Chung-Kwong CHAN()Huichun ZHANG()Xiping ZHU()
Department of Mathematics,Sun Yat-sen University,Guangzhou 510275,China
E-mail:chsongg@mail2.sysu.edu.cn;zhanghc3@mail.sysu.edu.cn;stszxp@mail.sysu.edu.cn
Abstract Monotonicity formulas play a central role in the study of free boundary problems.In this note,we develop a Weiss-type monotonicity formula for solutions to parabolic free boundary problems on metric measurecones.
Key words monotonicity formula;parabolic free boundary problem;RCD-spaces
Dedicated to Professor Banghe LI on the occasion of his 80th birthday
In [14],we considered free boundary problems onRCD(K,N)-spaces;this is a class of singular metric measure spaces with a generalized Riccicurvature bounded from below.The main example sincludefinited imensional Alexandrov spaces with curvature bounded from below [28,34],and the Gromov-Hausdorfflimit spaces of a family of Riemannian manifolds with Ricci curvature bounded from below uniformly and a dimension bounded from above uniformly [16,17].It is well-known that the singularities of anRCD(K,N)-space might be dense [27].We refer the readers to [6,7,25,26,30,31]and the survey [4]for the development of this generalized notion of the Riccicondition.
For a domain Ω on anRCD(K,N)space(X,d, µ),the one-phase parabolic free boundary problem on Ω is
whereIis an interval,φis a given nonnegative function onI×∂Ω ,and∂{u>0}∩(I× Ω )is the free boundary.
The existence of the solution to(1.1)with general given initial data is not obvious,even for the Euclidean caseX=Rn(with the Euclidean metric and the Lebesgue measure).Indeed,since the one-phase Bernoulli energy functional
is not convex,it is difficult to solve(1.1)via the gradient flow ofJ(u).The main approaches to constructing a solution of(1.1)are based on the approximations by a family of the following perturbations(see,for examples,[9,12,13,33]):
For the regularity of a solution to(1.1),and also the regularity of the free boundary,one usually performs a blow-up analysis at each point in the free boundary.Namely,one can study the limits on the tangentcone atx0,lying in the free boundary,arising from a blow-up sequence of solutions of(1.1).Since the tangentcones at almost every point in anRCD(K,N)-space areRCD(0,N)metric measurecones,it is natural to deal with the global solutions to a free boundary problem on anRCD(0,N)cone.
We now recall some notations.LetN≥2 and let(Σ,dΣ, µΣ)be a metric measure space satisfying a generalized Ricci curvature conditionRCD(N−1,N).The metric measurecone
is given byX:=CN(Σ)=[0,∞)×Σ/({0}×Σ),with the distance given by
and the measure given by
It is well known that(X,d, µ)is anRCD(0,N)-space.We refer readers to [18,23]for further propositions regrading thecones.We should mention that Allen and Chang Lara in [2]and Allen in [1]studied the free boundary problems on 2-dimensional and 3-dimensionalcones over smooth cross spaces.
Recall that the notions of the Sobolev spaceW1,2(X)on a metric measure space(X,d, µ)were introduced in [15];see also [5,10,29].SinceW1,2(X)is a Hilbert spaceon anRCD(K,N)-space,it provides the inner product RX〈∇f,∇g〉dµand a measure-valued Laplacian∆ffor allf∈in the sense of distributions(see [19]).
Letting∊>0,we consider a family of non-negative global solutions to the singular perturbation problems on theRCD(0,N)-cone(X,d, µ)=(CN(Σ),dC, µC),
whereI⊂R is an interval,in the sense of distributions,i.e.,
for allφ∈Lip0(I×X),where Lip0(I×X)is the space of Lipschitz continuous functions with compact support inI×X.
Monotonicity formulas play a central role in studies of free boundary problems(see [3,9,11,24,32,33]and so on).In the case whereX=Rn,the monotonicity formulas foru∊and for their limitsu=as∊j→0+were established by Weiss in [33].In this note,we will extend these Weiss-type monotonicity formulas to the corresponding equations on theRCD(0,N)-cone(X,d, µ)=(CN(Σ),dC, µC).
