Ann. I. H. Poincaré – AN 29 (2012) 83–108
www.elsevier.com/locate/anihpc
Regularity of flat free boundaries in two-phase problems for the p-Laplace operator
John L. Lewis
a,1, Kaj Nyström
b,∗,2aDepartment of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA bDepartment of Mathematics, Umeå University, S-90187 Umeå, Sweden Received 15 May 2010; received in revised form 13 June 2011; accepted 8 September 2011
Available online 16 September 2011
Abstract
In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for thep-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.
©2011 Elsevier Masson SAS. All rights reserved.
MSC:35J25; 35J70
Keywords:p-Harmonic function;p-Subharmonic; Free boundary; Two-phase; Boundary Harnack inequality; Hopf boundary principle; Lipschitz domain;-Monotone; Monotone; Regularity
1. Introduction
In [1–3] a theory for general two-phase free boundary problems for the Laplace operator was developed. In partic- ular, in [1] Lipschitz free boundaries were shown to beC1,γ-smooth for someγ∈(0,1)and in [2] it was shown that free boundaries which are well approximated by Lipschitz graphs are in fact Lipschitz. Finally, in [3] the existence part of the theory was developed. In [19] we initiated our study of the corresponding problems for thep-Laplace operator by generalizing the results in [1] to thep-Laplace operator whenp=2, 1< p <∞. In this paper we continue our study by establishing results similar to [2] for thep-Laplace operator when p=2, 1< p <∞. As in [19] we note that the generalization beyond the harmonic case, which corresponds top=2, is non-trivial due to the non-linear and degenerate character of thep-Laplace operator. In particular, our results and arguments rely heavily on the techniques developed in [14–20].
To properly state our results we need to introduce some notation. Points in Euclideann-space Rn are denoted byx=(x1, . . . , xn)or(x, xn)wherex=(x1, . . . , xn−1)∈Rn−1.LetE, ∂E,¯ diamEbe the closure, boundary, and
* Corresponding author.
E-mail addresses:[email protected] (J.L. Lewis), [email protected] (K. Nyström).
1 Lewis was partially supported by NSF grants DMS-0552281 and DMS-0900291.
2 Nyström was partially supported by grant VR-70629701 from the Swedish research council VR.
0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2011.09.002
diameter of E.Let ·,· denote the standard inner product onRn, |x| = x, x1/2, the Euclidean norm ofx, and let dx be Lebesgue n-measure on Rn. Given x∈Rn,r >0 ands >0, putB(x, r)= {y∈Rn: |x−y|< r}and Qr,s(x)= {y=(y, yn): |y−x|< r, |yn−xn|< s}.In caser=s,we writeQr(x)forQr,r(x).GivenE, F⊂Rn, let E+F denote the set{x+y: x∈E, y∈F}and let d(E, F )be the Euclidean distance fromE toF. In case E= {y},we writed(y, F ).Let
h(E, F )˘ =max
sup
y∈E
d(y, F ),sup
y∈F
d(y, E)
be the Hausdorff distance fromEtoF.
IfO⊂Rnis open and 1q∞,then byW1,q(O),we denote the space of equivalence classes of functionsf with distributional gradient∇f =(fx1, . . . , fxn),both of which areq-th power integrable onO.Let
fW1,q(O)= fLq(O)+ |∇f|Lq(O)
be the norm in W1,q(O)where · Lq(O) denotes the usual Lebesgueq-norm inO.Next letC0∞(O)be the set of infinitely differentiable functions with compact support inOand letC(E), be the set of continuous functions onE.
GivenD⊂Rn a bounded domain (i.e., a connected open set) and 1< p <∞,we say thatuisp-harmonic inD providedu∈W1,p(O)wheneverO¯ ⊂Dand
|∇u|p−2∇u,∇θdx=0 (1.1)
wheneverθ∈C0∞(D).Observe that ifuis smooth enough and∇u=0 inD,then
∇ ·
|∇u|p−2∇u
≡0 inD (1.2)
so u is a classical solution in D to the p-Laplace partial differential equation. Here, as in the sequel, ∇· is the divergence operator.uis said to bep-subharmonic (p-superharmonic) inDprovidedu∈W1,p(O)wheneverO¯ ⊂D and (1.1) holds with=replaced by() wheneverθ0 inD. Letu∈C(D)¯ and putD+(u)= {x∈D: u(x) >0}, F (u)=∂D+(u)∩D.LetD−(u)be the interior of{x∈D: u(x)0}and setu+=max{u,0},u−= −min{u,0}.
