Continuity of solutions of linear, degenerate elliptic equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, p. 103-116
We consider the simplest form of a second order, linear, degenerate, elliptic equation with divergence structure in the plane. Under an integrability condition on the degenerate function, we prove that the solutions are continuous.
Classification:  30C60,  35J15,  35J70
@article{ASNSP_2007_5_6_1_103_0,
     author = {Onninen, Jani and Zhong, Xiao},
     title = {Continuity of solutions of linear, degenerate elliptic equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {1},
     year = {2007},
     pages = {103-116},
     zbl = {1150.35055},
     mrnumber = {2341517},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_103_0}
}
Onninen, Jani; Zhong, Xiao. Continuity of solutions of linear, degenerate elliptic equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 103-116. http://www.numdam.org/item/ASNSP_2007_5_6_1_103_0/

[1] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 3 (1957), 25-43. | MR 93649 | Zbl 0084.31901

[2] E. De Giorgi, Congetture sulla continuità delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Unpublished, 1995.

[3] B. Franchi, R. Serapioni and F. Cassano, Irregular solutions of linear degenerate elliptic equations Potential Anal. 9 (1998), 201-216. | MR 1666899 | Zbl 0919.35050

[4] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order” 2nd ed., Springer-Verlag, New York, 1983. | MR 737190 | Zbl 0361.35003

[5] G. H. Hardy, J. E. Littlewood and G. Polya, “Inequalities”, 2nd ed., Cambridge University Press, Cambridge, 1952. | JFM 60.0169.01 | MR 46395 | Zbl 0634.26008

[6] T. Iwaniec, P. Koskela and J. Onninen, Mappings of finite distortion: monotonicity and continuity, Invent. Math. 144 (2001), 507-531. | MR 1833892 | Zbl 1006.30016

[7] O. A. Ladyzhenskaya and N. N. Ural'Tseva, “Linear and Quasilinear Elliptic Equations”, Academic Press, New York, 1968. | MR 244627 | Zbl 0164.13002

[8] H. Lebesgue, Sur le problème de Dirichlet. Rend. Circ. Mat. Palermo 27 (1907), 371-402. | JFM 38.0392.01

[9] J. J. Manfredi, Weakly monotone functions, J. Geom. Anal. 4 (1994), 393-402. | MR 1294334 | Zbl 0805.35013

[10] N. G. Meyers, An L p -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189-206. | Numdam | MR 159110 | Zbl 0127.31904

[11] C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166. | JFM 64.0460.02 | MR 1501936

[12] C. B. Morrey, Multiple integral problems in the calculus of variations and related topics. Univ. California Publ. Math. (N. S.) 1 (1943), 1-130. | MR 11537 | Zbl 0063.04107

[13] J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457-468. | MR 170091 | Zbl 0111.09301

[14] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. | MR 159138 | Zbl 0111.09302

[15] J. Nash, Continuity of solutions of elliptic and parabolic equations, Amer. J. Math. 80 (1958), 931-954. | MR 100158 | Zbl 0096.06902

[16] J. Onninen and X. Zhong, A note on mappings of finite distortion: the sharp modulus of continuity, Michigan Math. J. 53 (2005), 329-335. | MR 2152704 | Zbl 1086.30025

[17] L. C. Piccinini and S. Spagnolo, On the Hölder continuity of solutions of second order elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 391-402. | Numdam | MR 361422 | Zbl 0237.35028

[18] J. Serrin, On the strong maximum principle for quasilinear second order differential inequalities, J. Funct. Anal. 5 (1970), 184-193. | MR 259328 | Zbl 0188.41701

[19] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. | MR 463908 | Zbl 0353.46018

[20] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear, elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. | MR 226198 | Zbl 0153.42703

[21] N. S. Trudinger, On the regularity of generalized solutions of linear, non-uniformly elliptic equations, Arch. Rational Mech. Anal. 42 (1971), 51-62. | MR 344656 | Zbl 0218.35035

[22] K. O. Widman, On the Hölder continuity of solutions of elliptic partial differential equations in two variables with coefficients in L , Comm. Pure Appl. Math. 22 (1969), 669-682. | MR 251364 | Zbl 0183.11001