On subordination conditions for systems of minimal differential operators

D. V. Limanskyii ; M. M. Malamud

Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 1, pp. 121-149.

Résumé

In this paper, we provide a review of results on a priori estimates for systems of minimal differential operators in the scale of spaces $L^p(\Omega),$ where $p\in[1,\infty].$ We present results on the characterization of elliptic and $l$-quasielliptic systems using a priori estimates in isotropic and anisotropic Sobolev spaces $W_{p,0}^l(\mathbb R^n),$ $p\in[1,\infty].$ For a given set $l=(l_1,\dots,l_n)\in\mathbb N^n$ we prove criteria for the existence of $l$-quasielliptic and weakly coercive systems and indicate wide classes of weakly coercive in $W_{p,0}^l(\mathbb R^n),$ $p\in[1,\infty],$ nonelliptic, and nonquasielliptic systems. In addition, we describe linear spaces of operators that are subordinate in the $L^\infty(\mathbb R^n)$-norm to the tensor product of two elliptic differential polynomials.

Détail

Keywords: differential operator, a priori estimate, quasi-ellipticity, coercivity.

Bibliographie

[1] Besov O. V., “O koertsitivnosti v anizotropnom prostranstve S. L. Soboleva”, Mat. sb., 73:4 (1967), 585–599 | Zbl

[2] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1996 | MR

[3] Volevich L. R., “Lokalnye svoistva reshenii kvaziellipticheskikh sistem”, Mat. sb., 59 (1962), 3–52 | Zbl

[4] Volevich L. R., Gindikin S. G., Metod mnogogrannika Nyutona v teorii differentsialnykh uravnenii v chastnykh proizvodnykh, Editorial URSS, M., 2002

[5] Gorin E. A., “Ob issledovaniyakh G. E. Shilova po teorii kommutativnykh banakhovykh algebr i ikh dalneishem razvitii”, Usp. mat. nauk, 33:4 (1978), 169–188 | MR | Zbl

[6] Ilin V. P., “Ob usloviyakh spravedlivosti neravenstv mezhdu $L_p$-normami chastnykh proizvodnykh funktsii mnogikh peremennykh”, Tr. MIAN, 96, 1968, 205–242 | Zbl

[7] Kazaryan G. G., “Ob otsenkakh $L_p$-norm proizvodnykh cherez neregulyarnyi nabor differentsialnykh operatorov”, Diff. uravn, 5:5 (1969), 911–921 | Zbl

[8] Lizorkin P. I., “Predelnye sluchai teorem o $\mathfrak F L_p$-multiplikatorakh”, Tr. MIAN, 173, 1986, 164–180 | MR

[9] Limanskii D. V., “Ob otsenkakh dlya tenzornogo proizvedeniya dvukh odnorodnykh ellipticheskikh operatorov”, Ukr. mat. visn., 8:1 (2011), 101–111 | Zbl

[10] Limanskii D. V., “Umovi pidporyadkovanosti dlya tenzornogo dobutku dvokh zvichainikh diferentsialnikh operatoriv”, Dopov. NAN Ukr., 2012, no. 4, 25–29 | MR

[11] Limanskii D. V., Malamud M. M., “O slaboi koertsitivnosti sistem differentsialnykh operatorov v $L^1$ i $L^\infty$”, Dokl. RAN, 397:4 (2004), 453–458 | MR | Zbl

[12] Limanskii D. V., Malamud M. M., “Slabo koertsitivnye nekvaziellipticheskie sistemy differentsialnykh operatorov v $W_p^l(\mathbb R^n)$”, Dokl. RAN, 415:5 (2007), 583–588 | MR | Zbl

[13] Limanskii D. V., Malamud M. M., “Ellipticheskie i slabo koertsitivnye sistemy operatorov v prostranstvakh Soboleva”, Mat. sb., 199:11 (2008), 75–112 | DOI | MR

[14] Limanskii D. V., Malamud M. M., “Ob analoge teoremy de Lyu i Mirkila dlya operatorov s peremennymi koeffitsientami”, Mat. zametki, 83:5 (2008), 783–786 | DOI | MR | Zbl

[15] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, v. 1, Mir, M., 1971

[16] Lopatinskii Ya. B., “Ob odnom sposobe privedeniya granichnykh zadach dlya sistemy differentsialnykh uravnenii ellipticheskogo tipa k regulyarnym integralnym uravneniyam”, Ukr. mat. zh., 5 (1953), 123–151 | Zbl

[17] Malamud M. M., “Differentsialnye svoistva funktsii i koertsitivnost v prostranstvakh s ravnomernoi normoi”, Ukr. mat. zh., 34:5 (1982), 553–558 | MR

