×

zbMATH — the first resource for mathematics

Linear elliptic operators with measurable coefficients. (English) Zbl 0279.35025

MSC:
35J15 Second-order elliptic equations
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35B45 A priori estimates in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Campanato , S. , Proprietà di inclusione per spazi di Morrey , Ricerche Mat ., Vol. 12 ( 1963 ), 67 - 86 . MR 156228 | Zbl 0192.22703 · Zbl 0192.22703
[2] Chicco , M. , Principio di massimo forte per sottosoluzione di equazioni ellittiche di tipo variazionale , Boll. U.M.I. , Ser. 3 , Vol. 22 ( 1967 ), 368 - 372 . MR 224999 | Zbl 0149.32002 · Zbl 0149.32002 · eudml:195487
[3] Chicco , M. , Semicontinuità delle sottosoluzioni di equazioni ellittiche di tipo variazionale , Boll. U.M.I , Ser. 4 , Vol. 1 ( 1968 ), 548 553 . MR 235263 | Zbl 0159.39903 · Zbl 0159.39903
[4] Chicco , M. , Principio di massimo per soluzioni di problemi al contorno misti per equazioni ellittiche di tipo variazionale , Boll. U M.I. , Ser 4 , Vol. 3 ( 1970 ), 384 - 394 . MR 264218 | Zbl 0193.39102 · Zbl 0193.39102
[5] De Giorgi , E. , Sulla differenziabilità. e l’analiticità delle estremali degli integrali multipli regolari , Mem. Accad. Sci. Torino , Ser 3a , Vol. 3 ( 1957 ), 25 - 43 . MR 93649 | Zbl 0084.31901 · Zbl 0084.31901
[6] Donaldson , T.K. and Trudinger N.S. , Orlicz-Sobolev spaces and imbedding theorems , J . Functional Analysis , Vol. 8 ( 1971 ), 52 - 75 . MR 301500 | Zbl 0216.15702 · Zbl 0216.15702 · doi:10.1016/0022-1236(71)90018-8
[7] Hervé , R.M. and M. , Les fonctions surharmoniques associées à un opérateur. elliptique du second ordre a coefficients discontinus , Ann. Inst. Fourier , Vol. 19 ( 1969 ), 305 - 359 . Numdam | MR 261027 | Zbl 0176.09801 · Zbl 0176.09801 · doi:10.5802/aif.320 · numdam:AIF_1969__19_1_305_0 · eudml:73983
[8] John F. and Nirenberg L. , On functions of bounded mean oscillation , Comm. Pure Appl. Math . Vol. 14 ( 1961 ), 415 - 426 . MR 131498 | Zbl 0102.04302 · Zbl 0102.04302 · doi:10.1002/cpa.3160140317
[9] Kruzkov , S.N. , Certain properties of solutions of elliptic equations , (English translation) , Soviet Math . Vol. 4 ( 1963 ), 686 - 695 . Zbl 0148.35701 · Zbl 0148.35701
[10] Ladyzhenskaya O.A. and Ural’tseva N.N. , On the Hölder continuity of the solutions and their derivatives of linear and quasi linear equations of elliptic and parabolic type . (In Russian) . Trudy Mat. Inst. Steklov , Vol. 73 ( 1964 ), 173 - 190 . MR 173855
[11] Ladvzhenskaya O.A. and Ural’tseva N.N. , Linear and quasilinear equations of elliptic type , Izd. Nauka , Moscow , 1964 . English translation, Academic press , New York , 1968 . Zbl 0164.13002 · Zbl 0164.13002
[12] Littman , W. , Generalized subharmonic functions : Monotonic approximations and an improved maximum principle , Ann. Sc. Norm. Sup. Pisa , Vol. 17 ( 1963 ), 207 - 222 . Numdam | MR 177186 | Zbl 0123.29104 · Zbl 0123.29104 · numdam:ASNSP_1963_3_17_3_207_0 · eudml:83303
[13] Littman , W. , Stampacchia , G. , and Weinberger , H.F. , Regular points for elliptic equations with discontinuous coefficients , Ann. Sc. Norm. Sup. Pisa , Vol. 17 ( 1963 ), 43 - 77 . Numdam | MR 161019 | Zbl 0116.30302 · Zbl 0116.30302 · numdam:ASNSP_1963_3_17_1-2_43_0 · eudml:83299
[14] Meyers , G.N. and Serrin J.B. , H = W , Proc. Nat. Acad. Sci. , Vol. 51 ( 1964 ), 1055 - 1056 . MR 164252 | Zbl 0123.30501 · Zbl 0123.30501 · doi:10.1073/pnas.51.6.1055
[15] Morrey , C.B. Jr., Second order elliptic equations in several variables and Hölder continuity , Math. Zeit. , Vol. 72 ( 1959 ), 146 - 164 . MR 120446 | Zbl 0094.07802 · Zbl 0094.07802 · doi:10.1007/BF01162944 · eudml:183602
[16] Morrey , C.B. Jr., Mutiple integrals in the calculus of variations , Springer Verlag , New York , 1966 . MR 202511 | Zbl 0142.38701 · Zbl 0142.38701
