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Browder, Felix E.

On the spectral theory of elliptic differential operators. I. (English) Zbl 0104.07502

Math. Ann. 142, 22-130 (1961).
Reviewer: E. Heinz
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Cited in 1 Review
Cited in 217 Documents

MSC:

35P05 General topics in linear spectral theory for PDEs
35Jxx Elliptic equations and elliptic systems
47A10 Spectrum, resolvent
47F05 General theory of partial differential operators

Keywords:

partial differential equations
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Full Text: DOI EuDML

References:

[1] Agmon, S.: Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane. I. Communications Pure Appl. Math.10, 179-239 (1957). · Zbl 0081.09801 · doi:10.1002/cpa.3160100202
[2] Agmon, S.: The coerciveness problem for integro-differential forms. J. d’Analyse Math.6, 183-223 (1958). · Zbl 0119.32302 · doi:10.1007/BF02790236
[3] Browder, F. E.: The Dirichlet problem for linear elliptic equations of arbitrary even order with variable coefficients. Proc. Nat. Acad. Sci. U. S. A.,38, 230-235 (1952). · Zbl 0046.32302 · doi:10.1073/pnas.38.3.230
[4] Browder, F. E.: The Dirichlet and vibration problems for linear elliptic differential equations of arbitrary order. Proc. Nat. Acad. Sci.38, 741-747 (1952). · Zbl 0047.09501 · doi:10.1073/pnas.38.8.741
[5] Browder, F. E.: Strongly elliptic systems of differential equations. Ann. Math. Study33, 15-51 (1954). · Zbl 0057.32901
[6] Browder, F. E.: On the regularity properties of solutions of elliptic differential equations. Communs Pure Appl. Math.9, 351-361 (1956). · Zbl 0070.09601 · doi:10.1002/cpa.3160090307
[7] Browder, F. E.: La théorie spectrale des opérateurs aux dérivées partielles du type elliptique. C. R. Acad. Sci. Paris246, 526-528 (1958). · Zbl 0088.30503
[8] Browder, F. E.: Les opérateurs elliptiques et les problèmes mixtes. C. R. Acad. Sci. Paris246, 1363-1365 (1958). · Zbl 0088.30601
[9] Browder, F. E.: Eigenfunction expansions for nonsymmetric partial differential operators. I. Am. J. Math.80, 365-381 (1958). · Zbl 0084.30702 · doi:10.2307/2372790
[10] Browder, F. E.: Eigenfunction expansions for nonsymmetric partial differential operators. II. Am. J. Math.81, 1-22 (1959). · Zbl 0086.08102 · doi:10.2307/2372847
[11] Browder, F. E.: Eigenfunction expansions for nonsymmetric partial differential operators. III. Am. J. Math.81, 715-734 (1959). · Zbl 0103.31701 · doi:10.2307/2372924
[12] Browder, F. E.: On the eigenfunctions and eigenvalues of the general linear elliptic differential operator. Proc. Nat. Acad. Sci. U.S.A.39, 433-439 (1953). · Zbl 0050.32102 · doi:10.1073/pnas.39.5.433
[13] Browder, F. E.: On functional analysis and partial differential equations. I. Math. Ann.138, 55-79 (1959). · Zbl 0086.10301 · doi:10.1007/BF01369666
[14] Browder, F. E.: Estimates and existence theorems for elliptic boundary value problems. Proc. Nat. Acad. Sci.45, 365-372 (1959). · Zbl 0093.29402 · doi:10.1073/pnas.45.3.365
[15] Browder, F. E.: The spectral theory of strongly elliptic differential operators. Proc. Nat. Acad. Sci. U.S.A.,45, 1413-1421 (1959). · Zbl 0087.30403
[16] Browder, F. E.: A-priori estimates for solutions of elliptic boundary value problems. I., II. Koninkl. Ned. Akad. van Wetenschap,22, 145-159, 160-169 (1960). · Zbl 0096.30202
[17] Browder, F. E.: Functional analysis and partial differential equations, III. To appear. · Zbl 0191.44002
[18] Cacciopoli, R.: Limitazioni integrali per le soluzioni di un’equazione lineare ellitica a derivate parziali. Giorn. Mat. Battaglini80, 186-212 (1951). · Zbl 0043.31404
[19] Calderon, A. P.: Uniqueness in the Cauchy problem for partial differential equations. Am. J. Math.80, 16-36 (1958). · Zbl 0080.30302 · doi:10.2307/2372819
[20] Calderon, A. P., andA. Zygmund: On the existence of certain singular integrals. Acta Math.88, 85-139 (1952). · Zbl 0047.10201 · doi:10.1007/BF02392130
