Differential Equations, Proceedings of the Conference held by Ian W. Knowles and Roger T. Lewis (Eds.)

By Ian W. Knowles and Roger T. Lewis (Eds.)

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Extra info for Differential Equations, Proceedings of the Conference held at The University of Alabama in Birmingham

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Lo] J. Piepenbrink, A conjecture of Glazman, J. Diff. eqn. 24 (1977), 173-177 [ll] B. Simon, Schrodinger semigroups, B u l l . Amer. Math. S O C . (New Series) 7 (1982), 447-526. [12] G. Stampacchia, Le probleme de Dirichlet pour les equations elliptlques du second ordre a coefficients discontinus, Ann. Inst. Fourier 16 (1965), 189-258. [13] A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (19351, 767-776. T. (North-Holland), 1984 19 CLASSIFICATION OF I N I T I A L DATA FOR THE POROUS MEDIUM EQUATION I N RN N i c h o l a s D.

L e t K(T) denote t h e convex h u l l of r and l e t TI be a p o i n t i n K(7) which i s n o t a n extremal point. then t h e r e does not e x i s t a s o l u t i o n change sign) such t h a t uCx)e-<"'x' u 2 0 of Pu = 0 i n i s a bounded f u n c t i o n i n Rn . Rn (u may It follows from t h e theorem t h a t a l l p o i n t s of T are extremal p o i n t s of and t h u s i n p a r t i c u l a r i t follows t h a t no t h r e e p o i n t s of r are on t h e same l i n e . 2 t o e s t a b l i s h t h e s t r i c t convexity of K(T).

The theorem asserts we can c a l c u l a t e t h e moments EXAMPLE. 1. (4-x)) f o r 0 < x < 4. and dp(x) = l / ( n The n e x t step i s t o consider t h e orthogonal monic polynomials { P , ( Z ) } ~ = ~ where Pn(z) i s o f degree n, has u n i t leading c o e f f i c i e n t , and f Pn(z) dv(z) = 0 J f o r n # m, where the bar means the complex conjugate i s taken. When F(z) i s a polynomial o f degree g r e a t e r than one, i t i s always p o s s i b l e t o c a l c u l a t e r e c u r s i v e l y i n terms o f t h e c o e f f i c i e n t s i n F ( z ) an i n f i n i t e subsequence o f t h e associated orthogonal polynomials, [8, 17, 121.

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