Propriétés et existence des solutions de certaines classes d'équations du type elliptique
Gillis, Paul P.
Bulletin de la Société Mathématique de France, Tome 86 (1958), p. 283-297 / Harvested from Numdam
@article{BSMF_1958__86__283_0,
     author = {Gillis, Paul P.},
     title = {Propri\'et\'es et existence des solutions de certaines classes d'\'equations du type elliptique},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     volume = {86},
     year = {1958},
     pages = {283-297},
     doi = {10.24033/bsmf.1507},
     mrnumber = {21 \#3659},
     zbl = {0087.30104},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/BSMF_1958__86__283_0}
}
Gillis, Paul P. Propriétés et existence des solutions de certaines classes d'équations du type elliptique. Bulletin de la Société Mathématique de France, Tome 86 (1958) pp. 283-297. doi : 10.24033/bsmf.1507. http://gdmltest.u-ga.fr/item/BSMF_1958__86__283_0/

[1]Agmon (S.). - Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane (Comm. pure appl. Math., vol. 10, 1957, p. 179-239). | MR 21 #5057 | Zbl 0081.09801

[2]Aronsajn (N.). - On coercive integro-differential forms (Studies in Eigenvalue Problems, no 14, Lawrence Kansas, 1955, p. 94-106). | Zbl 0067.32702

[3]Aronsajn (N.). - A unique continuation theorem for all partial differential equations and inequalities of second order (Bull. Amer. math. Soc., vol. 62, Abstract 184, 1956, p. 154).

[4]Bers (L.). - Survey of local properties of solutions of elliptic partial differential equations (Comm. pure appl. Math., vol. 9, 1956, p. 339-350). | MR 18,398d | Zbl 0074.08001

[5]Bers (L.) et Nirenberg (L.). - Boundary value problems for nonlinear elliptic equations in two independent variables (Proc. Int. Congress Amsterdam, 1954, p. 84-85).

[6]Bers (L.) et Nirenberg (L.). - On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications (Convegno internazionale sulle equazioni derivate e parziali, 1954, p. 111-140). | MR 17,974d | Zbl 0067.32503

[7]Bers (L.) et Nirenberg (L.). - On linear and nonlinear elliptic boundary value problems in the plane (Convegno internationale sulle equazioni derivate e parziali, 1954, p. 141-167). | MR 17,974e | Zbl 0067.32504

[8]Browder (F. E.). - The Dirichlet problem for linear elliptic equations of arbitrary even order with variable coefficients (Proc. nat. Acad. Sc. U.S.A., vol. 38, no 3, 1952, p. 230-235). | MR 14,174c | Zbl 0046.32302

[9]Browder (F. E.). - The Dirichlet and vibration problems for linear elliptic differential equations of arbitrary order (Proc. nat. Acad. Sc. U.S.A., vol. 38, no 8, 1952, p. 741-747). | MR 14,473e | Zbl 0047.09501

[10]Browder (F. E.). - Assumption of boundary values and the Green's function in the Dirichlet problem for the general linear elliptic equation (Proc. nat. Acad. Sc. U.S.A., vol. 39, no 3, 1953, p. 179-184). | MR 14,984d | Zbl 0050.09701

[11]Browder (F. E.). - Linear parabolic differential equations of arbitrary order ; general boundary-value problems for elliptic equations (Proc. nat. Acad. Sc. U.S.A., vol. 39, no 3, 1953, p. 185-190). | MR 14,984f | Zbl 0050.09702

[12]Browder (F. E.). - Strongly elliptic systems of differential equations (Contributions to the Theory of partial differental equations, Ann. Math. Studies no 33, Princeton, 1954, p. 15-51). | MR 16,705f | Zbl 0057.32901

[13]Browder (F. E.). - Regularity properties of solutions of elliptic differential equations (Abstract 695, Bull. Amer. math. Soc., vol. 61, 1955, p. 528).

[14]Browder (F. E.). - On the regularity properties of solutions of elliptic differential equations (Comm. pure appl. Math., vol. 9, 1956, p. 351-361). | MR 19,862a | Zbl 0070.09601

[15]Calderon (A. P.) et Zygmund (A.). - On the existence of singular integrals (Acta Math., vol. 88, 1952, p. 85-139). | MR 14,637f | Zbl 0047.10201

[16]Carleman (T). - Sur les systèmes linéaires aux dérivées partielles du premier ordre à deux variables (C. R. Acad. Sc. Paris, t. 197, 1933, p. 471-474). | JFM 59.0469.03 | Zbl 0007.16202

[17]Carleman (T.). - Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes (Ark. Mat. Astr. Fys., vol. 26 B, no 17, 1939, p. 1-9). | JFM 65.0394.03 | MR 1,55f | Zbl 0022.34201