Assuming thatw∈F (u), and thatF (u)is smooth in a neighborhood ofw, we letν=ν(w)denote the unit normal, toF (u)atw, pointing intoD+(u). Moreover, we letu+ν(w)andu−ν(w)denote the normal derivatives ofu+andu− atwin the direction ofν.Note thatu+ν,−u−ν 0.In this paper we consider weak solutions, defined and continuous inD, to the following general two-phase free boundary problem,¯
(i) ∇ ·
|∇u|p−2∇u
=0 inD+(u)∪D−(u), (ii) u+ν(w)=G
−u−ν(w)
wheneverw∈F (u),
(iii) u=k∈C(∂D) on∂D. (1.3)
In (1.3)(ii) the function G:[0,∞)→ [0,∞)defines the free boundary condition and the interfaceF (u)is referred to as the free boundary. If we make no a priori classical regularity assumptions on the interfaceF (u)then the free boundary condition in (1.3)(ii) must be interpreted in a weak sense and in particular a notion of weak solutions to the problem in (1.3) has to be introduced. Let·,·+=max{·,·,0},·,·−= −min{·,·,0}. We will work with the following notion of weak solutions to the problem in (1.3).
Definition 1.4.Let D⊂Rn be a bounded domain,u∈C(D)¯ and 1< p <∞, be given.u is said to be a (weak) solution to the problem in (1.3) if u is p-harmonic in D+(u)∪D−(u), u=k, on ∂D and if the free boundary condition in (1.3)(ii) is satisfied in the following sense. Assume thatw∈F (u)and there exists a ballB(w,ˆ ρ),ˆ wˆ ∈ D+(u)∪D−(u) withw∈∂B(w,ˆ ρ). Ifˆ ν=(wˆ −w)/| ˆw−w|, then the following holds, as x→w, for some α, β∈ [0,∞]withα=G(β),
(i) ifB(w,ˆ ρ)ˆ ⊂D+(u), thenu(x)=αx−w, ν+−βx−w, ν−+o
|x−w| , (ii) ifB(w,ˆ ρ)ˆ ⊂D−(u), thenu(x)=αw−x, ν+−βw−x, ν−+o
|x−w| .
Letθ (ν,ν)˜ be the angle betweenν,ν˜=0 inRn. If|ν| =1, θ0∈(0, π/2],set Γ (ν, θ0):=
˜
ν: |˜ν| =1 andθ (ν,ν) < θ˜ 0 , C(ν, θ0):=
tν:˜ ν˜∈Γ (ν, θ0)and 0< t <∞
. (1.5)
Given >0 we say thatuis-monotone inO⊂Rn, with respect to the directions inΓ (ν, θ0)if sup
B(x,sinθ0)
u
y−ν
u(x) (1.6)
wheneverandx∈OwithB(x−ν, sinθ0)⊂O. uis said to be monotone inO⊂Rnwith respect to the directions inΓ (ν, θ0)if whenevery∈B(x, r)⊂Oand|yy−−xx|∈Γ (ν, θ0),it is true thatu(y)u(x).Note that ifuis monotone inOandB(x, r)⊂O,then (1.6) holds whenever 0< r/4.
Recall thatf:E→Ris Lipschitz onEprovided there existsb,0< b <∞,such that|f (z)−f (w)|b|z−w| wheneverz, w∈E. The infimum of all b such that this inequality holds is called the Lipschitz norm off onE, denoted byfLip(E).It is well known that ifE=Rn−1,thenf is differentiable almost everywhere onRn−1and fLip(Rn−1)= |∇f|L∞(Rn−1).In this paper we prove the following theorems.
Theorem 1. LetD⊂Rn be a bounded domain, assume that u∈C(D)¯ and that u is a weak solution in D, for some1< p <∞, to the problem in(1.3)in the sense of Definition1.4. Assume that the functionG0is strictly increasing and, for some N >0, that s→s−NG(s)is decreasing on (0,∞). Assume0∈F (u),Q¯1(0)⊂D, and thatθ¯∈(π/4, π/2). Then there exists¯= ¯(θ , p, n, N ) >¯ 0such that ifuis-monotone onQ1(0)with respect to the directions in the spherical cap,Γ (en,θ ), for some¯ ∈(0,), then¯ uis monotone inQ1/2(0)with respect to the directions inΓ (en,θ¯1)whereθ¯1> π/4.In particular,
D+(u)∩Q1/2(0)= x, xn
∈Rn: xn>f¯ x
∩Q1/2(0), F (u)∩Q1/2(0)=
x, xn
∈Rn: xn= ¯f x
∩Q1/2(0), wheref¯:Rn−1→Ris Lipschitz with ¯fLip(Rn−1)<1.
