Encuestas

Hypersurfaces Recall that Γ is ann-dimensionalCk hypersurface inRn+1 if for each x∈ Γ, there is an open set U ⊂_Rn+1 _{with x∈ U and a function Ψ∈C}k_(U) with∇Ψ6= 0 on Γ∩U and

Γ∩U ={x∈U |Ψ(x) = 0}.

Aparametrised Ck hypersurface inRn+1 is a map ψ∈Ck(Y;Rn+1) whereY ⊂Rn is a connected open set with rank(Dψ(y)) =nfor all y∈Y. Locally, parametrised hypersurfaces and hypersurfaces are the same [119, Chapter 15]. We call Γ a Ck hypersurface with boundary∂Γ if Γ\∂Γ is aCkhypersurface and if for everyx∈∂Γ, there exists an open setU ⊂Rn+1withx∈U and a homeomorphismψ:H→Γ∩U, whereH:=B1(0)∩ {y= (y1, ..., yn)∈Rn|yn≤0},with ψ(0) =x and

1. rank(Dψ(y)) =nfor all y∈H

2. ψ(B1(0)∩ {y= (y1, ..., yn)∈Rn|yn<0})⊂Γ\∂Γ 3. ψ(B1(0)∩ {y= (y1, ..., yn)∈Rn|yn= 0})⊂∂Γ.

See [119, Chapter 20]. A compact hypersurface has no boundary. We say Γ is a compact hypersurface with boundary ∂Γ if Γ is a hypersurface with boundary ∂Γ and Γ∪∂Γ is compact. Throughout this work we assume that Γ is orientable with unit normalν. We say Γ isflat if the normalν is same everywhere on Γ.

Sobolev spaces Suppose that Γ is ann-dimensional compactCk hypersurface in Rn+1 with k ≥ 2 and smooth boundary ∂Γ. We can define L2(Γ) in the natural way: it consists of the set of measurable functionsf: Γ→Rsuch that

kfk_L2_(Γ):= Z Γ |f(x)|2dσ(x) 1 2 <∞,

where dσ is the surface measure on Γ (which we often omit). We will use the notation∇Γ= (D1, ..., Dn+1) to stand for the surface gradient on a hypersurface Γ, and ∆Γ :=∇Γ· ∇Γ will denote the Laplace–Beltrami operator. The integration by

parts formula for functionsf ∈C1(Γ;Rn+1) is Z Γ ∇Γ·f = Z Γ f ·Hν+ Z ∂Γ f·µ

whereH is the mean curvature and µ is the unit conormal vector which is normal to∂Γ and tangential to Γ.Now ifψ∈C_c1(Γ), then this formula implies

Z Γ f D_iψ=− Z Γ ψD_if + Z Γ f ψHνi fori= 1, ..., n+ 1,

with the boundary term disappearing due to the compact support. This relation is the basis for defining weak derivatives. We say f ∈ L2(Γ) has weak derivative gi =:Dif ∈L2(Γ) if for every ψ∈Cc1(Γ), Z Γ f D_iψ=− Z Γ ψgi+ Z Γ f ψHνi

holds. Then we can define the Sobolev space

H1(Γ) ={f ∈L2(Γ)|D_if ∈L2(Γ), i= 1, ..., n+ 1}

withkfk2_H1_(Γ) := kfk_L22_(Γ)+k∇Γfk2L2_(Γ).The above applies to compact hypersur-

faces too; in this case the boundary terms in the integration by parts are simply not there. We write H−1(Γ) for the dual space of H1(Γ) when Γ is a compact hypersurface.

We shall also need a fractional-order Sobolev space. Let Ω ⊂ _Rn _{be a} bounded Lipschitz domain with boundary∂Ω. Define the space

H12(∂Ω) ={u∈L2(∂Ω)| Z ∂Ω Z ∂Ω |u(x)−u(y)|2 |x−y|n dσ(x)dσ(y)<∞}. This is a Hilbert space with the inner product

(u, v) H12(∂Ω) = Z ∂Ω u(x)v(x) dσ(x) + Z ∂Ω Z ∂Ω (u(x)−u(y))(v(x)−v(y)) |x−y|n dσ(x)dσ(y). See [109,§2.4] and [43, §3.2] for details. The notation

|u| H12_(∂Ω) = Z ∂Ω Z ∂Ω |u(x)−u(y)|2 |x−y|n dσ(x)dσ(y) 1₂

for the seminorm is convenient. Now, recall the standard Green’s formula: Z ∂Ω ∂v ∂νw= Z Ω ∇v∇w+ Z Ω w∆v ∀v ∈H2(Ω), ∀w∈H1(Ω).

When Ω is of class C1, this formula leads us to define a (weak) normal derivative for functions v ∈ H1(Ω) with ∆v ∈ L2(Ω) as the element ∂v/∂ν ∈ H−12(∂Ω) :=

(H12(∂Ω))∗ determined by ∂v ∂ν, w H−12(∂Ω),H12(∂Ω) := Z Ω ∇v∇E(w) + Z Ω E(w)∆v ∀w∈H 1 2(∂Ω), (2.1) where E(w) ∈ H1(Ω) is an extension of w ∈ H 1 2(∂Ω); the functional ∂v/∂ν is

independent of the extension used forw.See [43,§5.5.1] for more details on this.