[18] Malamud M. M., “Otsenka dlya differentsialnykh operatorov v ravnomernoi norme i koertsitivnost v prostranstvakh S. L. Soboleva”, Dokl. AN SSSR, 298:1 (1988), 32–36 | Zbl

[19] Malamud M. M., “Otsenki dlya sistem minimalnykh i maksimalnykh differentsialnykh operatorov v $L_p(\Omega)$”, Tr. Mosk. mat. ob-va, 56, 1995, 206–261

[20] Mityagin B. S., “O vtoroi smeshannoi proizvodnoi”, Dokl. AN SSSR, 123:4 (1958), 606–609 | Zbl

[21] Mityagin B. S., “O nekotorykh svoistvakh funktsii dvukh peremennykh”, Vestn. MGU. Ser. mat., 1959, no. 5, 137–152

[22] Mikhailov V. P., “O povedenii na beskonechnosti odnogo klassa mnogochlenov”, Tr. MIAN, 91, 1967, 59–80 | Zbl

[23] Mikhlin S. G., “O multiplikatorakh integralov Fure”, Dokl. AN SSSR, 109:4 (1956), 701–703 | Zbl

[24] Nirenberg L., Lektsii po nelineinomu funktsionalnomu analizu, Mir, M., 1977

[25] Spener E., Algebraicheskaya topologiya, Mir, M., 1971

[26] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973

[27] Khermander L., K teorii obschikh differentsialnykh operatorov v chastnykh proizvodnykh, Mir, M., 1959

[28] Shilov G. E., “O nekotorykh zadachakh obschei teorii kommutativnykh normirovannykh kolets”, Usp. mat. nauk, 12:1 (1957), 246–249 | MR | Zbl

[29] Yudovich V. I., “O nekotorykh otsenkakh, svyazannykh s integralnymi operatorami i resheniyami ellipticheskikh uravnenii”, Dokl. AN SSSR, 138:4 (1961), 805–808 | Zbl

[30] Agmon S., “The coerciveness problem for integro-differential forms”, J. Anal. Math., 6 (1958), 183–223 | DOI | MR | Zbl

[31] Aronszajn N., “On coercive integro-differential quadratic forms”, Conference on Partial Differential Equations, Univ. Kansas, Lawrence, 1954, 94–106

[32] Belinsky E. S., Dvejrin M. Z., Malamud M. M., “Multipliers in $L_1$ and estimates for systems of differential operators”, Russ. J. Math. Phys., 12:1 (2005), 6–16 | MR | Zbl

[33] Boman J., “Supremum norms for partial derivatives of functrions of several real variables”, Illinois J. Math., 16 (1972), 203–216 | DOI | MR | Zbl

[34] De Leeuw K., Mirkil H., “A priori estimates for differential operators in $L_\infty$ norm”, Illinois J. Math., 8 (1964), 112–124 | DOI | MR | Zbl

[35] Kazaniecki K., Stolyarov D. M., Wojciechowski M., “Anisotropic Ornstein non-inequalities”, Anal. PDE, 10:2 (2017), 351–366 | DOI | MR | Zbl

[36] Kazaniecki K., Wojciechowski M., On the analytic version of the Mityagin–de Leeuw–Mirkhil non-equality on bi-disc, 2023, arXiv: 2301.09526 [math.FA]

[37] Kirchheim B., Kristensen J., “On rank one convex functions that are homogeneous of degree one”, Arch. Ration. Mech. Anal., 221:1 (2016), 527–558 | DOI | MR | Zbl

[38] Kislyakov S. V., Maksimov D. V., Stolyarov D. M., “Differential expressions with mixed homogeneity and spaces of smooth functions they generate in arbitrary dimension”, J. Funct. Anal., 269:10 (2015), 3220–3263 | DOI | MR | Zbl

[39] Littman W., “The wave operator and $L^p$ norms”, J. Math. Mech., 12:1 (1963), 55–68 | MR | Zbl

[40] Ne\v{c as J.}, “Sur les normes équivalentes dans $W_{p}^{k}(\Omega)$ et sur la coercitivité des formes formellement positives”, Séminaire Equations aux Dérivées partielles, Univ. Montréal, Montréal, 1966, 102–128

[41] Ornstein D., “A non-equality for differential operators in the $L_1$ norm”, Arch. Ration. Mech. Anal., 11 (1962), 40–49 | DOI | MR | Zbl

[42] Schechter M., “Integral inequalities for partial differential operators and functions satisfying general boundary conditions”, Commun. Pure Appl. Math., 12 (1959), 37–66 | DOI | MR | Zbl

[43] Smith K. T., “Inequalities for formally positive integro-differential forms”, Bul. Am. Math. Soc., 67 (1961), 368–370 | DOI | MR | Zbl