Previous |  Up |  Next

Article

Keywords:
subharmonic function; extension theorem
Summary:
This note verifies a conjecture of Král, that a continuously differentiable function, which is subharmonic outside its critical set, is subharmonic everywhere.
References:
[1] Armitage, D. H., Gardiner, S. J.: Classical Potential Theory. Springer Monographs in Mathematics. Springer, London (2001). DOI 10.1007/978-1-4471-0233-5 | MR 1801253 | Zbl 0972.31001
[2] Caffarelli, L. A., Cabré, X.: Fully Nonlinear Elliptic Equations. Colloquium Publications 43. AMS, Providence (1995). DOI 10.1090/coll/043 | MR 1351007 | Zbl 0834.35002
[3] Crandall, M. G., Ishii, H., Lions, P.-L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., New Ser. 27 (1992), 1-67. DOI 10.1090/S0273-0979-1992-00266-5 | MR 1118699 | Zbl 0755.35015
[4] Juutinen, P., Lindqvist, P.: A theorem of Radó's type for the solutions of a quasi-linear equation. Math. Res. Lett. 11 (2004), 31-34. DOI 10.4310/MRL.2004.v11.n1.a4 | MR 2046197 | Zbl 1153.35324
[5] Juutinen, P., Lindqvist, P., Manfredi, J. J.: On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. 33 (2001), 699-717. DOI 10.1137/S0036141000372179 | MR 1871417 | Zbl 0997.35022
[6] Král, J.: Some extension results concerning harmonic functions. J. Lond. Math. Soc., II. Ser. 28 (1983), 62-70. DOI 10.1112/jlms/s2-28.1.62 | MR 0703465 | Zbl 0526.31003
[7] Král, J.: A conjecture concerning subharmonic functions. Čas. Pěst. Mat. 110 (1985), page 415 Czech.
[8] Pokrovskiĭ, A. V.: A simple proof of the Radó and Král theorems on removability of the zero locus for analytic and harmonic functions. Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 2015 (2015), 29-31. DOI 10.15407/dopovidi2015.07.029 | MR 3718287 | Zbl 1340.30135
[9] Rudin, W.: Real and Complex Analysis. McGraw-Hill Book Co., New York (1987). MR 0924157 | Zbl 0925.00005
Partner of
EuDML logo