Abstract
A uniqueness theorem is established for autonomous systems of ODEs, ˙ x = f (x) ,w heref is a Sobolev vector field with additional geometric structure, such as delta-monotonicity or reduced quasiconformality. Specifically, through every non-critical point of f there passes a unique integral curve.
Highlights
Introduction and OverviewLet f : Ω → Rn be a continuous vector field defined in a domain Ω ⊂ Rn
By virtue of Peano’s Existence Theorem, the system admits a local solution; that is, a solution defined in an open interval containing t0, in which we have x(t) ∈ Ω
The classical theory of ODEs tells us that Lipschitz vector fields admit unique local solutions; for less regular fields the solutions are seldom unique, see [8, Ch
Summary
Let f : Ω → Rn be a continuous vector field defined in a domain Ω ⊂ Rn. We shall consider the associated autonomous system of ordinary differential equations with given initial data (1.1). The uniqueness of integral curves, even for only Holder regular vector fields, is still possible under additional geometric conditions, like δ-monotonicity in Theorem 1.4. In this way every point z ∈ C0 is uniquely prescribed by its quasipolar coordinates associated with the vector field f ∈ FK (d) This is a pair (ρ, eiθ) ∈ R+ × S1 with ρ = |z| as the polar distance of z and θ as its quasipolar angle; for example, the identity map f = id : C → C gives the usual polar coordinates (ρ, eiθ) of z = ρeiθ. The conclusion of Theorem 1.10 fails for general quasiconformal maps [17], even for reduced ones [11] This is where the elementary but very useful concept of the modulus of monotonicity (1.19). We show that for reduced quasiconformal maps ∆f has the same quasisymmetric behavior as the modulus of continuity for general quasiconformal maps We exploit this property by computing ∆f at suitably chosen points on integral curves.
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