On the Banach Manifold of Simple Domains in the Euclidean Space and Applications to Free Boundary Problems

Abstract

In this paper we study the Banach manifold made up of simple C m+µ -domains in the Euclidean space R. This manifold is merely a topological or a C 0 Banach manifold. It does not possess a differentiable structure. We introduce the concept of differentiable point in this manifold and prove that it is still possible to introduce the concept of tangent vector and tangent space at a differentiable point. Consequent, it is possible to consider differential equations in this Banach space. We show how to reduce some important free boundary problems into differential equations in such a manifold and then use the abstract result that we established earlier to study these free boundary problems. AMS 2000 Classification: 35R35, 47J35, 58B99. Key words and phrases: Banach manifold; manifold of simple domains; free boundary problem.

Quasi-differentiable Banach manifold and phase-diagram of invariant parabolic differential equation in such manifold

“…Hence, the notion of equivariance in existing literatures such as [10] is a special situation of the notion of invariance here, or in another word, the concept of invariance defined here is an extension of the concept of equivariance for linear Lie group actions to general possibly nonlinear Lie group actions. Note that in [6], the phrase "quasi-invariance" rather than "invariance" as here is used.…”

“…Recall (cf. [6]) that an open set Ω ⊆ R n is said to be a simple C m+µ -domain if Ω is C m+µdiffeomorphic to the open unit sphere B(0, 1) in R n , i.e., there exists a bijective mapping Φ : B(0, 1) → Ω satisfying the following properties:…”

“…If instead of C m+µ the notation Ċm+µ is used in the above relations, then the notations Ḋm+µ (R n ) and Ṡm+µ (R n ) are used correspondingly. From the discussion in [6] we know that D m+µ (R n ) and Ḋm+µ (R n ) are Banach manifolds built on the Banach spaces C m+µ (S n−1 ) and Ċm+µ (S n−1 ), respectively, and each point Ω…”

Analysis of a free boundary problem modeling the growth of necrotic tumors

“…We use the notation Ḋm+µ (R 3 ) to denote the Banach manifold of all Ċm+µ -domains in R 3 ; cf. [9] for details. We denote…”

“…From [9] we know that M 0 is a C 3 -embedded Banach submanifold of M, so that every point in M 0 is C 3 -differentiable as a point of M. Given Ω ∈ M 0 , the equations (1.1) 1 -(1.1) 4 with Ω(t) replaced by Ω has a unique solution (σ, p) satisfying the following properties:…”