Theorem 2. LetD⊂Rn be a bounded domain, assume that u∈C(D)¯ and that u is a solution in D, for some 1< p <∞, to the problem in(1.3)in the sense of Definition1.4. Assume thatG0is strictly increasing, Lipschitz continuous withGLip(Rn−1)C,G(0) >0,and, for someN >0, thats−NG(s)is decreasing on(0,∞).If0∈ F (u)andQ¯1(0)⊂D, then there exist >ˆ 0,θˆ∈(π/4, π/2),both depending onp,n,C,N,G(0), andmaxQ¯1(0)u−, such that the following statement is valid. If0< ,ˆ θˆθπ/2,and ifu+is-monotone inQ1(0)with respect to the directions inΓ (en, θ ),thenu+is monotone inQ1/2(0)with respect to the directions inΓ (en,θˆ1)whereθˆ1> π/4.
As a corollary to Theorem 2 we also prove the following.
Corollary 1.Let D⊂Rn be a bounded domain, assume thatu∈C(D)¯ and that u is a solution in D, for some 1< p <∞, to the problem in(1.3)in the sense of Definition1.4. LetGbe as in the statement of Theorem2and assume that0∈F (u)andQ¯1(0)⊂D. Assume that there existsη1such that
η−1d
x, F (u)
u(x)ηd
x, F (u)
wheneverx∈D+(u)∩ ¯Q1(0).
Then there exist >ˆ 0,θˆ∈(π/4, π/2),both depending on p,n, C,N, G(0),maxQ¯1(0)u− and η, such that the following statement is valid. If0< ,ˆ θˆθπ/2,and if
h˘
F (u)∩ ¯Q1(0), Λ∩ ¯Q1(0)
,
whereΛ= {(x,f (x˜ )): x∈Rn−1}and ˜fLip(Rn−1)tan(π/2−θ ), thenu+is monotone inQ1/2(0)with respect to the directions inΓ (en,θˆ1)whereθˆ1> π/4.
As mentioned earlier, Theorems 1, 2, and Corollary 1 are part of the program initiated in [19]. In particular, in [19]
we proved the following theorem.
Theorem A.LetD⊂Rnbe a bounded domain, assume thatu∈C(D)¯ and thatuis a solution inD, for some1< p <
∞, to the problem in(1.3)in the sense of Definition1.4. Moreover, suppose thatG >0is strictly increasing on[0,∞) and, for someN >0, thats−NG(s)is decreasing on(0,∞). Assume that0∈F (u),B(0,¯ 2)⊂D,maxB(0,2)|u| =1 and that,
D+(u)∩B(0,2)=Ω∩B(0,2), F (u)∩B(0,2)=∂Ω∩B(0,2), Ω=
y= y, yn
∈Rn: yn> ψ y
,
in an appropriate coordinate system, whereψ is Lipschitz onRn−1withM= ψLip(Rn−1).Then there exists σ= σ (p, n, M, N )∈(0,1)such that ∇ψ is Hölder continuous of order σ on {x: (x, ψ (x))} ∈B(0,1/8).The Cσ- Hölder norm of∇ψdepends only onp, n, M, N.
Using Theorem A and invariance of thep-Laplacian under rotations, translations, and dilations, we deduce under the scenario of either Theorems 1, 2 or Corollary 1, thatF (u)∩B(0,1/16)is of classC1,σ. Furthermore, to indicate earlier work, forp=2,Theorem A is given in [1] while Theorems 1, 2 and Corollary 1 can be found in [2]. The work in [1,2] was generalized in [22,23], to solutions of fully non-linear concave PDE of the formF (D2u)=0,where F is homogeneous. Further analogues of the work in [1] were obtained for a class of nonisotropic operators in [7]
and in [8] the concavity assumption in [22] was removed for viscosity solutions to fully non-linear PDE of the form, F (D2u, Du)=0,whereF is homogeneous in both arguments. Moreover, generalizations of the results in [1] were made in [9] to fully non-linear PDE of the formF (D2u, x)=0 which have interiorC1,1-estimates. Generalizations of the work in [1], to linear divergence form PDE with variable Lipschitz continuous coefficients were obtained in [6,11].
Finally the work in [1,2] was generalized to viscosity solutions of certain linear nondivergence form elliptic PDE with drift term in [10]. However, Theorems A, 1, 2 and Corollary 1 are the first generalizations of [1,2] to divergence form operators ofp-Laplacian type.
The rest of the paper is organized in the following way. In Section 2, which partly is of a preliminary nature, we collect a number of results from [14,15,20,19] concerningp-harmonic functions in Lipschitz domains. In Section 3 we construct appropriatep-subsolutions to the free boundary problem under consideration using results from [2,19, 22]. We also prove that ifuis as in Theorem 1, thenusatisfies
c−1 u(y)
d(y, ∂Ω) ∇u(y) c u(y)
d(y, ∂Ω) (1.7)
wherec=c(p, n),at points sufficiently far away from∂Q1(0)∪F (u).Finally, in Section 3 we use (1.7) to show that u is monotone in the directionsΓ (en,θ )¯ at points sufficiently far away from∂Q1(0)∪F (u).In Section 4 we then prove Theorem 1 leaving the proofs of Theorem 2 and Corollary 1 for Section 5. At the end of Section 5 we mention possible generalizations of Theorems 1, 2 and Corollary 1.
