Conformally invariant systems of nonlinear PDE of Liouville type
Tóm tắt
We establish a strict isoperimetric inequality and a Pohozaev-Rellich identity for the system
i∈I = {1,...,N} under certain reasonable conditions on the γιJ
andu
ι. Thus we prove that under these conditions, all solutionsu
ι are radial symmetric and decreasing about some point.
Tài liệu tham khảo
[B]C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, 1980.
[Be]W. Beckner, to appear.
[Ben]W. H. Bennett, Magnetically self-focussing streams, Phys. Rev. 45 (1934), 890–897
[BrMe]H. Brezis, F. Merle, Uniform estimates and blow-up behavior of solutions of −Δu=V(x)e u in two dimensions. Commun. PDE 16 (1991), 1223–1253.
[CLMP]E. Caglioti, P.-L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanical description, Commun. Math. Phys. 143 (1992), 501–525.
[CaLo]E.A. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality, and Onofri's inequality onS n, Geom. Funct. Anal. 2 (1992), 90–104.
[ChKi]S. Chanillo, M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Commun. Math. Phys. 160:2 (1994), 217–238.
[ChLi]S. Chanillo, Y.Y. Li, Continuity of solutions of uniformly elliptic equations in ℝ2, Manuscr. Math. 77 (1992), 415–433.
[CheLi]W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622.
[ChoW]K.S. Chou, T.Y.H. Wan, Asymptotic radial symmetry for solutions of Δu+e u=0 in a punctured disc, Pac. J. Math. 163 (1994), 269–276.
[ESp]G. Eyink, H. Spohn, Negative temperature states and large-scale long-lived vortices in two-dimensional turbulence, J. Stat. Phys. 70 (1993), 833–886.
[GNNi]B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 (1979), 209–243.
[GiT]D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, New York (1983).
[KP]C. Kesavan, F. Pacella, Symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities, Appl. Anal. 54 (1994), 27–37.
[Ki]M.K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math. 46 (1993), 27–56.
[KiL]M.K.-H. Kiessling, J.L. Lebowitz, Dissipative Stationary Plasmas: Kinetic Modeling, Bennett's Pinch, and generalizations, Phys. Plasmas1 (1994), 1841–1849.
[Li]C.-M. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. PDE 16 (1991), 585–615.
[Lio]P.-L. Lions, Two geometrical properties of solutions of semilinear problems, Appl. Anal. 12 (1981), 267–272.
[Liou]J. Liouville, Sur l'equation aux différences partielles ∂2logλ/∂u∂v±∂/2a 2 = 0, J. Math. 18 (1853), 71–72.
[O]M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Diff. Geom. 6 (1971), 247–258.
[On]E. Onofri, On the positivity of the effective action in a theory of random surfaces, Commun. Math. Phys. 86 (1982), 321–326.
[OsPhS]B. Osgood, R. Phillips, P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.
[Po]S.I. Pohozaev, Eigenfunctions of the equation Δu+λf(u)=0. Sov. Math. Dokl. 5 (1965), 1408–1411.
[R]F. Rellich, Darstellung der Eigenwerte von Δu+λu=0 durch ein Randwertintegral, Math. Z. 46 (1940), 635–636.