Geodesic
Estimates on the boundary for differential operators with constant coefficients in a half-space
Izvestiya. Mathematics, Tome 8 (1974) no. 3, pp. 667-726 
I. V. Gel'man; V. G. Maz'ya. Estimates on the boundary for differential operators with constant coefficients in a half-space. Izvestiya. Mathematics, Tome 8 (1974) no. 3, pp. 667-726. http://geodesic.mathdoc.fr/item/IM2_1974_8_3_a11/
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Voir la notice de l'article provenant de la source Math-Net.Ru

For differential operators $A(D)$, $P_j(D)$ ($j=1,\dots,N$, $D=(\partial/i\partial x_1,\dots,\partial/i\partial x_{n-1};\partial/i\partial t)$) with constant complex coefficients in the half-space $\mathbf R^n_+=\{(x;t),x\in\mathbf R^{n-1},t\geqslant0\}$ we present a precise description of the “space of traces” $A(D)u|_{t=0}$ of elements $u$ in the completion of the space $C^\infty_0(\mathbf R^n_+)$ with respect to the metric $\sum_{j=1}^N\|P_j(D)u\|^2$ ($\|\cdot\|$ is the norm in $L_2(\mathbf R^n_+)$). We consider the case of the metric $\|P(D)u\|^2+\|u\|^2$ in detail. We establish necessary and sufficient conditions for validity of the inequality $$ \bigl\langle A(D)u\bigr\rangle_{s_0}^2\leqslant C\biggl(\sum_{j=1}^N\|P_j(D)u\|^2+\sum_{k=1}^r\langle B_k(D)u\rangle_{s_k}^2\biggr) $$ for all $u(x;t)\in C^\infty_0(\mathbf R^n_+)$ ($\langle\cdot\rangle$ is the norm in $\mathscr H_s(\partial\mathbf R^n_+)$).

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

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

[3] Schechter M., “On the dominance of partial differential operators”, Trans. Amer. Math. Soc., 107:2 (1963), 237–260 | DOI | MR | Zbl

[4] Schechter M., “On the dominance of partial differential operators, II”, Ann. Scuola Norm. Super., Pisa, XVII:3 (1964), 255–282 | MR

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

[6] Goulaouic Ch., Grisvard P., “Existence de traces pour les éléments d'éspaces de distributions définis comme domaines d'opérateurs maximaux”, Inventiones Math., 9:4 (1970), 308–317 | DOI | MR | Zbl

[7] Roitberg Ya. A., “O znacheniyakh na granitse oblasti obobschennykh reshenii ellipticheskikh uravnenii”, Matem. sb., 86:2 (1971), 248–267 | MR | Zbl

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

[9] Polia G., Sege G., Zadachi i teoremy iz analiza, GITTL, M., 1956

[10] Khermander L., Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, M., 1965 | MR

[11] Baiocchi C., “Définition d'opérateurs maximaux et applications”, Ann. Sci. Ecole Norm. Sup. (4), 2:4 (1969), 481–520 | MR | Zbl

[12] Slobodetskii L. N., “Obobschennye prostranstva S. L. Soboleva i ikh prilozheniya k kraevym zadacham”, Uch. zap. LGPI im. Gertsena, 197, 1958, 54–112