[18]Cordes (H. O). - Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen (Math. Ann., vol. 131, 1956, p. 278-312). | MR 19,961e | Zbl 0070.09604

[19]Courant (R.) et Hilbert (D.). - Methoden der mathematischen Physik, vol. 2, chap. 7, Berlin, Springer, 1937. | JFM 63.0449.05 | Zbl 0017.39702

[20]De Giorgi (E.). - Un esempio di non-unicità della soluzione del problema di Cauchy, relativo ad una equazione differenziale lineare a derivate parziali di tipo parabolico (Rend. Mat. e Appl., 5e série, vol. 14, 1955, p. 382-387). | MR 16,1119a | Zbl 0064.34504

[21]Douglis (A.). - Uniqueness in Cauchy problems for elliptic systems of equations (Comm. pure appl. Math., vol. 6, 1953, p. 291-298). | MR 16,257d | Zbl 0050.31902

[22]Douglis (A.) et Nirenberg (L.). - Interior estimates for elliptic systems of partial differential equations (Comm. pure appl. Math., vol. 8, 1955, p. 503-538). | MR 17,743b | Zbl 0066.08002

[23]Friedrichs (K. O.). - On differential operators in Hilbert spaces (Amer. J. Math., vol. 61, 1939, p. 523-544). | JFM 65.0508.02 | Zbl 0020.36802

[24]Kriedrichs (K. O.). - The identity of weak and strong extensions of differential operators (Trans. Amer. math. Soc., vol. 55, no 1, 1944, p. 132-151). | MR 5,188b | Zbl 0061.26201

[25]Friedrichs (K. O.). - Differentiability of solutions of elliptic differential equations (Comm. pure appl. Math., vol. 6, 1953, p. 299-326). | MR 15,430c | Zbl 0051.32703

[26]Gårding (L.). - Le problème de Dirichlet pour les équations aux dérivées partielles elliptiques homogènes à coefficients constants (C. R. Acad., Sc. Paris, t. 230, 1950, p. 1030-1032). | MR 11,521b | Zbl 0035.34704

[27]Gårding (L.). - Dirichlet's problem for linear elliptic partial differential equations (Math. Scand., vol. 1, 1953, p. 55-72). | MR 16,366a | Zbl 0053.39101

[28]Gillis (P. P.). - Sur certaines classes d'équations aux dérivées-partielles (Acad. Roy. Belgique, Bull. Cl. Sc., vol. 22, 1936, p. 835-854 et 941-947). | JFM 62.0558.02 | Zbl 0015.11201

[29]Gillis (P. P.). - Sur les problèmes réguliers du calcul des variations (Studia Mathematica, vol. 8, 1938, p. 68-77). | JFM 65.0448.01 | Zbl 0020.37105

[30]Gillis (P. P.). - Équations de Monge-Ampère à quatre variables indépendantes (Acad. Roy. Belgique, Bull. Cl. Sc., vol. 36, 1950, p. 474-484). | MR 12,415c | Zbl 0040.34502

[31]Gillis (P. P.). - Équations de Monge-Ampère à six variables indépendantes (Acad. Roy. Belgique, Bull. Cl. Sc., vol. 37, 1951, p. 229-240). | MR 12,830b | Zbl 0043.31403

[32]Guseva (O. V.). - On boundary problems for strongly elliptic systems (Doklady Akad. Nauk S. S. S. R., vol. 12, 1955, p. 1069-1072). | MR 17,161a | Zbl 0064.09705

[33]Haar (A.), - Zur Variationsrechnung (Abh. Math. Sem. Hamburgischen Univ. vol. 8, 1930, p. 1-27). | JFM 56.0437.03

[34]Hartman (P) et Wintner (A.). - On the local behavior of solutions of non-parabolic partial differential equations (Amer. J. Math., vol. 75, 1953, p. 449-476). | MR 15,318b | Zbl 0052.32201

[35]Hartman (P.) et Wintner (A.). - On the tocal behavior of solutions of non-parabolic partial differential equations. III : Approximations by spherical means (Amer. J. Math., vol. 77, 1955, p. 453-483). | MR 17,855a | Zbl 0066.08001

[36]Heinz (E.). - Über die Eindeutigkeit beim Cauchyschen Anfangswertproblem einer elliptischen Differentialgleichung zweiter Ordnung (Nachr. Akad. Wiss. Göttingen, Math. Phys., Kl. II a, n° 1, 1955, p. 1-12). | MR 17,626c | Zbl 0067.07503

[37]Hörmander (L.). - On the theory of general partial differential operators (Acta Math., vol. 94, 1955, p. 161-248). | MR 17,853d | Zbl 0067.32201