Concerning our proofs of Theorems 1, 2 and Corollary 1 we note that our argument combines the geometric approach developed in [1,2,23] with the analytic techniques forp-harmonic functions in Lipschitz domains, and in domains which are well approximated by Lipschitz graph domains in the Hausdorff distance sense, developed in [14–
16,18,19]. In particular, based on the results in our previous papers, we are able to proceed along the lines of [2] and [23] to complete the proofs. The most tricky part of the argument, as compared to the harmonic case in [2], is to obtain a contradiction when the graph of the solutionuand the graph of the carefully constructed subsolutionvt touch. In [19] we obtained a contradiction, and thus proved Theorem A, by using a boundary Harnack inequality – Hopf type maximum principle (see Theorem 2.9 below) as well as the fact that in [19]u satisfied the fundamental inequality (1.7) up toF (u).In the proof of Theorem 1, we can no longer assume (1.7) up toF (u). Instead we have to introduce several other comparison functions and prove these functions can be used to get the desired contradiction.
We emphasize that on the one hand this paper is not user friendly, as it relies heavily on previous rather technical work of the authors mentioned above. On the other hand, we state and give references for results which are used in this paper. In general our strategy in writing this paper has been to refer to previous work whenever possible, as well as, to provide details whenever our arguments differ from previous arguments. Thus as a minimum the interested reader should first be familiar with [2,19], and to have these papers at hand.
Finally the authors would like to thank the referee for some helpful comments.
2. Estimates forp-harmonic functions
We say thatΩ⊂Rnis a bounded Lipschitz domain if there exists a finite set of balls{B(xi, ri)}, withxi∈∂Ωand ri>0, such that{B(xi, ri)}constitutes a covering of an open neighborhood of∂Ωand such that, for eachi,
Ω∩B(xi,4ri)= x=
x, xn
∈Rn: xn> φi x
∩B(xi,4ri),
∂Ω∩B(xi,4ri)= x=
x, xn
∈Rn: xn=φi
x
∩B(xi,4ri), (2.1)
in an appropriate coordinate system and for a Lipschitz functionφi onRn−1.The Lipschitz constant ofΩis defined to beM=maxiφiLip(Rn−1).Ifw∈∂Ω, 0< r < r0, we let(w, r)=∂Ω∩B(w, r)be the naturally defined surface ball and we letei, 1in,denote the point inRnwith one in thei-th coordinate position and zeroes elsewhere.
Moreover, throughout the paper cwill denote, unless otherwise stated, a positive constant1, not necessarily the same at each occurrence, which only depends onp,n, andM. In general,c(a1, . . . , an)denotes a positive constant 1, not necessarily the same at each occurrence, which depends onp,n,M anda1, . . . , an. IfA≈BthenA/Bis bounded from above and below by constants which, unless otherwise stated, depend onp,nandMat most. Moreover, we let maxB(z,s)u,ˆ minB(z,s)uˆbe the essential supremum and infimum ofuˆonB(z, s)wheneverB(z, s)⊂Rnanduˆ is defined onB(z, s).
We first state a number of basic lemmas in Lipschitz domains. As a general reference for proofs of the following lemmas we refer to [14]. Lemmas 2.2, 2.3, 2.5, are classical interior type estimates for non-linear partial differential equations in divergence form. Lemma 2.4 is well known for harmonic functions while Theorems 2.6, 2.7, are recent results of the authors. Their proofs use deformations ofuˆ into v,ˆ similar to the one in (2.13), as well as uniform ellipticity estimates, similar to those in (2.14)–(2.17), and classical boundary Harnack inequalities for nondivergence uniformly elliptic partial differential equations.
