Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞
Tóm tắt
Here, all solutions of the form u=rkf(φ) to the p-harmonic equation, div(|∇u|p−2∇u)=0, (p>2) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions u=rkf(φ) of the limit equation for p=∞, u
x
2
uxx+2uxuyuxy+u
y
2
uyy=2, are constructed, some of which have a “strong” singularity at the origin (Theorem 2).
Tài liệu tham khảo
G. Aronsson, [AR1]: On the partial differential equation u 2x uxx+2uxuyuxy+u 2y uyy=0. Arkiv för matematik 7, 395–425 (1968)
— [AR2]: On certain singular solutions of the partial differential equation u 2x uxx+2uxuyuxy+u 2y uyy=0. manuscripta mathematica 47, 133–151 (1984)
H. Blum, [BL]: Zur Gitterverfeinerung für quasilineare Probleme auf Eckengebieten; Z. Angew. Math. u. Mech. 64, 264–266 (1984)
E. Di Benedetto, [DIB]: C1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis 7, 827–850 (1983)
M. Dobrowolski, [DOB]: Nichtlineare Eckenprobleme und Finite Elemente Methode, Z. Angew. Math. u. Mech. 64, 270–271 (1984)
M.S.P. Eastham, [EA]: Theory of Ordinary Differential Equations, van Nostrand, New York, 1970
L.C. Evans, [EV]: A new proof of local C1,α regularity for solutions of certain degenerate elliptic partial differential equations, J. Diff. Eq. 45, 356–373 (1982)
B. Fuglede, [FU]: A criterion of non-vanishing differential of a smooth map, Bull. London Math. Soc. 14, 98–102 (1982)
J. Lewis, [LE1]: Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66, 201–224 (1977)
— [LE2]: Smoothness of certain degenerate elliptic equations, Proc. Amer. Math. Soc. 80, 259–265 (1980)
— [LE3]: Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32, 849–858 (1983)
L.C. Piccinini, G. Stampacchia and G. Vidossich, [PSV]: Ordinary Differential Equations in Rn, Springer Verlag, New York, 1984
P. Tolksdorf, [TO]: Regularity for a more general class of quasilinear elliptic equations. Manuscript, Bonn 1982 or -83, to appear
K. Uhlenbeck, [UH]: Regularity for a class of non-linear elliptic systems, Acta Mathematica 138, 219–240 (1977)
N.N. Ural'tseva, [UR]: Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad Otdel. Mat. Inst. Steklov 7, 184–222 (1968) (in Russian)
B. Bojarski and T. Iwaniec. [BI]: p-harmonic Equation and Quasiregular Mappings, SFB 72, Report No 617, Bonn, 1983
S. Kichenassamy and L. Véron, [KV1]: Singularités isolées de l'équation div(|∇u|p−2∇u)=0, C.R. Acad. Sci. Paris 301, I, 149–151 (1985)
- [KV2]: Singular solutions of the p-Laplace equation. Manuscript, 1985