The coercivity of functional differential equations

L. E. Rossovskii

Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 1, Tome 45 (2012), pp. 122-131.

Résumé

New sufficient conditions for the Gårding inequality for variable-coefficient functional differential equations with expanded and contracted arguments of higher derivatives of the unknown function are found. These sufficient conditions generalize earlier known sufficient conditions.

Bibliographie

[1] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, Uspekhi mat. nauk, 19:3 (1964), 53–161 | MR | Zbl

[2] Vishik M. I., “O silno ellipticheskikh sistemakh differentsialnykh uravnenii”, Mat. sb., 29(71):3 (1951), 615–676 | MR | Zbl

[3] Rossovskii L. E., “Koertsitivnost funktsionalno-differentsialnykh uravnenii”, Mat. zametki, 59:1 (1996), 103–113 | DOI | MR | Zbl

[4] Rossovskii L. E., “Ob odnom klasse sektorialnykh funktsionalno-differentsialnykh operatorov”, Diff. uravn., 48:2 (2012), 227–237 | MR | Zbl

[5] Skryabin M. A., “Razbienie edinitsy i problema silnoi elliptichnosti dlya funktsionalno-differentsialnykh operatorov”, Sovrem. mat. Fundam. napravl., 34, 2009, 139–151 | MR

[6] Khërmander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 3, Psevdodifferentsialnye operatory, Mir, M., 1987 | MR

[7] Shamin R. V., “O prostranstvakh nachalnykh dannykh dlya differentsialnykh uravnenii v gilbertovom prostranstve”, Mat. sb., 194:9 (2003), 141–156 | DOI | MR | Zbl

[8] Auscher P., Hofmann S., McIntosh A., Tchamitchian P., “The Kato square root problem for higher order elliptic operators and systems on $\mathbb R^n$”, J. Evol. Equ., 1:4 (2001), 361–385 | DOI | MR | Zbl

[9] Gårding L., “Dirichlet's problem for linear elliptic partial differential equations”, Math. Scand., 1 (1953), 55–72 | MR

[10] Kato T., “Fractional powers of dissipative operators”, J. Math. Soc. Japan., 13:3 (1961), 246–274 | DOI | MR | Zbl

[11] Lions J. L., “Espaces d'interpolation et domaines de puissances fractionnaires d'operateurs”, J. Math. Soc. Japan., 14:2 (1962), 233–241 | DOI | MR | Zbl

[12] McIntosh A., “On the comparability of $A^{1/2}$ and $A^{*1/2}$”, Proc. Amer. Math. Soc., 32:2 (1972), 430–434 | MR | Zbl

[13] Skubachevskii A. L., Shamin R. V., “The mixed boundary value problem for parabolic differential-difference equation”, Functional Differential Equations, 8:3–4 (2001), 407–424 | MR | Zbl

[14] Skubachevskii A. L., “The first boundary value problem for strongly elliptic differential-difference equations”, J. Differential Equations, 63 (1986), 332–361 | DOI | MR

[15] Skubachevskii A. L., Elliptic Functional-Differential Equations and Applications, Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, 1997 | MR | Zbl