Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Local solvability and hypoellipticity for semilinear anisotropic partial differential equations
HTML articles powered by AMS MathViewer

by Giuseppe de Donno and Alessandro Oliaro PDF
Trans. Amer. Math. Soc. 355 (2003), 3405-3432 Request permission

Abstract:

We propose a unified approach, based on methods from microlocal analysis, for characterizing the local solvability and hypoellipticity in $C^\infty$ and Gevrey $G^\sigma$ classes of $2$-variable semilinear anisotropic partial differential operators with multiple characteristics. The conditions imposed on the lower-order terms of the linear part of the operator are optimal.
References
  • Antonio Bove and David S. Tartakoff, Propagation of Gevrey regularity for a class of hypoelliptic equations, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2533–2575. MR 1340171, DOI 10.1090/S0002-9947-96-01557-7
  • Jean-Michel Bony and Jean-Yves Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France 122 (1994), no. 1, 77–118 (French, with English and French summaries). MR 1259109, DOI 10.24033/bsmf.2223
  • Massimo Cicognani and Luisa Zanghirati, On a class of unsolvable operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), no. 3, 357–369. MR 1256073
  • A. Corli, On local solvability of linear partial differential operators with multiple characteristics, J. Differential Equations 81 (1989), no. 2, 275–293. MR 1016083, DOI 10.1016/0022-0396(89)90124-1
  • Andrea Corli, On local solvability in Gevrey classes of linear partial differential operators with multiple characteristics, Comm. Partial Differential Equations 14 (1989), no. 1, 1–25. MR 973268, DOI 10.1080/03605308908820589
  • G. De Donno and L. Rodino, Gevrey hypoellipticity for partial differential equations with characteristics of higher multiplicity, to appear in Rend. Sem. Mat. Univ. Politecnico Torino, (2000).
  • G. De Donno and L. Rodino, Gevrey hypoellipticity for equations with involutive characteristics of higher multiplicity, C. R. Acad. Bulgare Sci. 53 (2000), no. 7, 25–30. MR 1779525
  • B. Dehman, Résolubilité local pour des équations semi-linéaires complexes, Canad. J. Math. 42 (1990), no. 1, 126–140 (French). MR 1043515, DOI 10.4153/CJM-1990-008-x
  • Yuri V. Egorov and Bert-Wolfgang Schulze, Pseudo-differential operators, singularities, applications, Operator Theory: Advances and Applications, vol. 93, Birkhäuser Verlag, Basel, 1997. MR 1443430, DOI 10.1007/978-3-0348-8900-1
  • G. Garello, Inhomogeneous paramultiplication and microlocal singularities for semilinear equations, Boll. Un. Mat. Ital. B (7) 10 (1996), no. 4, 885–902 (English, with Italian summary). MR 1430158
  • G. Garello, Local solvability for semilinear equations with multiple characteristics, Proceedings of the Conference “Differential Equations” (Italian) (Ferrara, 1996), 1996, pp. 199–209 (1997). MR 1471025
  • Todor V. Gramchev, On the critical index of Gevrey solvability for some linear partial differential equations, Ann. Univ. Ferrara Sez. VII (N.S.) 45 (1999), no. suppl., 139–153 (2000) (English, with English and Italian summaries). Workshop on Partial Differential Equations (Ferrara, 1999). MR 1806494
  • T. Gramchev and P. Popivanov, Local solvability of semilinear partial differential equations, Ann. Univ. Ferrara Sez. VII (N.S.) 35 (1989), 147–154 (1990) (English, with Italian summary). MR 1079584
  • Todor V. Gramchev and Petar R. Popivanov, Partial differential equations, Mathematical Research, vol. 108, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. Approximate solutions in scales of functional spaces. MR 1747915
  • Todor Gramchev, Petar Popivanov, and Massafumi Yoshino, Critical Gevrey index for hypoellipticity of parabolic operators and Newton polygons, Ann. Mat. Pura Appl. (4) 170 (1996), 103–131. MR 1441616, DOI 10.1007/BF01758985
  • Todor Gramchev and Luigi Rodino, Gevrey solvability for semilinear partial differential equations with multiple characteristics, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2 (1999), no. 1, 65–120 (English, with Italian summary). MR 1794545
  • Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
  • Jorge Hounie and Paulo Santiago, On the local solvability of semilinear equations, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1777–1789. MR 1349231, DOI 10.1080/03605309508821151
  • Charles Hunt and Alain Piriou, Opérateurs pseudo-différentiels anisotropes d’ordre variable, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A28–A31 (French). MR 248567
  • Charles Hunt and Alain Piriou, Majorations $L^{2}$ et inégalité sous-elliptique pour les opérateurs pseudo-différentiels anisotropes d’ordre variable, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A214–A217 (French). MR 248568
  • K. Kajitani and S. Spagnolo, in progress, communicated to the meeting “Perturbative methods for nonlinear partial differential equations”, Cagliari 2000.
