Download On closed boundary value problems for equations of mixed by Lupo D., Payne K.R. PDF

By Lupo D., Payne K.R.

Show description

Read or Download On closed boundary value problems for equations of mixed elliptic-hyperbolic type PDF

Similar mathematics books

Additional info for On closed boundary value problems for equations of mixed elliptic-hyperbolic type

Sample text

The rest of the proof of the theorem is to check the gluing of the so defined metric and almost complex structure on the common definition set of any two k-orthogonal basis {X1 , . . , X2n } and {Y1 , . . , Y2n } which is so because of the construction made above and because of the uniqueness of the polar decomposition. We remark also that if (J, h) is a starting structure for the polarization of the fundamental form ω(X, Y ) = h(X, JY ), then the result of polarization is the same metric h and almost complex structure J.

Ti+1 . 2. Let M be real analytic manifold, ε > 0 and H(p, t) = ωt (p) homotopy operator for ω0 and ω1 consisting of closed differential 2-forms on M . Let B ⊆ E 1 (M ) be an arbitrary small convex Whitney neighbourhood ˜ t) = of the zero one-form on M . Then there exist a homotopy operator H(p, ˜ t)) < ε and a family of ω ˜ t (p) of the two-forms ω0 and ω1 , D(H(p, t) − H(p, ˜ one-forms βt = B(p, t) ∈ B, for each t ∈ I, continuous on the parameter t, ˜ t + dβt is real analytic for each t ∈ I.

Camassa-Holm equation – right invariant metric on the diffeomorphism group The construction described briefly in the previous section can be easily generalized in cases where the Hamiltonian is a left- or right-invariant bilinear form. Such an interesting example is the Camassa-Holm (CH) equation [2,12,17]. This geometric interpretation of CH was noticed firstly by Misiolek [23] and developed further by several other authors [7–9,13,14,18]. Let us introduce the notation u(g(x)) ≡ u ◦ g and let us consider the H 1 Sobolev inner product H(u, v) ≡ 1 2 M (uv + ux vx )dµ(x), with µ(x) = x (5) The manifold M is S1 or in the case when the class of smooth functions vanishing rapidly at ±∞ is considered, we will allow M ≡ R.

Download PDF sample

Rated 4.32 of 5 – based on 41 votes