[17] Moser , J.K. , A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations , Comm. Pure Appl. Math. , Vol. 13 ( 1960 ), 457 - 468 . MR 170091 | Zbl 0111.09301 · Zbl 0111.09301 · doi:10.1002/cpa.3160130308
[18] Moser , J.K. , On Harnack’s theorem for elliptic differential equations , Comm. Pure Appl. Math. , Vol. 14 ( 1961 ), 577 - 591 . MR 159138 | Zbl 0111.09302 · Zbl 0111.09302 · doi:10.1002/cpa.3160140329
[19] Moser , J.K. , On pointwise estimates for parabolic differential equations , Comm. Pure Appl. Math. , Vol. 24 ( 1971 ), 727 - 740 . MR 288405 | Zbl 0227.35016 · Zbl 0227.35016 · doi:10.1002/cpa.3160240507
[20] Murthy , M.K.V. , and Stampacchia , G. , Boundary value problems for some degenerate-elliptio operators , Ann. Mat. Pura Appl. , Vol. 80 ( 1968 ), 1 - 122 . MR 249828 | Zbl 0185.19201 · Zbl 0185.19201 · doi:10.1007/BF02413623
[21] Nash , J. , Continuity of solutions of elliptic and parabolic equations , Amer. J. Math. , Vol. 80 ( 1958 ), 931 - 954 . MR 100158 | Zbl 0096.06902 · Zbl 0096.06902 · doi:10.2307/2372841
[22] Serrin , J.B. , Local behavior of solutions of quasilinear elliptic equations , Acta Math. Vol. III ( 1964 ), 247 - 302 . MR 170096 | Zbl 0128.09101 · Zbl 0128.09101 · doi:10.1007/BF02391014
[23] Skaff , M.S. , Vector valued Orlicz spaces II , Pacific J. Math. , Vol. 28 ( 1969 ), 413 - 430 . Article | MR 415306 | Zbl 0176.11003 · Zbl 0176.11003 · doi:10.2140/pjm.1969.28.413 · minidml.mathdoc.fr
[24] Stampacchia , G. , Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellittiche , Ann. Sc. Norm. Sup. Pisa , Vol. 12 ( 1958 ), 223 - 245 . Numdam | MR 125313 | Zbl 0082.09701 · Zbl 0082.09701 · numdam:ASNSP_1958_3_12_3_223_0 · eudml:83215
[25] Stampacchia G. , Problemi al contorno ellittici con data discontinui dotati di soluzioni hölderiane , Ann. Mat. Pura Appl . Vol. 51 ( 1960 ), 1 - 38 . MR 126601 | Zbl 0204.42001 · Zbl 0204.42001 · doi:10.1007/BF02410941
[26] Stampacchia G. , Some limit cases of Lp-estimates for solutions of second order elliptic equations , Comm. Pure Appl. Math. Vol. 16 ( 1963 ), 505 - 510 . MR 152735 | Zbl 0147.09202 · Zbl 0147.09202 · doi:10.1002/cpa.3160160409
[27] Stampacchia G. , Le problème de Dirichlet pour les équations elliptique, du second ordre à coefficients discontinus , Ann. Inst. Fourier , Vol. 15 ( 1965 ), 189 - 258 . Numdam | MR 192177 | Zbl 0151.15401 · Zbl 0151.15401 · doi:10.5802/aif.204 · numdam:AIF_1965__15_1_189_0 · eudml:73861
[28] Trudinger N.S. , Second order linear and quasilinear elliptic equations in n-variables , Lecture notes , Stanford University 1971 .
[29] Trudinger N.S. , On Harnack type inequalities and their application to quasilinear elliptic equations , Comm. Pure Appl. Math. , Vol. 20 ( 1967 ), 721 - 747 . MR 226198 | Zbl 0153.42703 · Zbl 0153.42703 · doi:10.1002/cpa.3160200406
[30] Trudinger N.S. , On imbeddings into Orlicz spaces and some applications , J. Math. Mech. , Vol. 17 ( 1967 ), 473 - 484 . MR 216286 | Zbl 0163.36402 · Zbl 0163.36402
[31] Trudinger N.S. , On the regularity of generalized solutions of linear non-uniformly elliptic equations , Arch. Rat. Mech. Anal . Vol. 42 ( 1971 ), 51 - 62 . MR 344656 | Zbl 0218.35035 · Zbl 0218.35035 · doi:10.1007/BF00282317
[32] Trudinger N.S. , Generalized solutions of quasilinear, differential inequalities , I Elliptic operators , Bull. Amer. Math. Soc. , Vol. 77 ( 1971 ), 571 - 579 . Article | MR 277887 | Zbl 0235.35013 · Zbl 0235.35013 · doi:10.1090/S0002-9904-1971-12759-3 · minidml.mathdoc.fr
[33] Trudinger N.S. , Local and global behaviour of generalized solutions of quasilinear, differential inequalities , (to appear).
[34] Yosida , K. , Functional analysis , Springer-Verlag , New York , 1965 . · Zbl 0126.11504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.