  • K. Kajitani and S. Wakabayashi, Hypoelliptic operators in Gevrey classes, Recent developments in hyperbolic equations (Pisa, 1987) Pitman Res. Notes Math. Ser., vol. 183, Longman Sci. Tech., Harlow, 1988, pp. 115–134. MR 984364
  • Richard Lascar, Distributions intégrales de Fourier et classes de Denjoy-Carleman. Applications, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 9, A485–A488. MR 427876
  • Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
  • Otto Liess and Luigi Rodino, Inhomogeneous Gevrey classes and related pseudodifferential operators, Boll. Un. Mat. Ital. C (6) 3 (1984), no. 1, 233–323. MR 749292
  • O. Liess and L. Rodino, Linear partial differential equations with multiple involutive characteristics, Microlocal analysis and spectral theory (Lucca, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 490, Kluwer Acad. Publ., Dordrecht, 1997, pp. 1–38. MR 1451388
  • Michael Lorenz, Anisotropic operators with characteristics of constant multiplicity, Math. Nachr. 124 (1985), 199–216. MR 827898, DOI 10.1002/mana.19851240113
  • P. Marcolongo, Solvability and nonsolvability for partial differential equations in Gevrey spaces, Ph.D. Dissertation, Mathematics, University of Torino, 2000.
  • Paola Marcolongo and Alessandro Oliaro, Local solvability for semilinear anisotropic partial differential equations, Ann. Mat. Pura Appl. (4) 179 (2001), 229–262. MR 1848755, DOI 10.1007/BF02505957
  • Maria Mascarello and Luigi Rodino, Partial differential equations with multiple characteristics, Mathematical Topics, vol. 13, Akademie Verlag, Berlin, 1997. MR 1608649
  • A. Menikoff, On hypoelliptic operators with double characteristics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 689–724. MR 473490
  • P. R. Popivanov, The local solvability of a certain class of pseudodifferential equations with double characteristics, Trudy Sem. Petrovsk. Vyp. 1 (1975), 237–278 (Russian). MR 0427811
  • P. R. Popivanov, Local solvability of some classes of linear differential operators with multiple characteristics, Ann. Univ. Ferrara Sez. VII (N.S.) 45 (1999), no. suppl., 263–274 (2000). Workshop on Partial Differential Equations (Ferrara, 1999). MR 1806503
  • P. R. Popivanov, Microlocal properties of a class of pseudodifferential operators with double involutive characteristics, Partial differential equations (Warsaw, 1984) Banach Center Publ., vol. 19, PWN, Warsaw, 1987, pp. 213–224. MR 1055173
  • PetЪr R. Popivanov and Georgi St. Popov, Microlocal properties of a class of pseudodifferential operators with multiple characteristics, Serdica 6 (1980), no. 2, 167–181 (Russian). MR 601354
  • Gary B. Roberts, Quasisubelliptic estimates for operators with multiple characteristics, Comm. Partial Differential Equations 11 (1986), no. 3, 231–320. MR 822339, DOI 10.1080/03605308608820424
  • Luigi Rodino, Linear partial differential operators in Gevrey spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1249275, DOI 10.1142/9789814360036
  • L. Rodino, Local solvability in Gevrey classes, Hyperbolic equations (Padua, 1985) Pitman Res. Notes Math. Ser., vol. 158, Longman Sci. Tech., Harlow, 1987, pp. 167–185. MR 922088
  • N. A. Šananin, The local solvability of equations of quasiprincipal type, Mat. Sb. (N.S.) 97(139) (1975), no. 4 (8), 503–516, 633 (Russian). MR 0473446
  • F. Segàla, A class of locally solvable differential operators, Boll. Un. Mat. Ital. B (6) 4 (1985), no. 1, 241–251 (English, with Italian summary). MR 783342
  • Sergio Spagnolo, Local and semi-global solvability for systems of non-principal type, Comm. Partial Differential Equations 25 (2000), no. 5-6, 1115–1141. MR 1759804, DOI 10.1080/03605300008821543
  • Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
  • V. N. Tulovskiĭ, Propagation of singularities of operators with characteristics of constant multiplicity, Trudy Moskov. Mat. Obshch. 39 (1979), 113–134, 236 (Russian). MR 544943
  • Seiichiro Wakabayashi, Singularities of solutions of the Cauchy problem for hyperbolic systems in Gevrey classes, Japan. J. Math. (N.S.) 11 (1985), no. 1, 157–201. MR 877462, DOI 10.4099/math1924.11.157
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35S05
  • Retrieve articles in all journals with MSC (2000): 35S05
Additional Information
  • Giuseppe de Donno
  • Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
  • Email: dedonno@dm.unito.it
  • Alessandro Oliaro
  • Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
  • Email: oliaro@dm.unito.it
  • Received by editor(s): February 7, 2001
  • Received by editor(s) in revised form: October 8, 2002
  • Published electronically: April 11, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 3405-3432
  • MSC (2000): Primary 35S05
  • DOI: https://doi.org/10.1090/S0002-9947-03-03275-6
  • MathSciNet review: 1974694