Geodesic
On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 1, pp. 76-79.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that, for any compact set $K\subset\mathbb{R}^n$ ($n\ge 2$) of positive Lebesgue measure and any bounded domain $G\supset K$, there exists a function in the Hölder class $C^{1, 1}(G)$ that is a solution of the minimal surface equation in $G\setminus K$ and cannot be extended from $G\setminus K$ to $G$ as a solution of this equation.
Keywords: minimal surface equation, Hölder class, removable set, nonlinear mapping, Schauder theorem, fixed point. 
@article{FAA_2018_52_1_a8,
     author = {A. V. Pokrovskii},
     title = {On {Singular} {Points} of {Solutions} of the {Minimal} {Surface} {Equation} on {Sets} of {Positive} {Measure}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {76--79},
     publisher = {mathdoc},
     volume = {52},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2018_52_1_a8/}
}
TY  - JOUR
AU  - A. V. Pokrovskii
TI  - On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2018
SP  - 76
EP  - 79
VL  - 52
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2018_52_1_a8/
LA  - ru
ID  - FAA_2018_52_1_a8
ER  - 
%0 Journal Article
%A A. V. Pokrovskii
%T On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2018
%P 76-79
%V 52
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2018_52_1_a8/
%G ru
%F FAA_2018_52_1_a8
A. V. Pokrovskii. On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 1, pp. 76-79. http://geodesic.mathdoc.fr/item/FAA_2018_52_1_a8/

[1] E. P. Dolzhenko, UMN, 18:4(112) (1963), 135–142 | MR | Zbl

[2] Nguen Xuan Uy, Ark. Mat., 17:1 (1979), 19–27 | MR

[3] S. V. Khruschev, Zap. nauchn. sem. LOMI, 113 (1981), 199–203

[4] L. Carleson, Math. Scand., 12 (1963), 15–18 | DOI | MR | Zbl

[5] E. P. Dolzhenko, Izv. AN SSSR, ser. matem., 28:5 (1964), 1113–1130 | Zbl

[6] E. P. Dolzhenko, Izv. AN SSSR, ser. matem., 28:6 (1964), 1251–1270 | Zbl

[7] J. Verdera, Duke Math. J., 55:1 (1987), 157–187 | DOI | MR | Zbl

[8] A. V. Pokrovskii, Trudy mezhdunarodnoi konferentsii po differentsialnym uravneniyam i dinamicheskim sistemam (Suzdal, 2006), Sovremennaya matematika i ee prilozheniya, 57, 2007, 54–72

[9] A. V. Pokrovskii, Funkts. analiz i ego pril., 39:4 (2005), 62–68 | DOI | MR | Zbl

[10] L. Bers, Ann. of Math., 53 (1951), 364–386 | DOI | MR | Zbl

[11] E. De Giorgi, G. Stampacchia, Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., 38 (1965), 352–357 | MR | Zbl

[12] M. Miranda, “Sulla singolarità eliminabili delle soluzioni dell'equazione delle superfici minime”, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., Ser. IV, 4:1 (1977), 129–132 | MR | Zbl

[13] L. Veron, Singularities of Solutions of Second Order Quasilinear Elliptic Equations, Addison Wesley Longman, London, 1996

[14] H. Brezis, L. Nirenberg, Topol. Methods Nonlinear Anal., 9:2 (1997), 201–219 | DOI | MR | Zbl

[15] E. Dzhusti, Minimalnye poverkhnosti i funktsii ogranichennoi variatsii, Mir, M., 1989 | MR

[16] D. Gilbarg, N. Trudinger, Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR