A fourth order linear parabolic equation on conical surfaces

2021-12-09ZhangFangxu

中国科学技术大学学报 2021年6期

Zhang Fangxu

School of Mathematical Sciences, University of Science and Technology of China, Heifei 230026， China

Abstract: A parabolic equation of fourth order on surfaces with conical singularities is considered. By the analysis of energy and approximations, the existence and uniqueness of the solution of this equation in a special space that has some approximation property are proved. Finally, it’s proved that the property is equivalent to the finiteness of energy for some functions when β∈(-1,0).

Keywords: parabolic equation; Calabi flow; conical singularity

2020 Mathematics Subject Classification: 35G10;53C21

1 Introduction

LetMbe a smooth Riemann surface, andp1,…,plbe finitely many points onM. For each pointpi, we assign a weightβi> -1. We are interested in the class of metricsgwhich are smooth and compatible with the conformal structure ofMaway frompi. Assume that in some neighborhood ofpi,gis given by

g=|z|2βi|dz|2

(1)

wherezis a complex coordinate aroundpiandz(pi)=0. A metricgthat satisfies the above conditions is called a metric with conical singularities. Obviously, aroundpi, (M,g) is isometric to a flat cone metric with total cone angle 2π(βi+1), i.e.

r2βi(dr2+r2dθ2).

In this paper, we investigate a linear parabolic equation of fourth order:

(2)

where △ is the Laplacian ofg,aandK0are known coefficients. We will prove that under appropriate conditions the initial value problem of (2) has a solution, and the solution satisfies some estimates(see Theorem 1.1).

The purpose of discussing Equation (2) is to make preparations for investigating the conical Calabi flow. The Calabi flow was first proposed by Calabi[1]in 1982. Precisely, on a smooth surface, we define Calabi flow to be

(3)

whereKis the Gaussian curvature.For a smooth initial metricg0, ifg(t)=e2u(t)g0is a solution of (3), then

(4)

whereK0is the Gaussian curvature ofg0. Chrusciel[2], Chen[3]and Struwe[4]independently proved the long time existence and convergence of Calabi flow on smooth surfaces. And Li et al.[5]obtained convergence theorems of the Calabi flow on extremal Kähler surfaces, under the assumption of global existence of the Calabi flow solutions. In the mean time, the topic of Ricci flow with conical singularities attracts the attention of many researchers[6-9]. In particular, Yin[10]proved the long time existence of the conical Ricci flow for general cone angle. And Zheng[11,12]also did some research on the conical Calabi flow.

We may also investigate the conical Calabi flow. To dicuss Equation (4) on conical surfaces, we will first need to study the corresponding linear equation (2). Although Equation (2) is linear, (M,g) is incomplete. Hence, we need to define some special function spaces.

We assume without loss of generality that there is only one singular pointpof orderβ. In a neighborhood ofp, letx,ybe the real and imaginary part ofz. We define the coordinate (ρ,θ) by the following equations:

x=rcosθ,y=rsinθ

(5)

and

(6)

Definition 1.1[10]Let (ρ,θ) andUbe used as above, andl∈,α∈(0,1). For any functionu∈Cl,α(M(〗p}), we define

(7)

whereBris {(ρ,θ)|ρ

Similarly, we can define the parabolic version of the above weighted Hölder function space.

Definition 1.2[10]Ifuis a function defined onM{p}×[0,T] ,we define

‖u‖P l,α,[0,T]:=

‖u‖Cl,α(U×[0,T])

(8)

and Pl,α,[0,T]to be the set of functionsusatisfying ‖u‖P l,α,[0,T]<+∞.

It is not hard to see El,αand Pl,α,[0,T]are Banach spaces. Since the main tool used for proving the apriori estimate in this paper is the energy method, we need to add another constraint to the spaces above.

Definition 1.3For a functionudefined onM(〗p}, we define

(9)

For a functionvdefined on (M(〗p})×[0,T], we define

Definition 1.4Foru∈Cl,α(M(〗p}), we sayuhas the property of approximations, if there is a sequence of functionsuidefined onM(〗p} satisfying:

(i) For eachi, there is a neighborhood ofp, such thatuiin it are constant;

Similarly, forv∈Cl,α( (M(〗p})×[0,T]), we sayvhas the property of approximations, if there is a sequence ofvidefined in (M(〗p})×[0,T] satisfying:

(i′) For eachi, there is a neighborhood ofp, such that fort∈[0,T],vi(t) in it are constant;

(iv′) there is a constantc(independent ofv), such that

‖vi‖C0((M(〗p})×[0,T])≤c‖v‖C0((M(〗p})×[0,T]),

‖∂tvi‖C0((M(〗p})×[0,T])≤c‖∂tv‖C0((M(〗p})×[0,T]).

