A Rem ark on the Level Sets of the G raph of Harm onic Functions Bounded by Tw o Circles in Parallel Planes

KONG Shengli,XU Jinju2,?Department ofM athematics,University ofScienceand Technology ofChina, Hefei230026,China.

2Department ofM athematics,ShanghaiUniversity,Shanghai200444,China.

Received 14M arch 2015;A ccep ted 31 Ju ly 2015

A Rem ark on the Level Sets of the G raph of Harm onic Functions Bounded by Tw o Circles in Parallel Planes

KONG Shengli1,XU Jinju2,?1Department ofM athematics,University ofScienceand Technology ofChina, Hefei230026,China.

2Department ofM athematics,ShanghaiUniversity,Shanghai200444,China.

Received 14M arch 2015;A ccep ted 31 Ju ly 2015

.In thispaper,w e find tw o auxiliary functionsandm akeuseof them axim um p rincip le to study the level sets o f harm onic function defined on a convex ring w ith hom ogeneous Dirich let boundary cond itions in R2.In higher dim ensions,w e also have a sim ilar resu lt to Jagy’s.

Harm onic function;m axim um p rincip le;level set.

1 In troduction

Thegeom etry of the levelsetsof the solutionsofellip tic partial d ifferentialequations isa classicalsubject.For instance,Ahlfors[1]contains thew ell-know n result that levelcu rves of the Green function on a sim p ly connected convex dom ain in the p lane are convex Jordan cu rves.Motivated by catenoid or the ”Riem ann Stair-case” in m inim al surface theory,in 1956,Shiffm an[2]p roved the follow ing tw o resu lts:(i)if am inim al su rface S in R3is bounded by convex cu rves in parallel p lanes,and S is topologically an annu lus, then the intersections of S w ith allother parallel p lanes are also convex cu rves;(ii)if the boundariesare circles in parallelp lanes,then the intersectionsof S w ith allother parallel p lanesarealso circles.In 1957,Gabriel[3]p roved that the levelsetsof the Green function on a 3-d im ensional bounded convex dom ain are strictly convex.Later,in 1977,Lew is[4] extended Gabriel’s resu lt to p-harm onic functions in higher d im ensions.In 1990,Jagy[5]

Shiffm an used com p lex analysis to p rove the above result.In the p roof of case(i), Shiffm an introduced a cu rvatu re type function ψu(yù)(in page 80[2])of the level cu rve of them inim al su rfaces.He p roved that ψu(yù)isa harm onic function and it is non-negative if the level curvesare convex.In thep roofof case(ii),Shiffm an[2]introduced an im portant auxiliary function,now nam ed the Shiffm an function,w hich is a harm onic function on them inim al su rface.

Rem ark 1.1.In order to generalize the Shiffm an’s cu rvatu re type function ψu(yù)(see page 80 in[2])from m inim al surface to harm onic function,Talenti[7]got the follow ing resu lt. Suppose u is harm onic and has no critical points in a dom ain ? ? R2,K is the cu rvatu re of the level cu rves of u w ith respect to the norm ald irection ?u,defined as the follow ing then K/|?u|is a harm on ic function in ?.M ore recently,M a-Ou-Zhang[8]had generalized theabove resu lt to high d im ension harm onic function.

In this paper,for the harm onic function defined on p lane dom ain,w e find tw o harm onic functions correspond ing to Shiffm an function.Now w e state ou rm ain resu lt.

Theorem 1.1.Letu satisfy

where ?0and ?1arebounded smooth convex domains in R,and ?1??0,K is the curvature of the level curves ofu defined as in(1.1).Then

(i) ? =|?u|?3(K1u2?K2u1)isaharmonic function in ?.

(ii) ψ =|?u|?2K?1(K1u2?K2u1)satisfies the follow ing equation

where bi(i=1,2)are bounded continuous functions.

Corollary 1.1.Asa consequenceofTheorem 1.1,since(K1u2?K2u1)isthetangentialderivative of the curvature K of the level sets ofu on its level set,from themaximum principle,w e have if??0and ??1are circles,then the intermediate cross-sectionsmustbe circles.And w e have the follow ing correspond ing to the Jagy’s Theorem 1.1 in[5]to high d im ension harm onic functions.

Corollary 1.2.Letu satisfy

where ?0and ?1arebounded smooth convex domains in Rn(n ≥3).If??0and ??1are spheres and ifall the levelsurfaces ofu are spheres,then thegraph ofu isrotationally symmetric aboutan axiscontaining thecentersofall thespheres,so ?0and ?1mustbe theballsw ith thesamecenter.

The paper is organized as follow s.In Section 2,w e w ill com p lete the proof of Theorem 1.1.Then w e prove Corollary 1.2 in Section 3.In thep roofof Theorem 1.1,w euse the usual Euclidean coord inate to com p lete the calcu lation,certain ly w e can use the sim ilar com p lex analysism ethod as in Shiffm an[2]and Talenti[7].

2 Proof of Theorem 1.1

Shiffm an[2]introduced an im portantauxiliary function,now nam ed the Shiffm an function(see the form u la(8)in page 79,the function β in[2]),w hich is a harm onic function. In this section,w e find the correspond ing harm onic function ? for 2-d im harm onic function.M oreover w e also get the new function ψ.In Theorem 1.1,the norm al d irection o f the level sets of u is ?u,so(K1u2?K2u1)is the tangential derivative of the cu rvatu re K of the levelsetsof u on its levelset.By the classical resu lt in[9],the cu rvature K is strictly positive.