Lemma 2.2.Givenp,1< p <∞,letuˆbe a positivep-harmonic function inB(w,2r). Then (i) max
B(w,r)uˆc min
B(w,r)u.ˆ
Furthermore, there existsα=α(p, n)∈(0,1)such that ifx, y∈B(w, r), then (ii) u(x)ˆ − ˆu(y) c
|x−y| r
α
max
B(w,2r)u.ˆ
Lemma 2.3.LetΩ ⊂Rn be a bounded Lipschitz domain andpgiven,1< p <∞. Letw∈∂Ω,0< r < r0, and suppose thatuˆ is a non-negativep-harmonic function inΩ∩B(w,2r).Assume also thatuˆ is continuous onΩ¯ ∩ B(w,2r)withuˆ≡0 on(w,2r). There existc=c(p, n, M)1and α=α(p, n, M)∈(0,1)such that ifx, y∈ Ω∩B(w, r), then
u(x)ˆ − ˆu(y) c
|x−y| r
α
Ω∩maxB(w,2r)u.ˆ
Lemma 2.4.LetΩ ⊂Rn be a bounded Lipschitz domain andp given,1< p <∞.Letw∈∂Ω,0< r < r0, and suppose that uˆ is a non-negativep-harmonic function inΩ∩B(w,2r).Assume also thatuˆ is continuous inΩ¯ ∩ B(w,2r)withuˆ≡0on(w,2r). There existsc=c(p, n, M),1c <∞, such that ifr˜=r/c,then
max
Ω∩B(w,r)˜ uˆcuˆ ar˜(w)
,
wherear˜(w)is any point inΩ∩B(w,r)˜ withd(ar˜(w), ∂Ω)r/c.˜
Lemma 2.5.LetΩ⊂Rnbe a bounded Lipschitz domain andpgiven,1< p <∞.Letw∈∂Ω,0< r < r0and sup- pose thatuˆis a non-negativep-harmonic function inΩ∩B(w,2r).Assume also thatuˆis continuous inΩ¯∩B(w,2r) withuˆ≡0on(w,2r). Extenduˆ toB(w,2r)by defininguˆ≡0onB(w,2r)\Ω. Thenuˆ has a representative in
W1,p(B(w,2r))with Hölder continuous partial derivatives inΩ∩B(w,2r). In particular, there exists σ∈(0,1], depending only onpandn, such that ifx, y∈B(w,˜ r/2),˜ B(w,˜ 4r)˜ ⊂Ω∩B(w,2r),then
c−1 ∇ ˆu(x)− ∇ ˆu(y)
|x−y|/r˜σ
max
B(w,˜r)˜ |∇ ˆu|cr˜−1
|x−y|/r˜σ
ˆ u(w).˜ Moreover, if for someβ∈(1,∞),
ˆ u(y)
d(y, ∂Ω)β ∇ ˆu(y) for ally∈B(w,˜ 2r),˜
thenuˆ∈C∞(B(w,˜ 2r))˜ and given a positive integerkthere existsc¯1,depending only onp,n,β,k,such that max
B(w,˜ r2˜)
Dkuˆ c¯ u(ˆ w)˜
d(w, ∂Ω)˜ k whereDkuˆdenotes an arbitraryk-th order derivative ofu.ˆ Next we state two theorems proved in [14,15].
Theorem 2.6. LetΩ ⊂Rn be a bounded Lipschitz domain with constantsM, r0. Given p,1< p <∞, w∈∂Ω, 0< r < r0, suppose that uˆ and vˆ are positive p-harmonic functions inΩ∩B(w,2r). Assume also that u,ˆ vˆ are continuous in Ω¯ ∩B(w,2r)and thatuˆ≡0≡ ˆv on(w,2r). Under these assumptions there existsc1,1c1<
∞, γ >0,depending only onp,n,andM, such that ifx, y∈Ω∩B(w, r/c1),then log
u(y)ˆ ˆ v(y)
−log u(x)ˆ
ˆ v(x)
c1
|x−y| r
γ
.
Theorem 2.7. Let Ω ⊂Rn be a bounded Lipschitz domain with constants M,r0.Let w∈∂Ω, 0< r < r0,and suppose that (2.1)holds withxi, ri, φi,replaced byw,r,φ.Givenp,1< p <∞,letuˆ be a positivep-harmonic function in Ω∩B(w,2r).Assume also thatuˆ is continuous inΩ¯ ∩B(w,2r)withuˆ≡0 on(w,2r).Then there existsθ0∈(0, π/2]and1c2<∞,both depending only onp,n,M,such that
c−21 u(y)ˆ d(y, ∂Ω)
∇ ˆu(y), ξ
∇ ˆu(y) c2 u(y)ˆ d(y, ∂Ω) whenevery∈Ω∩B(w, r/c2)andξ∈Γ (en, θ0).
We note that in [14] we proved, under the assumptions stated in Theorem 2.6, that log
u(y)ˆ ˆ v(y)
−log u(x)ˆ
ˆ v(x)
c wheneverx, y∈Ω∩B(w, r/c). (2.8)
Herec=c(p, n, M).Using (2.8) we then obtained, essentially simultaneously, Theorems 2.6 and 2.7 in [15]. Further- more, in [19] we also proved the following theorem.
Theorem 2.9.LetΩ⊂Rnbe a bounded Lipschitz domain with constantsM,r0. Letw∈∂Ω,0< r < r0,1< p <∞, and suppose that u,ˆ vˆ are positive p-harmonic functions in Ω∩B(w,2r)with vˆu.ˆ Assume also that u,ˆ vˆ are continuous inΩ¯ ∩B(w,2r)withuˆ≡ ˆv≡0on(w,2r).Then there existsc3,1< c3<∞,such that
ˆ
u(y)− ˆv(y) ˆ
v(y) c3u(x)ˆ − ˆv(x) ˆ
v(x) wheneverx, y∈Ω∩B(w, r/c3).
