Radially Symmetric Functions as Fixed Points of some Logarithmic Operators
Tóm tắt
In this note we show that for f ∈ C((0,∞); R+) ∩ C1 ((0,∞)) with support in [0,∞), if a function u ∈ C1(R2) is such that support (u+) is compact and u(x) = ∫R2 f(u(y)) log 1/(|x-y|)dy ∀ x, then u is radial. This result is important for some free boundary problems in R2 or some axisymmetric ones in Rn.
Tài liệu tham khảo
Amick, C.J. and Frankel, L.E.: ‘The uniqueness of Hill's spherical vortex’, Arch. Rat. Mech. Anal. 92 (1986), 91–119.
Caffarelli, L.A. and Frieman, A.: ‘Asymptotic estimates for the plasma problem’, Duke Math. J. 47(3) (1980), 705–742.
Gilbarg, D. and Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, 1983.
Protter, M.H. and Weinberger, H.F.: Maximum principles in differential equations. Prince Hall Inc.
Tadie: ‘Problèmes elliptiques à frontière libre axi-symétriques: Estimation du diamétre de la section au moyen de la capacité’, Potential Anal. 5 (1996), 61–72.
Tadie: ‘On the bifurcation of steady vortex rings from a Green function’, Math. Proc. Camb. Phil. Soc. 116 (1994), 555–568.