Citation Context

...ss W k (locally square integrable up to the k-th derivative) by A k and G k , respectively. Here k is a nonnegative integer. For generalities on Sobolev spaces of cross sections in fibre bundles, see =-=[29]-=-. Provided 2k > dim M, A k is an affine Hilbert space and G k+1 is a Hilbert Lie group acting smoothly from the right on A k [33, 27, 24]. We shall even assume 2k > dim M + 2. Then, by the Sobolev Emb...

Citation Context

... V) suggests that this domain is independent of A, and in fact it is. (As a consequence of the compactness of M, the Sobolev spaces do not even depend on the metric of the tangent bundle or of V; see =-=[28]-=-.) In addition, there is always some usable operator A – for example, make metric and connection choices and take the Laplacian ∇∗∇. Though the Sobolev space L2 r(M, V) is Hilbertizable, it is more pr...

Citation Context

... V) suggests that this domain is independent of A, and in fact it is. (As a consequence of the compactness of M, the Sobolev spaces do not even depend on the metric of the tangent bundle or of V; see =-=[28]-=-.) In addition, there is always some usable operator A – for example, make metric and connection choices and take the Laplacian ∇∗∇. Though the Sobolev space L2 r(M, V) is Hilbertizable, it is more pr...

Citation Context

...e restriction of the mapping Hu does indeed satisfy the hypotheses of the Banach Contraction Theorem provided ‖u‖ and ∆t are sufficiently small. The space we will choose for V is the Sobolev Hilbert space Hm = Hm(S1, V ), with m > 12 . We recall that this is the Hilbert space of all functions u in L2(S1, V ) such that ‖u‖2m = ∑ k(1 + k 2) m 2 |u(k)|2 is finite, where as before, u(k) are the Fourier coefficients of u. The principal property of these spaces that we shall need is that Hm(S1,R) is a commutative Banach algebra under pointwise multiplication when m > 12 (cf. [A], Theorem 5.23, or [P]). As a first consequence, it follows that if P : V → V is a polynomial mapping, then u 7→ P (u) is a map of Hm to itself, and moreover ‖P (u)‖m < C ‖u‖ r m, where r is the degree of P . We will permit ourselves the abuse of notation of denoting this latter map by P , and it is now elementary to see that it is Frechet differentiable, and in fact that DPu(v) = P ′(u)v, where P ′ is the derivative of P . (This will follow if we can show that there is an algebraic identity of the form P (X + Y ) = P (X) + P ′(X)Y +Q(X,Y )Y 2, for some polynomial Q in X and Y . But it is clearly enough to check th...

Citation Context

...Adσ → GAdσ , Adσ(g)(x) = gxσ(g −1). The classifying space BLσG is homotopy equivalent to the homotopy quotient EG×G GAdσ (Lemma 4.2). 1We work with continuous loops throughout, but by work of Palais (=-=[P]-=- Thm 13.14) the homotopy type of BLσG is unchanged if we work for C k loops for k ≥ 0 or Lkp loops for k > 1/p 1 There is another description of GAdσ that we make use of. If σ has order n, then form t...