no.~7, 303ff., approx. 7 pp.},
ISSN = {1073-7928},
MRCLASS = {35L70 (35D05)},
MRNUMBER = {1283026},
MRREVIEWER = {Yun Mei Chen},
ZBLNUMBER = {0830.35086},
DOI = {10.1155/S1073792894000346},
} -
[sm98]
H. F. Smith, "A parametrix construction for wave equations with $C^{1,1}$ coefficients," Ann. Inst. Fourier $($Grenoble$)$, vol. 48, iss. 3, pp. 797-835, 1998.
@article {sm98, MRKEY = {1644105},
AUTHOR = {Smith, Hart F.},
TITLE = {A parametrix construction for wave equations with {$C\sp {1,1}$} coefficients},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {48},
YEAR = {1998},
NUMBER = {3},
PAGES = {797--835},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {35L15 (35S30)},
MRNUMBER = {1644105},
MRREVIEWER = {Peter R. Popivanov},
ZBLNUMBER = {0974.35068},
DOI = {10.5802/aif.1640},
} -
[smso95]
H. F. Smith and C. D. Sogge, "On the critical semilinear wave equation outside convex obstacles," J. Amer. Math. Soc., vol. 8, iss. 4, pp. 879-916, 1995.
@article {smso95, MRKEY = {1308407},
AUTHOR = {Smith, Hart F. and Sogge, Christopher D.},
TITLE = {On the critical semilinear wave equation outside convex obstacles},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {8},
YEAR = {1995},
NUMBER = {4},
PAGES = {879--916},
ISSN = {0894-0347},
MRCLASS = {35L70 (58G16 58G20 58G25)},
MRNUMBER = {1308407},
MRREVIEWER = {Jean-Marc Delort},
ZBLNUMBER = {0860.35081},
DOI = {10.2307/2152832},
} -
[smso06]
H. F. Smith and C. D. Sogge, "On the $L^p$ norm of spectral clusters for compact manifolds with boundary," Acta Math., vol. 198, iss. 1, pp. 107-153, 2007.
@article {smso06, MRKEY = {2316270},
AUTHOR = {Smith, Hart F. and Sogge, Christopher D.},
TITLE = {On the {$L\sp p$} norm of spectral clusters for compact manifolds with boundary},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {198},
YEAR = {2007},
NUMBER = {1},
PAGES = {107--153},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {58J50 (35J15 35P15 47F05 47G10)},
MRNUMBER = {2316270},
MRREVIEWER = {Elena A. Mazepa},
ZBLNUMBER = {1189.58017},
DOI = {10.1007/s11511-007-0014-z},
} -
[stein3] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton Univ. Press, 1993, vol. 43.
@book {stein3, MRKEY = {1232192},
AUTHOR = {Stein, Elias M.},
TITLE = {Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals},
SERIES = {Princeton Math. Ser.},
VOLUME = {43},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1993},
PAGES = {xiv+695},
ISBN = {0-691-03216-5},
MRCLASS = {42-02 (35Sxx 43-02 47G30)},
MRNUMBER = {1232192},
MRREVIEWER = {Michael Cowling},
ZBLNUMBER = {0821.42001},
} -
[tat02]
D. Tataru, "Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III," J. Amer. Math. Soc., vol. 15, iss. 2, pp. 419-442, 2002.
@article {tat02, MRKEY = {1887639},
AUTHOR = {Tataru, Daniel},
TITLE = {Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. {III}},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {15},
YEAR = {2002},
NUMBER = {2},
PAGES = {419--442},
ISSN = {0894-0347},
MRCLASS = {35L10 (35A22 35B40 35L70 35S05)},
MRNUMBER = {1887639},
MRREVIEWER = {Luigi Rodino},
ZBLNUMBER = {0990.35027},
DOI = {10.1090/S0894-0347-01-00375-7},
}