[21] Calderon, A. P., andA. Zygmund: On singular integrals. Am. J. Math.78, 289-309 (1956). · Zbl 0072.11501 · doi:10.2307/2372517
[22] Douglis, A., andL. Nirenberg: Interior estimates for elliptic systems of partial differential equations. Communs Pure Appl. Math.8, 503-538 (1955). · Zbl 0066.08002 · doi:10.1002/cpa.3160080406
[23] Dunford, N., andJ. T. Schwartz: Linear Operators, Vol. I. New York 1958. · Zbl 0084.10402
[24] DuPlessis, N.: Some theorems on the Riesz fractional integral. Trans. Am. Math. Soc.80, 124-134 (1955). · Zbl 0067.28101 · doi:10.1090/S0002-9947-1955-0086938-3
[25] Fichera, G.: Su un principio di dualità per talune formule di maggiorazione relativa alle equazioni differenziali. Rend. Acad. Naz. Lincei19, 411-418 (1955). · Zbl 0071.31801
[26] Friedrichs, K. O.: On the differentiability of solutions of linear elliptic differential equations. Communs Pure Appl. Math.6, 299-326 (1953). · Zbl 0051.32703 · doi:10.1002/cpa.3160060301
[27] Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. RicercheMa t.7, 102-137 (1958). · Zbl 0089.09401
[28] Gagliardo, E.: Un osservazione sul problema di Dirichlet per un equazione lineare ellitica del secondo ordine. Rend. Accad. Napoli25, 1-15 (1958).
[29] Garding, L.: Le problème de Dirichlet pour les équations aux dérivées partielles elliptiques homogènes à coefficients constants. C. R. Acad. Sci. Paris230, 1030-1032 (1950). · Zbl 0035.34704
[30] Garding, L.: Le problème de Dirichlet pour les équations aux dérivées partielles elliptiques linéaires dans les domaines bornés. C. R. Acad. Sci. Paris233, 1554-1556 (1951).
[31] Garding, L.: Dirichlet’s problem for linear elliptic partial differential equations. Mathematica scand.1, 55-72 (1953). · Zbl 0053.39101
[32] Grothendieck, A.: La Théorie de Fredholm. Bull. Soc. Math. France84, 319-384 (1956). · Zbl 0073.10101
[33] Guseva, O. V.: On boundary problems for strongly elliptic systems. Doklady Akad. Nauk S.S.S.R.102, 1069-1072 (1955).
[34] Hille, E., andR. S. Phillips: Functional Analysis and Semi-Groups. Am. Math. Soc. Colloquium Publications31, Providence 1957. · Zbl 0078.10004
[35] Hopf, E.: Über den funktionalen, insbesondere den analytischen Charakter der Lösungen elliptischer Differentialgleichungen zweiter Ordnung. Math. Z.34, 191-233 (1931). · Zbl 0002.34003
[36] Hörmander, L.: On the regularity of the solutions of boundary problems. Acta Math.99, 225-264 (1958). · Zbl 0083.09201 · doi:10.1007/BF02392427
[37] John, F.: General properties of solutions of linear elliptic partial differential equations. Proc. Symposium on Spectral Theory and Differential Problems. Oklahoma: Still-water 1951.
[38] Kato, T.: Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Am. Math. Soc.70, 195-211 (1951). · Zbl 0044.42701
[39] Kato, T.: Perturbation of continuous spectra by trace class operators. Proc. Japan Acad.33, 260-264 (1957). · Zbl 0090.33101 · doi:10.3792/pja/1195525063
[40] Koshelev, A. I.: On the boundedness inL p of the derivatives of solutions of elliptic differential equations. Mat. Sbornik38, 359-372 (1956).
[41] Koshelev, A. I.: On a-priori estimates inL p of generalized solutions of elliptic equations and systems. Uspekhi Mat. Nauk13, 29-88 (1958).
[42] Kuroda, S. T.: Perturbation of continuous spectrum by unbounded operators. To appear. · Zbl 0094.09303
[43] Ladyzenskaya, O.: On the closure of an elliptic operator. Doklady Akad. Nauk S.S.S.R.79, 723-725 (1951).
[44] Lions, J. L.: Problèmes aux limites en théorie des distributions. Acta Math.94, 13-153 (1955). · Zbl 0068.30902 · doi:10.1007/BF02392490
[45] Lions, J. L.: Lectures on elliptic partial differential equations. Tata Institute lecture notes. Bombay 1957.
[46] Marcinkiewicz, J.: Sur les multiplicateurs des séries de Fourier. Studia Math.8, 78-91 (1939). · Zbl 0020.35403
[47] Miranda, C.: Equazioni alle derivate parziali di tipo ellittico. Ergeb. Math. N.S.2, Berlin (1955). · Zbl 0065.08503