[38]John (F.). - On linear partial differential equations with analytic coefficients (Comm. pure appl. Math., vol. 2, 1949, p. 209-253). | MR 12,185d | Zbl 0035.34601

[39]John (F.). - The fundamental solution of linear elliptic differential equations with analytic coefficients (Comm. pure appl. Math., vol. 3, 1950, p. 273-304). | MR 13,40h | Zbl 0041.06203

[40]John (F.). - Derivatives of continuous weak solutions of linear elliptic equations (Comm. pure appl. Math., vol. 6, 1953, p. 327-335). | MR 15,431a | Zbl 0051.32801

[41]John (F.). - Plane waves and spherical means applied to partial differential equations, New-York, Interscience Publ., 1955. | MR 17,746d | Zbl 0067.32101

[42]Lax (P. D.). - On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations (Comm. pure appl. Math., vol. 8, 1955, p. 615-633). | MR 17,1212c | Zbl 0067.07502

[43]Lax (P. D.). - A stability theorem for solutions of abstract differential equations and its application to the study of the local behavior of solutions of elliptic equations (Comm. pure appl. Math., vol. 9, 1956, p. 747-766). | MR 19,281a | Zbl 0072.33004

[44]Leray (J.) et Schauder (J.). - Topologie et équations fonctionnelles (Ann. scient. Éc. Norm. Sup., t. 51, 1934, p. 45-78). | JFM 60.0322.02 | Numdam | Zbl 0009.07301

[45]Lions (J. L.). - Sur les problèmes de dérivées obliques (C. R. Acad. Sc. Paris, t. 240, 1955, p. 266-268). | MR 16,1028d | Zbl 0064.10004

[46]Lions (J. L.). - Problèmes aux limites en théorie des distributions (Acta Math., vol. 94, 1955, p. 13-153). | MR 17,745d | Zbl 0068.30902

[47]Lopatinskii (Ya. B.). - On a method of reducing boundary problems for a system of differential equations of elliptic type to regular equations (Ukrain. Mat. Ž., t. 5, 1953, p. 123-151). | MR 17,494b

[48]Malgrange (B.). - Exixtence et approximation des solutions des équations aux dérivées partielles et des équations de convolution (Thèse Sc. math., Paris, 1955).

[49]Milgram (A.) et Rosenbloom (P. C.). - Harmonic forms and heat conduction. | Zbl 0044.31703

Milgram (A.) et Rosenbloom (P. C.) I : Closed Riemannian manifols ;

Milgram (A.) et Rosenbloom (P. C.) II : Heat distribution on complexes and approximation theory (Proc. Nat. Acad. Sc. U. S. A., vol. 37, 1951, p. 180-184 et 435-438). | MR 13,160a | Zbl 0054.04202

[50]Miranda (C). - Formule di maggiorazione e teorema di esistenza per le funzioni biarmoniche di due variabili (Giorn. Mat. Battaglini, vol. 78, 1948-1949, p. 97-118). | MR 10,706f | Zbl 0037.07103

[51]Miranda (C.). - Sui sisteme di tipo ellittico di equazioni lineari a derivate parziali del primo ordine, in n variabili indipendenti (Atti delle Ac. naz. dei Lincei., 8e série, 1 a, vol. 3, 1952, p. 85-121). | MR 14,175b | Zbl 0049.35001

[52]Miranda (C.). - Equazioni alle derivate parziali di tipo ellitico, Berlin, Springer, 1955. | Zbl 0065.08503

[53] Morrey (C. B.). - Multiple integral problems in the calculus of variations and related topics (Univ. Calif. Publ. Math., nouv. série, vol. 1, 1943, p. 1-130). | MR 6,180b | Zbl 0063.04107

[54] Morrey (C. B.). - Second order elliptic systems of differential equations (Contribution to the theory of partial differential equations, Ann. math. Studies n° 33, Princeton, 1954, p. 101-159). | MR 16,827e | Zbl 0057.08301

[55] Morrey (C. B.). - A variational method in the theory of harmonic integrals, II (Amer. J. Math., vol. 78, 1956, p. 137-170). | MR 19,408a | Zbl 0070.31402

[56] Morrey (C. B.) et Eells (J.). - A variational method in the theory of harmonic integrals, I (Ann. of Math., vol. 63, 1956, p. 91-128). | MR 19,407b | Zbl 0070.09901

[57] Morrey (C. B.) et Nirenberg (L.). - On the analyticity of the solutions of linear elliptic systems of partial differential equations (Comm. pure appl. Math., vol. 10, 1957, p. 271-290). | MR 19,654b | Zbl 0082.09402

