Elliptic and weakly coercive systems of operators in Sobolev spaces

D. V. Lymanskyi   ; M. M. Malamud

Sbornik. Mathematics, Tome 199 (2008) no. 11, pp. 1649-1686

Résumé

It is known that an elliptic system $\{P_j(x,D)\}_1^N$ of order $l$ is weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^n)$, that is, all differential monomials of order $\leqslant l-1$ on $C_0^\infty(\mathbb R^n)$-functions are subordinated to this system in the $L^\infty$-norm. Conditions for the converse result are found and other properties of weakly coercive systems are investigated. An analogue of the de Leeuw-Mirkil theorem is obtained for operators with variable coefficients: it is shown that an operator $P(x,D)$ of $n\geqslant 3$ variables with constant principal part is weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^n)$ if and only if it is elliptic. A similar result is obtained for systems $\{P_j(D)\}_1^N$ with constant coefficients under the condition $n\geqslant 2N+1$ and with several restrictions on the symbols $P_j(\xi)$. A complete description of differential polynomials of two variables which are weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^2)$ is given. Wide classes of systems with constant coefficients which are weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^n)$, but non-elliptic are constructed. Bibliography: 32 titles.

Bibliographie

[1] O. V. Besov, V. P. Il'in, S. M. Nikol'skiĭ, Integral representations of functions and imbedding theorems, vol. I, II, Winston, Washington, DC; Wiley, New York–Toronto, ON–London, 1979 | MR | MR | Zbl | Zbl

[2] L. R. Volevich, “Lokalnye svoistva reshenii kvaziellipticheskikh sistem”, Matem. sb., 59(101){(dopolnitelnyi)} (1962), 3–52 | MR | Zbl

[3] S. Gindikin, L. R. Volevich, The method of Newton's polyhedron in the theory of partial differential equations, Math. Appl. (Soviet Ser.), 86, Kluwer Acad. Publ., Dordrecht, 1992 | MR | Zbl

[4] Ya. B. Lopatinskiǐ, “A method of reducing boundary problems for a system of differential equations of elliptic type to regular integral equations”, Amer. Math. Soc. Transl. Ser. 2, 89 (1970), 149–183 | MR | Zbl | Zbl

[5] M. S. Agranovich, “Partial differential equations with constant coefficients”, Russian Math. Surveys, 16:2 (1961), 23–90 | DOI | MR | Zbl

[6] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, 1968 | MR | MR | Zbl | Zbl

[7] H. Triebel, Theory of function spaces, Monogr. Math., 78, Birkhäuser, Basel–Boston–Stuttgart, 1983 | MR | MR | Zbl | Zbl

[8] O. V. Besov, “On coercivity in nonisotropic Sobolev spaces”, Math. USSR-Sb., 2:4 (1967), 521–534 | DOI | MR | Zbl

[9] L. Hörmander, “On the theory of general partial differential operators”, Acta Math., 94:1 (1955), 161–248 | DOI | MR | Zbl

[10] M. M. Malamud, “Differential properties of functions and coerciveness in spaces with uniform norm”, Ukrainian Math. J., 34:5 (1982), 444–449 | DOI | MR | Zbl

[11] M. M. Malamud, “An estimate for differential operators in the uniform norm, and coercivity in Sobolev spaces”, Soviet Math. Dokl., 37:1 (1988), 25–29 | MR | Zbl

[12] M. M. Malamud, “Estimates for systems of minimal and maximal differential operators in $L_p(\Omega)$”, Trans. Moscow Math. Soc., 1995, 159–202 | MR | Zbl

[13] D. Ornstein, “A non-inequality for differential operators in the $L_1$ norm”, Arch. Rational Mech. Anal., 11:1 (1962), 40–49 | DOI | MR | Zbl

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

[15] D. V. Limanskii, M. M. Malamud, “Weak coercivity of systems of differential operators in $L^1$ and $L^\infty$”, Russian Acad. Sci. Dokl. Math., 70:1 (2004), 584–588 | MR

[16] K. de Leeuw, H. Mirkil, “A priori estimates for differential operators in $L_\infty$ norm”, Illinois J. Math., 8:1 (1964), 112–124 | MR | Zbl

[17] P. I. Lizorkin, “Obobschennoe liuvillevskoe differentsirovanie i funktsionalnye prostranstva $L_p^r(E_n)$. Teoremy vlozheniya”, Matem. sb., 60(102):3 (1963), 325–353 | MR | Zbl

[18] S. G. Mikhlin, “O multiplikatorakh integralov Fure”, DAN, 109:4 (1956), 701–703 | MR | Zbl

[19] D. V. Limanskii, M. M. Malamud, “Weakly coercive non-quasi-elliptic systems of differential operators in $W_p^l(\mathbb R^n)$”, Russian Acad. Sci. Dokl. Math., 76:1 (2007), 583–588

[20] E. H. Spanier, Algebraic topology, McGraw-Hill, New York–Toronto, ON–London, 1966 | MR | MR | Zbl | Zbl

[21] L. Nirenberg, Topic in nonlinear functional analysis, Courant Institute of Math. Sciences, New York Univ., New York, 1974 | MR | MR | Zbl | Zbl

[22] J. Boman, “Supremum norm estimates for partial derivatives of functions of several real variables”, Illinois J. Math., 16:2 (1972), 203–216 | MR | Zbl

[23] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton Univ. Press, Princeton, NJ, 1970 | MR | MR | Zbl | Zbl

[24] O. V. Besov, “On Hörmander's theorem on Fourier multipliers”, Proc. Steklov Inst. Math., 173 (1987), 1–12 | MR | Zbl

[25] P. I. Lizorkin, “Limiting cases of theorems on $\mathscr{F}L_p$-multipliers”, Proc. Steklov Inst. Math., 173 (1987), 177–194 | MR | Zbl

[26] R. M. Trigub, “Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus”, Math. USSR-Izv., 17:6 (1981), 567–593 | DOI | MR | Zbl | Zbl

[27] R. M. Trigub, E. S. Belinsky, Fourier analysis and approximation of functions, Kluwer Acad. Publ., Boston–Dordrecht–London, 2004 | MR | Zbl

[28] L. R. Volevich, S. G. Gindikin, “The Newton polyhedron and local solvability of linear partial differential equations”, Trans. Moscow Math. Soc., 1986, 227–276 | MR | Zbl

[29] Ju. M. Berezans'kiǐ, Expansions in eigenfunctions of selfadjoint operators, Amer. Math. Soc., Providence, RI, 1968 | MR | MR | Zbl | Zbl

[30] A. I. Markovskii, “Ob ansamblyakh minimalnykh differentsialnykh operatorov”, Teor. i prikl. matem. Lvov, 1963, no. 2, 48–52 | MR

[31] M. A. Neumark, Lineare Differentialoperatoren, Akademie-Verlag, Berlin, 1960 | MR | MR | Zbl

[32] W. Littman, C. McCarthy, N. Rivière, “The non-existence of $L^p$ estimates for certain translation-invariant operators”, Studia Math., 30:2 (1968), 218–229 | MR | Zbl