With these definitions, we can state the main theorem of this paper as follows.

Theorem 1.1Let (M,g) be a closed surface with conical metric and assume thatpis the only cone point. Assume thata∈P2,α ,[0,T], ∂ta∈C0(M×[0,T]),K0∈E2,α,u0∈E4,α, and

(i)K0is identically 0 in a neighborhoodUKofp;

(ii) [a]X,T, [u0]X<∞;

(iii)aandu0have the property of approximations,

then there exists a solutionu∈P4,α,[0,T]satisfying

(10)

andu(0)=u0such that

‖u‖P 4,α,[0,T]≤C(‖u0‖E 4,α,

‖a‖P 4,α,[0,T], ‖K0‖E 2,α)

(11)

and

[u]X,T≤C(‖a‖C0, ‖∂ta‖C0,

‖K0‖E 2,α, [a]X,T, [u0]X,T)

(12)

To prove this theorem, we use a sequence of surfaces with boundary to approximate the surface with conical point. Specifically, we consider

In this surface with boundary, we consider the same initial value problem, with some special boundary conditions:

By the boundary conditions, we can use the energy method to get some uniform apriori estimates (see Section 2). Based on this result, in Section 3, we finish the proof of Theorem 1.1 by takingk→∞. Finally, in Section 4, we discuss the property of approximations stated in Definition 1.4 and prove that the condition (iii) in Theorem 1.1 can be removed whenβ∈(-1,0) and △uis bound. Specifically,

Theorem 1.2Letβ∈(-1,0). Ifu∈C4,α(M(〗p}) satisfies [u]X<∞ and △uis bounded, thenuhas the property of approximations defined as Definition 1.4. Similarly, ifa∈C2,α((M(〗p})×[0,T]) satisfies [a]X,T<∞ and △a(t) is bounded ,∀t∈[0,T], thenahas the property of approximations.

2 Estimates of boundary value problem

In this section,Mis a compact surface with nonempty boundary and a smooth Riemannian metricg. Consider the linear boundary value problem

(13)

whereνis the outward normal vector to the boundary.

Theorem 2.1Leta∈C2,α(M×[0,T]),K0∈C2,α(M),u0∈C4,α(M). Assume thata(t) andu0are constants in a neighborhood of ∂M. Then there is a unique solutionu(x,t)∈C4,α(M×[0,T]) to Equation (13) satisfying the initial conditionu(0)=u0. IfM′ containing the support ofK0is a smooth domain with boundary satisfyingM′∩∂M=Ø, then we have the following uniform estimate:

C(‖a‖C0,‖K0‖C2,‖∂ta‖C0,[a]X,T,[u0]X,T,M′)

(14)

whereCdepends on the geometric property of (M′,g), such as Sobolev inequality, the coefficient ofLpestimates, but is independent ofM.

ProofSinceu0is constant around ∂M, the compatibility condition

(15)

holds. The existence and uniqueness of the solutionuis well known from the classical theroy[13].

Next, we prove some uniform estimates of theL2norm of △u,u,u.

Differentiating directly and using integration by parts by boundary conditions, we have

Took one of the bags of gold under his arm: Some versions of the story, such as Tabart s version, make Jack a righteous trickster character by justifying55 his thievery from the giant

whereCais a constant depending on ‖a‖C0and ‖∂ta‖C0. In the calculation below, each timeCaappears it may represent a different constant. If necessary, to specify a constant, we use double subscripts, such asCa1.

Use integration by parts once again, substitute Equation (2) into the inequality, and apply Young’s inequality, we get

(16)

where the meaning of subscript inCK0andCa,K0is understood in a similar way as inCa. The items in parentheses above, except |a|4, are controlled by [a]X,T. Using Sobolev’s embedding theorem andL2estimates in the support ofK0, we have

CK0CM′(‖△a‖L2(M′)+‖a‖L2(M′)+‖a‖L2(M′))4≤

Base on the above, we have

Similarly, using Equation (2) and integration by parts twice(use the condition thatais constant around ∂Mand boundary condition), we have

In the last line above, we drop a negative term, and use the Schwarz’s inequality. The first term on the far right of the above inequality is already estimated, and the second term is already discussed in (16).

To get the estimate ofL2norm ofu, noticing that ∂νa=∂νK0=∂ν(△u)=∂νu=0, we use integration by parts twice,

By the Young’s inequality,

By the Gronwall’s inequality, and the proof is done.

