×

zbMATH — the first resource for mathematics

Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. (English) Zbl 0552.35032
The authors extend the classical De Giorgi theorem by proving the Hölder regularity of the weak solutions of \(Lu=0\) where \(L=\sum^{n}_{i,j=1}\partial_ i(a_{i,j}\partial_ j)\) is a degenerate ellipic operator in divergence form; precisely they prove the Theorem: Let \(\Omega\) be a \(\lambda\)-connected (i.e. for every \(x,y\in R^ n\) it is possible to join x and y by a continuous curve which is a piecewise integral curve of the vector fields \(\pm \lambda_ 1\partial_ 1,...,\pm \lambda_ n\partial_ n)\) open subset of \(R^ n\). If \(u\in W_{\lambda}^{loc}(\Omega)\) and \(L(u)=0\) in \(\Omega\) then u is locally Hölder-continuous in \(\Omega\). \((\lambda_ 1,\lambda_ 2,...,\lambda_ n\) are real continuous nonnegative functions such that the quadratic form \(\sum^{n}_{j=1}\lambda^ 2_ j(x)\xi^ 2_ j\) is equivalent to \(\sum^{n}_{i,j=1}a_{i,j}n(x)\xi_ i\xi_ j\) and, in addition, satisfy suitable conditions.)
Reviewer: R.Salvi

MSC:
35J70 Degenerate elliptic equations
35J15 Second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] N. Burger , Espace des fonctions à variation moyenne bornée sur un espace de nature homogène , C. R. Acad. Sci. Paris Sér. A , 236 ( 1978 ), pp. 139 - 142 . MR 467176 | Zbl 0368.46037 · Zbl 0368.46037
[2] H. Busemann , The Geometry of Geodesics , Academic Press , New York , 1955 . MR 75623 | Zbl 0112.37002 · Zbl 0112.37002
[3] R.R. Coifman - G. Weiss , Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes , Springer , Berlin - Heidelberg - New York , 1971 . MR 499948 | Zbl 0224.43006 · Zbl 0224.43006 · doi:10.1007/BFb0058946
[4] R.R. Coifman - G. Weiss , Extensions of Hardy Spaces and Their Use in Analysis , Bull. Amer. Math. Soc. , 83 ( 1977 ), pp. 569 - 645 . Article | MR 447954 | Zbl 0358.30023 · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5 · minidml.mathdoc.fr
[5] E. De Giorgi , Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari , Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. , 3 ( 3 ) ( 1957 ), pp. 25 - 43 . MR 93649 | Zbl 0084.31901 · Zbl 0084.31901
[6] E.B. Fabes - C.E. Kenig - R.P. Serapioni , The Local Regularity of Solutions of Degenerate Elliptic Equations , Comm. Partial Differential Equations 7 ( 1 ) ( 1982 ), pp. 77 - 116 . MR 643158 | Zbl 0498.35042 · Zbl 0498.35042 · doi:10.1080/03605308208820218
[7] C. Fefferman - D. Phong , Subelliptic Eigenvalue Problems , Preprint 1981 . MR 730094 · Zbl 0503.35071
[8] B. Franchi - E. Lanconelli , De Giorgi’s Theorem for a Class of Strongly Degenerate Elliptic Equations , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. , 72 ( 8 ) ( 1982 ), pp. 273 - 277 . MR 728257 | Zbl 0543.35041 · Zbl 0543.35041
[9] B. Franchi - E. Lanconelli , Une métrique associée à une classe d’opérateurs elliptiques dégénérés , Proceedings of the meeting Linear Partial and Pseudo Differential Operators , Torino ( 1982 ), Rend. Sem. Mat. Univ. e Politec . Torino , to appear. MR 728257 | Zbl 0553.35033 · Zbl 0553.35033
[10] B. Franchi - E. Lanconelli , An Embedding Theorem for Sobolev Spaces Related to Non-Smooth Vector Fields and Harnack Inequality , to appear. Zbl 0589.46023 · Zbl 0589.46023 · doi:10.1080/03605308408820362
[11] Gilbarg - N.S. Trudinger , Elliptic Partial Differential Equations of Second Order , Springer , Berlin - Heidelberg - New York , 1977 . MR 473443 | Zbl 0361.35003 · Zbl 0361.35003
[12] L. Hörmander , Hypoelliptic Second-Order Differential Equations , Acta Math. 119 ( 1967 ), pp. 147 - 171 . MR 222474 | Zbl 0156.10701 · Zbl 0156.10701 · doi:10.1007/BF02392081
[13] I.M. Kolodii , Qualitative Properties of the Generalized Solutions of Degenerate Elliptic Equations , Ukrain. Math. Z. , 27 ( 1975 ), pp. 320 - 328 = Ukrainian Math. J. , 27 ( 1975 ), pp. 256 - 263 . MR 412576 | Zbl 0337.35009 · Zbl 0337.35009 · doi:10.1007/BF01092093
[14] S.N. Kruzkov , Certain Properties of Solutions to Elliptic Equations , Dokl. Akad. Nauk SSSR , 150 ( 1963 ), pp. 470 - 473 = Soviet Math. Dokl ., 4 ( 1963 ), pp. 686 - 690 . MR 150442 | Zbl 0148.35701 · Zbl 0148.35701
[15] J. Moser , A New Proof of De Giorgi’s Theorem Concerning the Regularity Probem for Elliptic Differential Equations , Comm. Pure Appl. Math. , 13 ( 1960 ), pp. 457 - 468 . MR 170091 | Zbl 0111.09301 · Zbl 0111.09301 · doi:10.1002/cpa.3160130308
[16] M.K.V. Murthy - G. Stampacchia , Boundary Value Problems for Some Degenerate-Elliptic Operators , Ann. Mat. Pura Appl. , 80 ( 4 ) ( 1968 ), pp. 1 - 122 . MR 249828 | Zbl 0185.19201 · Zbl 0185.19201 · doi:10.1007/BF02413623
[17] J. Nash , Continuity of Solutions of Parabolic and Elliptic Equations , Amer. J. Math. , 80 ( 1958 ), pp. 931 - 954 . MR 100158 | Zbl 0096.06902 · Zbl 0096.06902 · doi:10.2307/2372841
[18] N.S. Trudinger , Linear Elliptic Operators with Measurable Coefficients , Ann. Scuola Norm. Sup. Pisa , ( 3 ) 27 ( 1973 ), pp. 265 - 308 . Numdam | MR 369884 | Zbl 0279.35025 · Zbl 0279.35025 · numdam:ASNSP_1973_3_27_2_265_0 · eudml:83635
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.