The Euler-Lagrange Equation

We now derive the Euler-Lagrange equation satisfied by the minimiser of the PFC func- tional (3.1) (the Euler-Lagrange equation for a more general functional is given in [Eva10, Subsection 8.1.2]). Initially we state the weak form of this equation.

Definition 3.3.1 (Euler-Lagrange Equation (Weak Form)). A functionuinH¯u2(Ω) solves

the Euler-Lagrange equation if

3.3. EULER-LAGRANGE EQUATION CHAPTER 3. PFC MODEL ANALYSIS Existence of a solution to this equation is given by Theorem 3.2.1 and the following lemma.

Lemma 3.3.1. The minima of the PFC functional (3.1) satisfy the Euler-Lagrange equa- tion, Definition 3.3.1.

Proof. Let usolve (P) (page 37) and choosev in H_#2(Ω), then leti() =F[u+v] where clearlyu+v∈H2

¯

u(Ω) and thus ihas a critical point at = 0. Since by Lemma 3.1.4i is Fr´echet differentiable soi0() is well-defined andi0(0) = 0 which translates to

δF[u, v] = d dF[u+v] ₌₀ =i0(0) = 0,

and hence the first variation vanishes.

We now wish to prove that solutions of the Euler-Lagrange equation (Definition 3.3.1) are smooth. We first prove an auxiliary regularity result for a linear PDE.

Lemma 3.3.2. If η∈H_#2(Ω),f ∈Hperm (Ω), m≥0 and

h∆η,∆vi=hf, vi, ∀v∈H_#2(Ω) (3.14)

thenη∈H_#m+4(Ω).

Proof. Recall Parseval’s Theorem (on periodic domains)

hf, gi=hf ,ˆ ˆgi

where ˆf and ˆgare the Fourier transforms off andgrespectively (this follows from [BO12, Equation (1.1)] withs= 0 applied tof+g). Using this on (3.14) we have

hF[∆η],F[∆v]i=hf ,ˆvˆi ∀vˆsuch thatF−1[ˆv]∈H_#2(Ω). Using that derivatives become multiples bykin Fourier space (2.22) we have

h−|k|2_{η,ˆ −|k|}2_{ˆvi=hf ,ˆ ˆvi ∀ˆvsuch thatF}−1_[ˆv]∈H2

#(Ω).

Therefore we have

h|k|4_{η,ˆ ˆvi=hf ,ˆvˆi ∀vˆsuch thatF}−1_[ˆv]∈H2

#(Ω).

The inner product in Fourier space can be written as a sum (this follows from [Gra08, Equation (3.1.5)] with rescaling to generalise fromTd to Ω). Hence we have

X k∈Zd |k|4_{ηˆ(k)ˆv(k) =} X k∈Zd ˆ f(k)ˆv(k).

3.3. EULER-LAGRANGE EQUATION CHAPTER 3. PFC MODEL ANALYSIS Consider ˆ v(0) = Z Ω v(x)ei0xdx= Z Ω v(x)dx= 0.

Hence the first Fourier mode corresponds to the average and is thus already specified. Therefore we no longer consider thek= 0 case and we have

|k|4_{ηˆ(k) = ˆf(k) ∀k∈}

Zd/{0}. (3.15)

It is easy to see that

g∈H_pers (Ω) iff |k|s_ˆg∈l2₍

Zd). (3.16)

Therefore sincef ∈Hs

per(Ω) we have from (3.15)

|k|s+4_η(k)∈l2₍

Zd\{0}). Hence from (3.16) we have that

η∈H_#s+4(Ω).

We can now prove thatuis smooth.

Lemma 3.3.3. Any solution to the Euler-Lagrange equation, Definition 3.3.1, belongs to

C_per∞(Ω).

Proof. We show this result by using induction to show thatu∈Hu¯k(Ω) for allk. From(P)

(page 37) we already know thatu∈H2 ¯

u(Ω), our inductive step is to show that, if we know thatu∈Hk

¯

u(Ω), then we can deduceu∈H k+2 ¯

u (Ω) for allk≥2.

Consider the weak Euler-Lagrange equation, Definition 3.3.1, which we re-arrange to give

h∆η,∆vi=hf, vi, ∀v∈H_#2(Ω) whereη=u−¯u∈H_#k+2(Ω) for allk≥0 and

f =−2∆u+ (δ−1)u−u3. (3.17)

SinceHperk+2(Ω) is a Banach algebra fork≥0 (see [Pel11, Appendix B.1]) it follows thatu3∈

Hk+2

per (Ω). Therefore, from (3.17),f ∈ Hperk (Ω). Lemma 3.3.2 implies that η ∈ Hperk+4(Ω),

and hence u∈ H_uk_¯+4(Ω). Starting from k = 0 induction gives that u ∈ Hk+4

per (Ω) for all

k≥0 taking the limitk→ ∞gives thatuis smooth.

3.3. EULER-LAGRANGE EQUATION CHAPTER 3. PFC MODEL ANALYSIS anywinH2

per(Ω),wcan be written as

w= ¯w+ ˜w, where

Z

Ω

˜ wdx= 0.

Using this formulation in the weak form of the Euler-Lagrange equation, Definition 3.3.1, we have 0 =δF[u,w˜], =δF[u, w]−δF[u,w¯], =δF[u, w]− Z Ω (∆ +I)u(∆ +I) ¯w−δuw¯+u3w¯dy. Using thatu∈C∞(Ω), which follows from Lemma 3.3.3, we have

0 =δF[u, w]− Z Ω (∆ +I)2u R Ωwdx |Ω| −δu R Ωwdx |Ω| +u 3 R Ωwdx |Ω| dy, =δF[u, w]− 1 |Ω| Z Ω Z Ω (∆y+I)2u(y)w(x)−δu(y)w(x) +u(y)3w(x)dxdy, =δF[u, w]− Z Ω w(x) 1 |Ω| Z Ω (∆ +I)2u(y)−δu(y) +u(y)3dydx, = Z Ω (∆ +I)u(∆ +I)wdx−δ Z Ω uwdx+ Z Ω u3wdx−λ Z Ω wdx,

where since 1/|Ω| ∈H2(Ω),λis given by λ=δF= δF[η], 1 |Ω| = (1−δ)¯u+− Z Ω u3dx.

We can now give the strong form of this equation.

Definition 3.3.2(Euler-Lagrange Equation (Strong Form)). A functionu∈H4_{(Ω), solves} the strong form of the Euler-Lagrange equation if

   (∆ +I)2u−δu+u3_{−λ= 0,} R udx= ¯u.

It is clear that, if this equation is satisfied, then its weak form, Definition 3.3.1 is satisfied as well. Since from Lemma 3.3.3 the solutions of the weak form are sufficiently regular (u∈H4_{(Ω)) the reverse relation is true as well.}

Remark 3.3.1. The Euler-Lagrange equation can also be obtained by finding the critical points of the functional

I[η,λ˜] =F[η]−˜λ

Z

Ω

ηdx− |Ω|u¯

3.4. CONCLUSION CHAPTER 3. PFC MODEL ANALYSIS

over the spaceη ∈H2

per(Ω)andλ˜∈R.

3.4

Conclusion

In this chapter we have formulated the minimisation problem for the PFC functional which we will focus on solving throughout the rest of this thesis. To achieve this we defined the space from which the minimiser is sought and proved properties of the PFC functional which will prove important in this and subsequent chapters. We also showed that the minimisation problem has a solution, i.e., there is a minimum of the PFC functional under the constraint that mass is conserved. The PDE problem associated with this minimisation is also formulated and we prove that the solution to this problem is smooth.