[58] Müller (C). - On the behavior of solutions of the differential equation Δu = F(x, u) in the neighbourhood of a point (Comm. pure appl. Math., vol. 7, 1954, p. 505-515). | Zbl 0056.32201

[59] Nash (J). - Parabolic equations (Proc. nat. Acad. Sc. U.S.A., vol. 43, 1957, p. 754-758). | MR 19,749b | Zbl 0078.08704

[60] Nirenberg (L.). - On nonlinear elliptic partial differential equation and Hölder continuity (Comm. pure appl. Math., vol. 6, 1953, p. 103-156). | MR 16,367c | Zbl 0050.09801

[61] Nirenberg (L.). - On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations (Contribution to the theory of partial differential equations, Ann. math. studies, n° 33, Princeton, 1954, p. 95-100). | MR 16,592a | Zbl 0057.08604

[62] Nirenberg (L). - Remarks on strongly elliptic partial differential equations (Comm. pure appl. Math., vol. 8, 1955, p. 649-676). | MR 17,742d | Zbl 0067.07602

[63] Nirenberg (L.). - Estimates and existence of solutions of elliptic equations (Comm. pure appl. Math., vol. 9, 1956, p. 509-552). | MR 19,962b | Zbl 0070.32301

[64] Nirenberg (L.). - Uniqueness in Cauchy problems for differential equations with constant leading coefficients (Comm. pure appl. Math., vol. 10, 1957, p. 89-105). | MR 19,147a | Zbl 0077.09402

[65] Pleijel (A.). - On Green's functions for elastic plates with clamped, supported and free edges (Proc. Symposium on spectral theory and differential problems, Oklahoma, 1951, p. 413-437). | MR 13,948f | Zbl 0068.08502

[66] Plis (A.). - The problem of uniqueness for the solution of a system of partial differential equations (Bull. Acad. polon. Sc., Cl. III, 2, 1954, p. 55-57). | MR 16,258a | Zbl 0055.32304

[67] Rado (T.). - Über zweidimensionale reguläre Variationsprobleme ... (Math. Ann., vol. 101, 1929, p. 620-632). | JFM 55.0899.01

[68] Schauder (J.). - Über lineare elliptische Differentialgleichungen zweiter Ordnung (Math. Z., vol. 38, 1934, p. 257-282). | JFM 60.0422.01 | Zbl 0008.25502

[69] Schauder (J.). - Numerische Abschätzungen in elliptischen linearen Differentialgleichungen (Stud. Math., vol. 5, 1934, p. 34-42). | JFM 60.1129.03 | Zbl 0010.20701

[70] Schwartz (L.). - Théorie des distributions, Paris, Hermann, 1950-1951. | Zbl 0037.07301

[71] Sobolev (S. L.). - On a theorem of functional analysis (Mat. Sbornik, nouv. série, vol. 4, 1938, p. 471-497). | JFM 64.1100.02 | Zbl 0131.11501 | Zbl 0022.14803

[72] Van Hove (L.). - Sur l'extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues (Indag. Math., vol. 7, n° 1, 1947, p. 3-8). | MR 8,522a | Zbl 0029.26802

[73] Van Hove (L.). - Sur le signe de la variation seconde des intégrales multiples à plusieurs fonctions inconnues (Acad. Roy. Belgique, Cl. Sc., Mémoires, vol. 24, fasc. 5, p. 1-68). | MR 11,730g | Zbl 0036.34501

[74] Vekua (N). - On a method of solving boundary value problems for partial differential equations (Doklady Akad. Nauk. S. S. S. R., vol. 101, 1955, p. 593-596). | Zbl 0064.34601

[75] Vishik (M. I.). - The method of orthogonal and direct decomposition in the theory of elliptic differential equations (Math. Sbornik, nouv. série, vol. 25, 1949, p. 189-234). | MR 11,520e | Zbl 0039.32003

[76] Vishik (M. I.). - On strongly elliptic systems of differential equations (Mat. Sbornik, nouv. série, vol. 29, 1951, p. 615-676). | MR 14,174a | Zbl 0044.09503

[77] Vishik (M. I.). - On general boundary problems for elliptic differential equations (Trudy Moskov. Mat. Obsč., vol. 1, 1952, p. 187-246). | MR 14,473d | Zbl 0131.32301

[78] Von Neumann (J.). - Über einen Hilfssatz der Variationsrechnung (Abh. Mat. Sem. Hamburgischen Univ., vol. 8, 1930, p. 28-31). | JFM 56.0440.04

[79] Weyl (H.). - The method of orthogonal projection in potential theory (Duke Math. J., vol. 7, 1940, p. 411-444). | JFM 66.0444.01 | MR 2,202a | Zbl 0026.02001