[48] Miranda, C.: Teorema del massimo modulo e teorema di esistenza e di unicità per il problema di Dirichlet relativo alle equazioni ellitiche in due variabili. Ann. Mat. pur. appl.46, 265-312 (1958). · Zbl 0093.29403 · doi:10.1007/BF02412919
[49] Morrey jr., C. B.: Second-order elliptic systems of differential equations. Ann. Math. Study33, 101-159 (1954). · Zbl 0057.08301
[50] Morrey jr., C. B.: On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. II. Am. J. Math.80, 219-237 (1958). · Zbl 0081.09402 · doi:10.2307/2372831
[51] Morrey jr., C. B., andL. Nirenberg: On the analyticity of solutions of linear elliptic systems of partial differential equations. Communs Pure Appl. Math.10, 271-290 (1957). · Zbl 0082.09402 · doi:10.1002/cpa.3160100204
[52] Nirenberg, L.: Remarks on strongly elliptic partial differential equations. Communs Pure Appl. Math.8, 649-675 (1955). · Zbl 0067.07602
[53] Nirenberg, L.: Estimates and existence of solutions of elliptic equations. Communs Pure Appl. Math.9, 509-530 (1956). · Zbl 0070.32301 · doi:10.1002/cpa.3160090322
[54] Rosenblum, M.: Perturbation of the continuous spectrum and unitary equivalence. Pacific J. Math.7, 997-1010 (1957). · Zbl 0081.12003
[55] Ruston, A. F.: The Fredholm theory of integral equations for operators belonging to the trace class of a general Banach space. Proc. London Math. Soc.53, 109-124 (1951). · Zbl 0054.04906 · doi:10.1112/plms/s2-53.2.109
[56] Schauder, J.: Über lineare elliptische Differentialgleichungen zweiter Ordnung. Math. Z.38, 257-282 (1934). · Zbl 0008.25502 · doi:10.1007/BF01170635
[57] Schauder, J.: Sur les équations linéaires du type elliptique à coefficients continus. C. R. Acad. Sci. Paris199, 1366-1368 (1934). · Zbl 0010.35502
[58] Schechter, M.: On estimating elliptic partial differential operators in theL 2-norm. Am. J. Math.79, 431-443 (1957). · Zbl 0079.11701 · doi:10.2307/2372690
[59] Schechter, M.: Coerciveness of linear partial differential operators for functions satisfying zero Dirichlet-type boundary data. Communs Pure Appl. Math.11, 153-174 (1958). · Zbl 0101.31202 · doi:10.1002/cpa.3160110202
[60] Schechter, M.: Solution of the Dirichlet problem for equations not necessarily strongly elliptic. Bull. Am. Math. Soc.64, 371-372 (1958). · Zbl 0087.30202 · doi:10.1090/S0002-9904-1958-10239-6
[61] Schechter, M.: Integral inequalities for partial differential operators and functions satisfying general boundary conditions. Communs Pure Appl. Math.12, 37-66 (1959). · Zbl 0093.29401 · doi:10.1002/cpa.3160120104
[62] Schechter, M.: General boundary value problems for elliptic partial differential equations. Bull. Am. Math. Soc.65, 70-72 (1959). · Zbl 0087.30203 · doi:10.1090/S0002-9904-1959-10279-2
[63] Schwartz, L.: Théorie des distributions. Paris 1951. · Zbl 0042.11405
[64] Smulyan, Yu. L.: Completely continuous perturbations of operators. Doklady Akad. Nauk S.S.S.R. (N.S.)101, 35-38 (1955) (Am. Math. Soc. Translations, Series 2, 10).
[65] Sobolev, S. L.: On a theorem of functional analysis. Mat. Sbornik N.S.4, 471-497 (1938).
[66] Sobolev, S. L.: Some applications of functional analysis to mathematical physics. Leningrad 1950. · Zbl 0041.52307
[67] Stummel, F.: Singuläre elliptische Differentialoperatoren in Hilbertschen Räumen. Math. Ann.132, 150-176 (1956). · Zbl 0070.34603 · doi:10.1007/BF01452327
[68] van Hove, L.: Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues. Koninkl. Ned. Akad. van Wetenschap50, 18-23 (1947). · Zbl 0029.26802
[69] Visik, M. I.: On strongly elliptic systems of differential equations. Mat. Sbornik29, 615-676 (1951).
[70] Wienholtz, E.: Halbbeschränkte partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus. Math. Ann.135, 50-80 (1958). · Zbl 0142.37701 · doi:10.1007/BF01350827
[71] Wolf, F.: Essential spectrum of partial differential boundary problems. Communs Pure Appl. Math.12, 211-228 (1959). · Zbl 0087.30501 · doi:10.1002/cpa.3160120202
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