By Berti M., Bolle P.
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Extra resources for Multiplicity of periodic solutions of nonlinear wave equations
1ff is a formal power series in which is symmetric in x], [can be expressed in terms of the elementary symmetric func- tions in 4. For the kth elementary symmetric function, substitute so that I represents a polynomial in the Pontrjagin classes. 5]]. Appendix II. THEOREM. The transversal elliptic index is a distribution. PR00E Let A : —+ E, F complex vector bundles over M be transversal elliptic relative to a compact Lie group action G. Let denote the natural Laplacian on G and let denote the induced operators on and That is, = — relative to an orthonormal base of an invariant inner product on g, the Lie algebra of G.
Thus g1) which by definition is p,tr has of the form no pole at 0 or and hence is constant. -s Part (b) of the theorem is obtained by evaluating the constant at t = 1. The theorems above and methods of proof by Kosniowski and Lusztig suggested to Atiyah and Hirzebruch that they try the same idea on the Dirac operator where for spin manifolds. They found THEOREM . fold of dim 4k. If A(M) + 0, then any compact subgroup of the group of diffeomorphisms of M is finite. METHOD OF PROOF. One shows that if S' acts nontrivially on M, then A(M) = 0.
An equivalent definition of subellipticity is that, to every compact subset K of Q, there is a number > 0 and a positive constant such that, (or all u E with support in K, + IIuIIo}. 1) holds with (5 = 1, P(x, D) is elliptic at x0. If P(x, D) is subelliptic at a point, it is so in a full neighborhood of this point. 1. A simple first-order equation. 1) are the first-order ones whose principal part does not vanish at any point. Among the latter, the simplest nontrivial examples are provided by the operators of the form L = 3/3t + ib(t) t3/3x, where x and t are real variables and b(t) a real-valued function in some open interval — T < t < T(T> 0).