We note that Theorem 2.9 implies (2.8), as follows easily from the fact that the p-Laplacian is invariant under multiplication by a constant. Thus replacingvˆbyδvˆin the above display, multiplying both sides byδ,and then letting δ→0,we get (2.8). A slightly more involved argument (see Section 6 of [20]) also gives Theorem 2.6. Also for later use we observe from Theorem 2.6 thatvcan be replaced byuin the denominator of the display in Theorem 2.9.
Next we state Lemma 2.10 which gives a useful criteria for determining when a positive p-harmonic function satisfies the last inequality in Theorem 2.7 at a point. Note that Lemma 2.10 is similar to Lemmas 4.3 and 5.4 in [15], Lemma 3.1 in [18], and Lemma 3.18 in [20]. Hence we do not include a proof of this lemma here.
Lemma 2.10.LetObe an open set, and suppose thatu,ˆ vˆare positivep-harmonic functions inO.Leta1,y∈O,
|ξ| =1,and assume that 1
a ˆ v(y) d(y, ∂O)
∇ ˆv(y), ξ
∇ ˆv(y) a v(y)ˆ d(y, ∂O).
Let˜−1=(ca)(1+σ )/σ,whereσ is as in Lemma2.5and c=c(p, n). Then the following statement is true forc= c(p, n)suitably large. If
(1− ˜)L˜ vˆ ˆ
u(1+ ˜)L˜
inB(y,14d(y, ∂O))for someL,˜ 0<L <˜ ∞,then 1
ca ˆ u(y) d(y, ∂O)
∇ ˆu(y), ξ
∇ ˆu(y) ca u(y)ˆ d(y, ∂O).
To continue our basic estimates, we list some results for the difference of twop-harmonic functions. To this end, letu,ˆ vˆbe positivep-harmonic functions in an open setO,satisfying 1u/ˆ vˆc4.Suppose also thatvˆsatisfies, for someδ >ˆ 1,the fundamental inequality
δˆ−1 v(x)ˆ
d(x, ∂O) ∇ ˆv(x) δˆ v(x)ˆ
d(x, ∂O), wheneverx∈B(w, r), (2.11)
whereB(w,¯ 2r)⊂O.Define
e(x)= ˆu(x)− ˆv(x) wheneverx∈B(w, r). (2.12)
and put
u(x, τ )=τu(x)ˆ +(1−τ )v(x)ˆ wheneverx∈B(w, r), andτ∈ [0,1]. (2.13) Clearly,e(x)=u(x,1)−u(x,0).Usingp-harmonicity ofu,ˆ vˆand that
|ξ|p−2ξ− |η|p−2η= 1 0
d{|t ξ+(1−t )η|p−2[t ξ+(1−t )η]}
dt dt
wheneverξ, η∈Rn\ {0},it follows thateis a weak solution to ˆ
Le:=
n
i,j=1
∂
∂yi
bˆij(y)eyj(y)
=0 whenevery∈B(w, r) (2.14)
where
bˆij(y)= 1 0
bij(y, τ ) dτ,
bij(y, τ )= ∇u(y, τ ) p−4
(p−2)uyi(y, τ )uyj(y, τ )+δij ∇u(y, τ ) 2
. (2.15)
Herei, j∈ {1, . . . , n}andδij is the Kroneckerδ.Ify∈B(w, r),then from (2.14) we observe thate= ˆu− ˆv is the solution in to a symmetric divergence form PDE with ellipticity constants aty estimated by
min{p−1,1}|ξ|2λ(y)ˆ n
i,j=1
bˆij(y)ξiξjmax{p−1,1}|ξ|2λ(y),ˆ (2.16)
wheneverξ ∈Rn, and where
λ(y)ˆ = 1 0
∇u(y, τ ) p−2dτ≈ ∇ ˆu(y) + ∇ ˆv(y) p−2≈ ˆ
v(y)/d(y, ∂O)p−2
. (2.17)
The right-hand side inequality in (2.17) was obtained by using Lemma 2.5 to estimate |∇ ˆu(·)| in terms of ˆ
u(·)/d(·, ∂O), the assumption thatuˆc4v,ˆ (2.11), and the fact that 1/2d(x, ∂O)
d(z, ∂O) 2 forx, z∈B(w, r)
sinceB(w,¯ 2r)⊂O.In (2.17) the constants of proportionality depend only onp,n,δ,ˆ andc4.In Sections 4 and 5 we will need the following interior Harnack inequality.
Lemma 2.18.Letu,ˆ v,ˆ be positivep-harmonic functions inOwith1u/ˆ vˆc4and suppose thatvˆsatisfies(2.11) withr=2r, w˜ = ˜w,whereB(¯ w,˜ 4r)˜ ⊂O.Ife= ˆu− ˆv,then there exists a constantc=c(p, n,δ, cˆ 4) >1such that
B(maxw,˜r)˜ ec min
B(w,˜ r)˜ e.