3 Solution of the linear equation via approximations

The purpose of this section is to prove Theorem 1.1 by Theorem 2.1.

Recall the definition

(17)

Due to the assumption (iii) in Theorem 1.1, by taking a subsequence, we can assume without loss of generality that there areu0;k∈C4,α(X(〗p}) such thatu0;kis constant around ∂Mk, similarly,ak∈C2,α( (M(〗p})×[0,T]) satisfying thatakis constant around ∂Mkfort∈[0,T]. Furthermore, due to the assumption (i) in Theorem 1.1, we can assume thatM′:=MUKis compactly contained inMkwhenkgoes to infinity.

By the above discussion, we can apply Theorem 2.1 to the boundary value problem

(18)

and denote the solution byuk. It is defined inMk×[0,T], and satisfies the uniform estimate (14):

C(‖ak‖C0,‖K0‖C2,‖∂tak‖C0,

[ak]X,T,[u0;k]X,T,M′)

(19)

Meanwhile, for any fixed compact setW⊂M(〗p},akconverges toainC2,α(W×[0,T]),u0;kconverges tou0inC2,α(W). After taking subsequence if necessary, we might as well call ituk, it converges to a functionu(x,t) defined in (M(〗p})×[0,T], anduis a classical solution to the initial value problem of Equation (2).

Sincea∈P2,α,[0,T],u0∈E4,αandK0∈E2,α, we may apply the Schauder interior estimates to (2) to obtain thatu∈P4,α,[0,T]and (11).

Meanwhile, sinceuksatisfies (14), by the definition of the property of approximations, we have

‖ak‖C0≤c‖a‖C0,

‖∂tak‖C0≤c‖∂ta‖C0,

Letkgo to ∞, and we get (12). Hence we finish the proof of Theorem 1.1.

4 About the property of approximations

The aim of this section is to prove Theorem 1.2. We will give the approximation sequence by explicit construction. The proof is divided into two parts. First, we prove the theorem foru∈C4,α(M(〗p}).

In order to define the approximation sequence, we first give some properties of the functionunear the cone pointp. Although the function classC4,α(M(〗p}) puts few restrictions on the properties of the function nearp, the condition

implies a lot, which is summarized in the following lemma.

First,we need a lemma about the integration by parts.

Lemma 4.2Letusatisfy the requirements of Theorem 1.2, then

(i) there isγ∈(0,1) which depends only onβ, such thatuis inC0,γin the coordinatez;

ProofWe denote the flat metric dx2+dy2bygsand writeW2,p(gs) for the Sobolev space with respect togs.

Lettingf=△u, in the neighborhoodB:={ρ<1/2} ofp, by (1), we have

△gsu=|z|2βf

(20)

(21)

By Sobolev’s embedding theorem, ∂xuand ∂yuare bounded. Hence, |u|2is bounded because

|u|2=|z|-2β(|∂xu|2+|∂yu|2).

g=dρ2+(1+β)2ρ2dθ2.

Consider another cone of orderβ0, whose metric is given by

To define the approximation sequence, we need a sequence of cut-off functionsφisatisfying:

(C2) there isδi>0,φi≡1 in [0,δi];

(C4)φiis smooth, and

(C5) there existsc>0, such that

We claim thatφisatisfying the above conditions exists. To see this, for anym>1, we choose a smooth functionψ:→such that

ψ(s)≡1 ∀x≥m+1,

ψ(s)≡0 ∀x≤m.

Naturally we have

For anyi, we choosemwhich is large enough (dependent ofi), and define

φi(ρ)=ψ(log(-logρ)).

Due to the equations

and

we obtain that ifmis large enough, (C1)-(C5) hold.

By Lemma 4.2, we can write

whereu(p) is the value ofuatp, and

(22)

for someα>1. This is very important for later estimates.

We define

(23)

We just need to verify thatuimeets the requirements of Definition 1.4, where (i) and (ii) therein are direct consequences of (C2) and (C3). Hence, it suffices to show

(24)

By the dominated convergence theorem,

is obvious. Meanwhile,

By (C4), we get that the right hand side of the above equation goes to 0 wheni→∞.

Finally,

It is obvious that the integral of the last term in the right hand side of the above inequality goes to 0, and the integral of the second term also goes to 0 because of (ii) in Lemma 4.2 and (C4). To estimate the first one, we use (C5) and (22),

Letasatisfy the assumptions of Theorem 1.2. Naturally,a(t) as a function defined onM(〗p}, satisfies that [a(t)]Xis finite, then Lemma 4.2 holds fora(t). Hence we can write