Set
Denote byothe vertex of the cone(X,d, µ)=(CN(Σ),dC, µC)and setρ(x):=dist(o,x)for any pointx=(ρ, ξ)∈X.Establish a function by
for all(s,x):=(s,ρ, ξ)∈(0,+∞)×X,and a measure on(−∞,0)×Xby
whereu∊,r:=(u∊)r.Here and in the sequel,for a functionv(t,x),weal ways denote there scaling withr>0 by
By passing to a limit as∊j→0+,we will deduce a monotonicity formula for limit solutions as follows:
Remark 1.3One of the differences between the Rn(or the cones over smooth cross spaces,like those cones in [1,2])and the generalRCD-cones is that we are not able to assume a prioriC2-regularity and aC1/2,1-bound(uniform with respect to∊)of solutions to(1.6)in the generalRCD-cones.To get the monotonicity formulas in Theorems 1.1 and 1.2,we need to pay great attention to the sometimes subtle regularity issues on generalRCD-cones.
Let(X,d, µ)be anRCD(K,N)-space withK∈R,N∈[2,+∞).We recall the notion of Sobolev spaces;see [5,6,15,21,29].Given a continuous functionfonX,the point wise Lipschitz constant [15]offatxis defined by
It is clear that Lipfisµ-measurable.Let Ω ⊂Xbe an open domain and let 1≤p<+∞.TheW1,p-norm of a locally Lipschitz functionf∈Liploc( Ω )on Ω is defined by
For anyf∈W1,p( Ω )wherep∈(1,∞),it is well-known that there is a function|∇f|∈Lp( Ω ),the so-called generalized minimal upper gradient,such that
Since(X,d, µ)is anRCD(K,N)-space,the spaceW1,2( Ω )is a Hilbert space.It was shown in [19,Sect.4]that,for anyf,g∈W1,2( Ω ),the limit
exists inL1( Ω ).Moreover,this inner product〈∇f,∇g〉is bi-linear,and satisfies Cauchy-Schwarz inequality,the Chain rule and the Leibniz rule(see [19,Sect.4]).
for allφ∈Lip0( Ω ).If there is a signed Radon measureνsuch that∆f(φ)=RΩφdνfor allφ∈Lip0( Ω ),then we say that∆f=νon Ω in the sense of distributions.
This Laplacian on Ω is linear,and it satisfies the Chain rule,and the Leibniz rule [19].
When Ω =X,the inner product(2.2)provides a Dirichlet form E(f,g):=RX〈∇f,∇g〉dµonX.This Dirichlet form(E,W1,2(X))has an infinitesimal generator∆Ewith domainD(∆E)⊂W1,2(X),i.e.,
It has been shown [19]that iff∈W1,2(X)andg∈L2(X),then
We need the following regularity result for Poisson equations:
see,for instance,[8]for the existence of such a cut-offfunction.By the Leibniz rule for the measured-value Laplacian(see [19,Sect.4]),we have that
ProofTake any functionφ(t,x)∈Lip0(I× Ω ).Then,for eacht∈I,the functionφ(t,·)is in Lip0( Ω ).For almost allt∈I,since the functionw(t,·)is a solution to the Poisson equation(2.7)on Ω ,we have that
Now the desired result follows from the Fubini Theorem. □
We refer readers to [20]for more properties of Sobolev spaces on product spaces.
From this point forward,we always assume that(X,d, µ)=(CN(Σ),dC, µC)is anRCD(0,N)-cone withN≥2.
Recall thatρ(x)=dist(o,x)for any pointx=(ρ, ξ)∈X,whereξ∈Σ andρ=ρ(x).It is easy to check that
and hence that
whereGis given by(1.7).
wherewris given by(1.11).
ProofSinceG(−t,x)is bounded onI×X,the function
Hence,we can rewrite Φψ(r)as
where we have used(3.4),the fact thatβ(rwr)wr≥0,and the Leibniz rule for the inner product〈∇lnG,∇wr〉(by [19,Sect.4]).By combining(3.5),(3.6),and
This,by the Cauchy-Schwarz inequality,implies the desired estimate,and hence the proof is finished. □
Now we are in position to prove Theorem 1.1.
Now we present the proof of Theorem 1.2.
