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Abstract
A geometric formulation of singular partial differential equations (PDEs) is considered. Surgery techniques and integral bordism groups are utilized, following previous works by Prástaro on PDEs, in order to build global solutions crossing also singular points and to study their stability properties.
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Acknowledgement
The work was partially supported by Italian grants MURST ‘Geometry of PDEs and Applications’.
Notes
Notes
1. For basic informations on differential topology and algebraic topology, see, e.g. Citation3,Citation4,Citation21–33. For general informations on the geometric theory of PDEs, see, e.g. Citation1,Citation2,Citation20–22,Citation29–31,Citation34–38. Interesting applications of the Prástaro's PDEs theory can be found also in Citation39,Citation40.
2. By using the natural embedding , we can consider PDRs (resp. PDEs), E
k
⊂ JD
k
(W) as contained in
too.
3. For a p-dimensional integral manifold V of E k , 0 ≤ p ≤ n, with boundary ∂V (or eventually with ∂V = ∅), we mean an element V∈C p (E k+h ), h ≥ 0, such that TV ⊂ E k+h . Here C p (E k+h ) is the space of p-chains in E k+h . So, if V = ∑ i a i u i , a i ∈ℝ, one has ∂V = ∑ i (−1) i a i ∂ i u.
4. If S is the set of non-zero divisors of A, Q(A) ≡ S
−1
A ≡ A × S/∼, where ∼ is the following equivalence relation: (a, s) ∼ (a′, s′) ⇔ as′ − a′s = 0. One denotes by the equivalence class of (a, s) (for more details, see Citation10).
5. If is any ideal of A, the radical of
is the following ideal
. An ideal
such that
is called radical-ideal (or perfect). Furthermore, the radical of A, is the intersection, rad(A), of all maximal ideal
. (for more details, see Citation10).
6. This means that , iff
(see Citation10 for notations).
7. .
8. H(E
∞) has a natural structure of Hopf algebra if is a finite group, otherwise it is an extension of a Hopf subalgebra Citation8,Citation9.
9. Note that the bifurcation does not necessarily imply that the tangent planes in the points of V ij ⊂ V to the components V i and V j should be different.
10. We shall assume that E k is encoded by local equations that can be solved with respect to the vertical coordinates representing k-derivatives.
11. But does not necessitate to be asymptotically stable. Note that the average stability for solutions of ODEs coincides with the asymptotic stability. For a geometric theory of stability of PDEs and PDEs solutions, related to integral bordism groups, see Citation14,Citation16–19.
12. We adopt the notation that vertical coordinates in JD
k
(W) are written in the form , with 0 ≤ α ≤ k, and we set
.
13. In the case where (E k ) q is undetermined, there is no solution passing through the singular point q∈E k . However, if in the neighbourhood of q there exist solutions tending to such a point q, we can, for continuity, prolong such solutions.
14. This is the equation for circular membrane subject to a normal uniform pressure, where x = r is the radial coordinate and y(r) is the radial stress.
15. See, e.g. Citation43.
16. In fact, if the overdetermined Equation (Equation11) should be integrable, we could surgery a solution of this with a solution of A 1 passing through q (1) to obtain a singular solution of (Equation10).
17. Note that such solutions are smooth in A 1, but singular in E 2. In fact, the tangent space to such solutions in q (0), have zero projection on T 0 M, via the canonical projection (π2,0)* : T q (0) V → T 0 M.
18. This is the case when the integral curve approaches q (0) such that in the plane (x, y′) the velocity at x = 0 is along the y′-axis (see (Equation12)).