Estimates for differential operators with constant coefficients in a half-space

V. G. Maz'ya   ; I. V. Gel'man

Sbornik. Mathematics, Tome 25 (1975) no. 2, pp. 225-258

Résumé

Necessary and sufficient conditions (and also more explicit sufficient conditions) are obtained for the validity of the following estimates for differential operators with constant coefficients in the half-space $\mathbf R_+^n=\{(x,t):x\in\mathbf R^{n-1},\ t\geqslant0\}$: \begin{gather*} \|\mathscr R(D)u\|^2\leqslant C\|\mathscr P(D)u\|^2,\qquad u\in C_0^\infty(\mathbf R_+^n),\quad (\mathscr Q_j(D)u)(x;0)=0\ (j=1,\dots,N), \\ \|\mathscr R(D)u\|^2\leqslant C\biggl(\|\mathscr P(D)u\|^2+\sum_{j=1}^N\langle\!\langle\mathscr Q_j(D)u\rangle\!\rangle _{s_j}^2\biggr), \end{gather*} where ${\|\cdot\|}$ and $\langle\!\langle\,\cdot\,\rangle\!\rangle$ are the norms in $L_2(\mathbf R_+^n)$ and $H_s(\partial\mathbf R_+^n)$, $$ D=\biggl(\frac1i\,\frac\partial{\partial x_1},\dots,\frac1i\,\frac\partial{\partial x_{n-1}};\frac1i\,\frac\partial{\partial t}\biggr), $$ and $C_0^\infty(\mathbf R_+^n)$ is the space of restrictions to $\mathbf R_+^n$ of functions in $C_0^\infty(\mathbf R^n)$. Bibliography: 18 titles.

Bibliographie

[1] N. Aronszajn, “On coercive integro-differential quadratic forms”, Conference on partial differential equations (Univ. of Kansas), no. 14, 1954, 94–106

[2] S. Agmon, “The coerciveness problem for integro-differential forms”, J. d'Analyse Math., 6:2 (1958), 183–223 | DOI | MR | Zbl

[3] F. E. Browder, “On regularity properties of solutions of elliptic differential equations”, Comm. Pure Appl. Math., 9 (1956), 351–361 | DOI | MR | Zbl

[4] L. Nirenberg, “Remarks on strongly elliptic partial differential equations”, Comm. Pure Appl. Math., 8 (1955), 649–675 | DOI | MR | Zbl

[5] B. Malgrange, “Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution”, Ann. Inst. Fourier (Grenoble), 6 (1955–1956), 271–335 | MR

[6] L. Khermander, K teorii obschikh differentsialnykh operatorov v chastnykh proizvodnykh, IL, Moskva, 1959

[7] J. Peetre, “On estimating the solutions of hypoelliptic differential equations near the plane boundary”, Math. Scandin, 9:2 (1961), 337–351 | MR | Zbl

[8] M. Schechter, “On the dominance of partial differential operators, II”, Ann. Scuola Norm. Super. Pisa, sci fis.-mat., ser. III, 18:2 (1964), 255–282 | MR | Zbl

[9] M. Schechter, “Systems of partial differential equations in a half-space”, Comm. Pure Appl. Math., 17:4 (1964), 423–434 | DOI | MR | Zbl

[10] C. F. Schubert, “Overdetermined systems on $L^2(\mathbf{R}_+^n)$”, Lecture Notes on Math., 183 (1971), 221–225 | DOI | MR | Zbl

[11] I. V. Gelman, V. G. Mazya, “Otsenki na granitse dlya differentsialnykh operatorov s postoyannymi koeffitsientami v poluprostranstve”, Izv. AN SSSR, seriya matem., 38 (1974), 663–720 | MR

[12] T. Matsuzawa, “On quasi-elliptic boundary problems”, Trans. Amer. Math. Soc., 133:1 (1968), 241–265 | DOI | MR | Zbl

[13] C. Parenti, “Valutazioni a priori e regolarita per soluzioni di equazioni quasi-ellittiche”, Rend. Semin, Matem. Univ. Padova, 45 (1971), 1–70 | MR | Zbl

[14] L. R. Volevich, B. P. Paneyakh, “Nekotorye prostranstva obobschennykh funktsii i teoremy vlozheniya”, Uspekhi matem. nauk, XX:1(121) (1965), 3–74

[15] I. V. Gelman, V. G. Mazya, “Otsenki dlya differentsialnykh operatorov s postoyannymi koeffitsientami v poluprostranstve”, DAN SSSR, 202:4 (1972), 751–754 | MR

[16] I. S. Louhivaara, Ch. G. Simader, “Uber nichtelliptische lineare partielle Differentialoperation mit konstanten Koeffizienten”, Ann. Acad. Sci. Fenn. ser., A1, 1972, no. 513, 3–22 | MR

[17] V. E. Katsnelson, “O nekotorykh operatorakh, deistvuyuschikh v prostranstvakh, porozhdennykh funktsiyami $(t-z_k)^{-1}$”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, no. 4, izd. KhGU, 1967, 58–66

[18] D. K. Faddeev, I. S. Sominskii, Sbornik zadach po vysshei algebre, izd-vo «Nauka», Moskva, 1972