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.
@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}, pages = {103--116}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {1}, year = {2007}, mrnumber = {2341517}, zbl = {1150.35055}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2007_5_6_1_103_0/} }
TY - JOUR AU - Onninen, Jani AU - Zhong, Xiao TI - Continuity of solutions of linear, degenerate elliptic equations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 SP - 103 EP - 116 VL - 6 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2007_5_6_1_103_0/ LA - en ID - ASNSP_2007_5_6_1_103_0 ER -
%0 Journal Article %A Onninen, Jani %A Zhong, Xiao %T Continuity of solutions of linear, degenerate elliptic equations %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 103-116 %V 6 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2007_5_6_1_103_0/ %G en %F 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://archive.numdam.org/item/ASNSP_2007_5_6_1_103_0/
[1] 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 | Zbl
,[2] Congetture sulla continuità delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Unpublished, 1995.
,[3] Irregular solutions of linear degenerate elliptic equations Potential Anal. 9 (1998), 201-216. | MR | Zbl
, and ,[4] “Elliptic Partial Differential Equations of Second Order” 2nd ed., Springer-Verlag, New York, 1983. | MR | Zbl
and ,[5] “Inequalities”, 2nd ed., Cambridge University Press, Cambridge, 1952. | JFM | MR | Zbl
, and ,[6] Mappings of finite distortion: monotonicity and continuity, Invent. Math. 144 (2001), 507-531. | MR | Zbl
, and ,[7] “Linear and Quasilinear Elliptic Equations”, Academic Press, New York, 1968. | MR | Zbl
and ,[8] Sur le problème de Dirichlet. Rend. Circ. Mat. Palermo 27 (1907), 371-402. | JFM
,[9] Weakly monotone functions, J. Geom. Anal. 4 (1994), 393-402. | MR | Zbl
,[10] An -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 | Zbl
,[11] On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166. | JFM | MR
,[12] Multiple integral problems in the calculus of variations and related topics. Univ. California Publ. Math. (N. S.) 1 (1943), 1-130. | MR | Zbl
,[13] 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 | Zbl
,[14] On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. | MR | Zbl
,[15] Continuity of solutions of elliptic and parabolic equations, Amer. J. Math. 80 (1958), 931-954. | MR | Zbl
,[16] A note on mappings of finite distortion: the sharp modulus of continuity, Michigan Math. J. 53 (2005), 329-335. | MR | Zbl
and ,[17] 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 | Zbl
and ,[18] On the strong maximum principle for quasilinear second order differential inequalities, J. Funct. Anal. 5 (1970), 184-193. | MR | Zbl
,[19] Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. | MR | Zbl
,[20] On Harnack type inequalities and their application to quasilinear, elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. | MR | Zbl
,[21] On the regularity of generalized solutions of linear, non-uniformly elliptic equations, Arch. Rational Mech. Anal. 42 (1971), 51-62. | MR | Zbl
,[22] On the Hölder continuity of solutions of elliptic partial differential equations in two variables with coefficients in , Comm. Pure Appl. Math. 22 (1969), 669-682. | MR | Zbl
,