Proof. From (2.16), (2.17), and (2.11), we see thatLˆ is uniformly elliptic inB(w,˜ 2r)˜ with bounded measurable co- efficients. Constants depend only onp,n,δ,ˆ c4.The stated Harnack inequality now follows from classical arguments, see [21]. 2
Finally in this section we prove a lemma concerning properties of a positive minimalp-harmonic function in a cone.
More specifically, if 0< θ0< π,recall the definition of the coneC(en, θ0)in (1.5). We writeC(θ0)for C(en, θ0).
Givenp, 1< p <∞,we say thatuˆis a minimal positivep-harmonic function inC(θ0),relative to∞,provideduˆis a positivep-harmonic function inC(θ0)with continuous boundary value zero on∂C(θ0).
Lemma 2.19.Givenθ0∈(0, π],and1< p <∞,there exists a unique minimal positivep-harmonic function,uˆ= ˆ
u(·, θ0),inC(θ0)withu(eˆ n)=1.Moreover, ifr= |x|,xn=rcosθ,0θ < θ0,are spherical coordinates ofx,then there existψ∈C∞(cosθ0,1)andγ >0such that
ˆ
u(x)= ˆu(r, θ )=rγψ (cosθ ), 0θ < θ0.
Also,γis a decreasing positive continuous function ofθ0∈(0, π )withγ (π/2)=1.
Proof. We note that in [12], homogeneousp-harmonic functions of the above form are constructed in cones. To begin the proof of Lemma 2.19, existence of a minimal positive p-harmonic functionuˆ relative to∞ inC(θ0)is easily shown. For example one can take uˆ to be the limit of a subsequence of(um)∞1 whereum is a positivep-harmonic function inC(θ0)∩B(0, m)with continuous boundary value 0 on∂C(θ0)∩B(0, m)andum(en)=1.Existence ofum, m=1,2, . . . ,follows from a calculus of variations argument. Applying Lemmas 2.2–2.5 toum,m=1,2, . . . ,and using Ascoli’s theorem we getu.ˆ To prove uniqueness ofu,ˆ letvˆbe another minimal positivep-harmonic function inC(θ0)withv(eˆ n)=1.Using Theorem 2.6 withΩ=C(θ0)∩B(0,2r),w=0,we get, upon lettingr→ ∞,that
ˆ
v= ˆu. Thusuˆ is the unique minimal positivep-harmonic function inC(θ0)withu(eˆ n)=1.To obtain the desired form foruˆwe first note that uniqueness ofuˆand invariance of thep-Laplace equation under rotations imply thatuˆis symmetric about the xn axis. Thus we writeu(r, θ )ˆ foru(x)ˆ whenx∈ ¯C(θ0)andr= |x|,xn=rcosθ, 0θθ0. Also since thep-Laplacian is invariant under dilations it follows from uniqueness ofuˆthat
ˆ
u(λx)= ˆu(λen)u(x)ˆ wheneverλ >0 andx∈C(θ0). (2.20) Differentiating (2.20) with respect toλand evaluating atλ=1 we find that
ruˆr(r, θ )=
x,∇ ˆu(x)
=
∇ ˆu(en), en ˆ u(r, θ ).
Dividing this equality by ru(r, θ ),ˆ integrating, and then exponentiating, we getu(r, θ )ˆ =rγψ (cosθ ) whereγ = en,∇ ˆu(en).Continuity ofγ once again follows from uniqueness ofu(ˆ ·, θ0)and Lemmas 2.2–2.5. Also,γ (θ0)is decreasing forθ0∈(0, π ),as follows easily from comparing solutions in different cones and using the maximum principle forp-harmonic functions. Finallyu(x)ˆ =xn=rcosθwhenθ0=π/2,soγ (π/2)=1. 2
3. Preliminary reductions for Theorem 1
Recall that given a bounded domain D and 1< p <∞, we say that u is p-subharmonic inD provided u∈ W1,p(Ω)wheneverΩ is an open set withΩ¯ ⊂Dand
|∇u|p−2∇u,∇θdx0 (3.1)
wheneverθ∈C0∞(D)andθ0 onD. We say thatuisp-superharmonic provided−uisp-subharmonic. Moreover, we letC2(D)denote the set of functions which have continuous second partial derivatives inD.Forφ∈C2(D)we let∇2φ(x)denote the Hessian matrix ofφatx∈D.
LetS(n)denote the set of all symmetricn×nmatrices and letP be the Pucci type extremal operator (see [4]) defined, forM∈S(n), as
P (M)= inf
A∈Ap
n
i,j=1
aijMij. (3.2)
HereApdenotes the set of all symmetricn×nmatricesA= {aij}which satisfy min{p−1,1}|ξ|2
n
i,j=1
aijξiξjmax{p−1,1}|ξ|2 wheneverξ∈Rn. (3.3) For the proof of the following lemma we refer to [19, Lemma 3.5].
