By Ian W. Knowles, Roger T. Lewis
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Extra info for Differential Equations: Conference Proceedings
Similarly when G is abelian one can establish a representation theorem for positive solutions of Pu = 0 in M which generalizes the representation theorem given at the end of section 2. 1 need not be true. To this end we assume now that M is the universal cover of the We shall also take the compact Riemannian manifold M so that G = nl(M). operator P to be -A where A is the Laplace-Beltrami operator on M. We denote : by A the bottom of the spectrum of -A A = inf I - M 2 IVV\ / I _ 1v12 1. o M where the infimum is taken over all smooth functions v M.
262 (1980) 551-563. S . Kamin, S i m i l a r s o l u t i o n s and t h e a s y m p t o t i c s of f i l t r a t i o n e q u a t i o n , Arch. B a t l . Mech. Anal. (1976) 171-183. V. , The sphericallysymmetric three-dimensional case is treated by the methods of ordinary differential equations using an outgoing wave boundary condition to define resonance energies. A detailed analysis of the associated Riccati equation, S(x) = 1 - (Wn(x) - E)S(Xl2, plays a major role in our approach; here W (XI is the barri- er potential which is assumed to approach infinity as n for almost all x in the barrier region.
I = 1,. ,n) i s a continuous F. map which t a k e s K i n t o i t s e l f . Hence, applying t h e Schauder-Tychonoff f i x e d p o i n t theorem [ 1 3 ] t o F1 we conclude t h a t F1 has a f i x e d p o i n t i n K which implies t h a t there e x i s t s a positive solution X1. some p o s i t i v e number subset of N e x t denote by c o n s i s t i n g of a l l K commutes with in K u TIU1 = Xlul for t h e non-empty closed convex K1 such t h a t F2 maps i t follows t h a t T~ such t h a t u1 X u.
Differential Equations: Conference Proceedings by Ian W. Knowles, Roger T. Lewis