Evolving hypersurfaces We say that{Γ(t)}_t∈[0,T] is anevolving hypersurface if for every t0 ∈ [0, T], there exist open sets I = (t0−δ, t0+δ) for some δ > 0 and U ⊂_Rn+1 _{and a map Ψ : I×U →}

R such that∇Ψ(t, x)6= 0 forx∈Γ(t) and t∈I, and

Γ(t)∩U ={x∈U |Ψ(t, x) = 0} fort∈I.

The normal velocity of a hypersurface Γ(t) :={x ∈Rn+1 |Ψ(x, t) = 0} defined by a (global) level set function is given by

wν =−

Ψt

|∇Ψ| ∇Ψ

|∇Ψ|.

Remark 2.2.1. It is important to note thatthe normal velocity is sufficient to define the evolution of a compact hypersurface. However, a parametrised hypersurface would require the prescription of the full velocity of the parametrisation.

Remark 2.2.2. Consider an evolving hypersurface with boundary. In this case, we need the normal velocity of the surface and the conormal velocity of the boundary in order to describe the evolution. The normal velocity of the surface must agree with the normal velocity of the boundary.

Remark 2.2.3. An evolving bounded domain {Ω(t)} in Rn can be viewed as an evolving flat hypersurface with boundary{Ω(ˆ t)}inRn+1 (though we choose not to use this viewpoint in this thesis). If we embed each Ω(t) into the same hyperplane ofRn+1 (for example, ˆΩ(t) ={(x1, ..., xn,0)|(x1, ..., xn)∈Ω(t)}), then the normal velocitywν of ˆΩ(t) is zero.

In order to describe the evolution of a hypersurface, it is also useful to assume that there exists a map F(·, t) : Γ(0) → Γ(t) which is a diffeomorphism for each t∈[0, T] satisfyingF(·,0)≡Id and _dtdF(·, t) =w(F(·, t), t).Here we say thatw is thematerial velocity field and write

w=wν+wa (2.2)

wherewν is the given normal velocity of the evolving hypersurface andwais a given tangential velocity field.

In the next two definitions, we suppose thatuis a sufficiently smooth function defined on{Γ(t)}_t∈[0,T](see §2.4.1 later).

Definition 2.2.4 (Normal time derivative). Suppose that the hypersurface{Γ(t)}

evolves with a normal velocitywν. The normal time derivative is defined by ∂◦u:=ut+∇u·wν.

Definition 2.2.5(Material derivative). Suppose that the hypersurface Γ(t) evolves with a normal velocitywν. Given a tangential velocity fieldwa, withwas in (2.2), thematerial derivative is defined by

∂•u:=ut+∇u·w. (2.3) We also write ˙u for∂•u. See [33, 35].

Remark 2.2.6(Velocity fields). It is useful to note that there are different notions of velocities for an evolving hypersurface.

• Suppose that the velocity w of an evolving compact hypersurface is purely tangential (sow·ν = 0). In this case, material points on the initial surface get transported across the surface over time butthe surface remains the same. One can see this for a sufficiently smooth initial surface Γ0 by supposing that Γ0 is the zero-level set of a function Ψ :Rn+1→R:

Γ0 ={x∈_Rn+1 |Ψ(x) = 0}.

Let P be a material point on Γ0 and γ(t) denote the position of P at time t, with γ(t) ∈Γ(t). Then a purely tangential velocity means that ∇Ψ(γ(t))·

γ0(t) = 0,but this is precisely d

Γ

Γ(t)

Figure 2.2.1: A sketch of the evolution of two material points on an evolving curve. The normal motion is given by the blue arrows and the tangential motion is given by the red arrows.

so the point persists in being a zero of the level set. SinceP was arbitrary, we conclude that Γ(t) coincides with Γ0 for all t∈[0, T],i.e.,

Γ(t) ={x∈_Rn+1|Ψ(x) = 0}.

• In applications, there may be aphysical velocity

wν+wτ,

wherewν is the normal component andwτ is the tangential component. The tangential velocity may be associated with the motion of physical material points and may be relevant to the mathematical models of processes on the surface.

• The velocity field (2.2) defines the path that points on the initial surface take with respect to the mappingF. In finite element analysis, it may be necessary to choose the tangential velocitywain an ALE approach so as to yield a shape- regular or adequately refined mesh. See [59] and [50,§5.7] for more details on

this. One may wish to use this physical tangential velocity to define the map F. In writing down PDEs on evolving surfaces it is important to distinguish these notions.

• In certain situations, it can be useful to consider on an evolving surface a boundary velocity wb which we can extend (arbitrarily) to the interior. In the case of flat hypersurfaces withwν ≡0 (this is the case when an evolving domain inRnis viewed like in Remark 2.2.3), the conormal component of the arbitrary velocity must agree with the conormal component of the boundary velocitywb, otherwise the velocities map to two different surfaces.