Proof.Setting

w e on ly need to prove the follow ing equality for suitable choice of b,c,

To p rove(2.1)at an arbitrary poin t x0∈ ?,w e m ay choose the coord inate such that u1(x0)=0 and u2(x0)=|?u|＞ 0.If w e can establish(2.1)at x0under the above assum p tions,then going back to the original coord inate,w e find that(2.1)rem ains valid.

Thus it su ffi ces to estab lish(2.1)under the above assum p tions.Denote by

Therefore

Differentiating the Eq.(2.3),w e have

Differentiating the Eq.(2.4),w e get

Pu tting(2.3)-(2.4)in to(2.5),w e have

w here

Du ring the follow ing calculation,w eoften app ly the follow ing form u las:

We shall com p lete the com pu tation in three steps.

Step 1:We fi rst calcu late I1and shall get the form u la(2.11).Firstly,w e calcu late the follow ing form u las.

So w e have

App lying(2.7)-(2.10),w e get

Step 2:We com pute I2and shallget the form u la(2.18).Taking the fi rstderivativesof (2.2),w e have

So by(2.7)and(2.12),w e obtain

Taking the second derivativesof(2.2),

By(2.14),w e also get

From(2.9),(2.13)and(2.16),w e have

Using(2.7)and(2.17),it follow s that

Step 3:Wew ill calcu late I3and get the form u la(2.24).Taking the third derivative of(2.2) and using(2.14),w e have

w here

Using(2.7)-(2.8)and(2.13),w eget

From(2.15),w e obtain

Using(2.7)-(2.10)and(2.17),it follow s that

App lying(2.19)-(2.21),w eobtain

Since

from(2.22)-(2.23),w e have

By(2.17),w e obtain

Therefore,by(2.11),(2.18),(2.24)and(2.25),w e get

Now w divide two casesaccord ing to the d ifferent choicesof b,c.

Case I:Let b=0,c=3.For ? =|?u|?3(K1u2?K2u1),it follow s that

Case II:From(2.4),w e have

Then pu t the form u la(2.28)into(2.26),w eobtain

Let b= ?1,c=2.It follow s that

satisfi es the follow ing equation

w here bi(i=1,2)are bounded continuous functions.Hence,w e com p lete the p roof of Theorem 1.1.

3 The Proof of Corollary 1.2

In this section,asm otivated by the Jagy’s[5]theorem on m inim al su rfaces,w e give the correspond ing resu lt for harm onic function in high d im ension dom ain.Ou r p roof follow s Jagy’s[5]p roofonm inim alsu rfaces.The key poin tof Jagy’s proofw as the resu ltof Schoen[10]that is based on them axim um p rincip le for ellip tic d ifferentialequations.As inm inim alsu rface,p lease see Theorem 2 in Schoen[10],essentially w e know theboundary B of the graph of the harm onic function in Rn+1enjoys som e reflection symm etries, so does the graph of theharm onic function itself.

Proof.Let M ? Rn+1be the graph of the harm onic function u in ? ? Rn.For hyperp lanes π0={xn+1=0}and π1={xn+1=1}that intersect M.We can arrange that the centers o f both the spheres M Tπ0and M Tπ1lie in the x1xn+1p lane.

We need to show that the center of every parallel spherical cross-section lies in the x1xn+1p lane.We consider B=(π0T M)S(π1

T M)to be the boundary of the subset of M lying betw een the tw o p lanes π0and π1.B is invariant under each of the refl ections x2→ ?x2,...,xn→ ?xn.As Schoen’s Theorem 2 in[10],M itself inherits these refl ection symm etries.We conclude that all of M does indeed inherit the symm etriesm entioned. In particu lar,the cen ter ofevery spherical cross-section of M lies in the sam e2-p lane,that w here x2=0,...,xn=0.

Since u=u(x1,...,xn),w e need tw o functions o f u,r(u)and c(u),to denote(resp.)the rad ius and the x1coord inate of the cen ter of the sphere in hyperp lane xn+1=u.M is the setof poin tsw here

Next,w e need to calcu late Δu.Firstly,taking the fi rst and second derivatives of(3.1)to x1,xi(i≥ 2)respectively,w e have

and

So w e obtain

The functions r,c,r′,c′,r′′,c′′are functions o f u=xn+1on ly,or(m ore to the point)have constant values on any fixed p lane xn+1=u.We m ay therefore exam ine the values o fthese functions at d ifferent locations in M and find equations that hold sim u ltaneously. In particu lar,ifw eevaluate theequation thatdescribesm inim ality along the threesubsets of M w herein

w eget the follow ing three(sligh tly rearranged)equations:

Taking half the sum of the fi rst tw o equations,w e have

Subtracting the third equation,w e fi nally obtain

Since n≥3,w e conclude that c(u)isconstan tand hence M isahypersurfaceof revolu tion.

This com p letes the p roofof Corollary 1.2.

Acknow ledgem en t

Theauthorsw ou ld like to thank Prof.Xi-Nan M a forsuggesting this problem and help fu l d iscussions.The au thors also w ou ld like to thank Prof.Xu-Jia Wang to comm unicate w ith us the possibility of Corollary 1.2 in Ju ly 2012.

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[8]Ma Xi-Nan,Ou Qianzhong and Zhang Wei,Gaussian cu rvature estim ates for the convex level sets of p-harm onic functions.Comm.Pure Appl.M ath.63(7)(2010),935-971.

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?Correspond ing au tho r.Emailaddresses:kongs@ust c.edu.cn(S.L.Kong),j j xujane@shu.edu.cn(J.J.Xu)

AMSSub ject Classifi cations:35B45,35J92,35B50

Chinese Library Classifi cations:O175.25