Lemma 3.4.Letφ >0be inC2(D),∇φL∞(D)1/2, pfixed,1< p <∞,and suppose that φ(x)P
∇2φ(x)
50pn ∇φ(x) 2 wheneverx∈D.
Letube continuous in an open setOcontaining the closure of
x∈DB(x, φ(x))and define v(x)= max
B(x,φ (x))¯
u
wheneverx∈D. Ifuisp-harmonic inO\ {u=0},thenvis continuous andp-subharmonic in{v=0} ∩D.
Next we consider the asymptotic development, near F (v), of the p-subharmonic function constructed in Lemma 3.4.
Lemma 3.5.LetD,u,φ,O,andv=vφbe as in the statement of Lemma3.4and letGbe as in Theorem1. Suppose also that(i), (ii)of Definition1.4hold for someα,β,wheneverw∈O∩∂{u >0}and there existsB(w,ˆ ρ)ˆ ⊂O\
∂{u >0}withw∈∂B(w,ˆ ρ).ˆ Ifw˜ ∈F (v),then there existw∗∈D+(v)andρ∗>0such thatB(w∗, ρ∗)⊂D+(v)and
˜
w∈∂B(w∗, ρ∗). Also, there existα,˜ β˜∈ [0,∞), such that the following hold, asx→ ˜w, withν˜=(w∗− ˜w)/|w∗− ˜w|, (a) v(x)α˜x− ˜w,ν˜+− ˜βx− ˜w,ν˜−+o
|x− ˜w| ,
(b) α˜
1− |∇φ(w)˜ | G
β˜ 1+ |∇φ(w)˜ |
.
Proof. The proof of Lemma 3.5 for p=2 can be found in Lemmas 10, 11 of [1]. The proof is based on a purely geometric argument using smoothness ofφ,and the asymptotic expansion ofuin balls tangent toF (u).Hence it is also valid here. 2
We will also use the following lemma.
Lemma 3.6.LetD,φ,be as in Lemma3.4. Assume >0small,pfixed,1< p <∞,θ0∈(π/4, π/2)and letO˜ be an open set containing
y: |y−z|2for somezin the closure of
x∈D
B
x, φ(x) .
Assume thatuis continuous, and-monotone inO˜ with respect to the directions inΓ (en, θ0).Assume also thatuis p-harmonic inO˜\∂{u >0}and satisfies(as in Lemma3.5) (i), (ii)of Definition1.4at points ofO˜∩∂{u >0}.LetG be as in Theorem1and definevrelative tou,φas in Lemma3.5. If0< θθ0,12sinθ0< φ(x) < ,and
sinθ 1
1+ |∇φ|(x)
sinθ0−
2φ(x)(cosθ0)2− |∇φ|(x)
for allx∈D,thenvis monotone inD with respect to the directions inΓ (en, θ)andF (v)∩D is the graph of a Lipschitz function with constantM,Mcotθ.
Proof. A proof for p=2 is given in [2, Lemma 2]. The proof involves a purely geometric argument so can be repeated here. However for p=2, 1< p <∞,certain issues should be clarified. Indeed, let xˆ∈D and suppose
ˆ
y∈∂B(x, φ(ˆ x))ˆ withu(y)ˆ =v(x).ˆ We consider several cases. Ifyˆ∈ ˜O\∂{x: u(x) >0}and∇u(y)ˆ =0,thenuis p-harmonic, consequently infinitely differentiable in a neighborhood ofy,ˆ so we can argue as in Lemma 2 of [2] to get thatvis increasing atxˆin the directions given byΓ (en, θ).Ifyˆ∈ ˜O∩∂{x: u(x) >0}andα˜ =0 in Lemma 3.5(a), we can once again use the geometric argument in [2] to conclude thatvis increasing atxˆin the directions given by Γ (en, θ)∩ {x: |x| =1}.Hence it remains to consider the cases when (a)yˆ∈ ˜O\∂{x: u(x) >0},∇u(y)ˆ =0,and (b)yˆ∈ ˜O∩∂{x:u(x) >0},α˜=0. In case (a) it follows from the Hopf boundary maximum principle and the fact that u(y)ˆ =max{u(z): z∈ ¯B(x, φ(ˆ x))ˆ }thatu≡u(y)ˆ inB(¯ x, φ(ˆ x)).ˆ In this case we note that
2φ(x)ˆ sinθ0(1−sinθ0)+(√ 2−1).
Thus B
ˆ x+
φ(x)ˆ −
en, sinθ0
∩B ˆ x, φ(x)ˆ
= ∅
and so it follows from the definition of -monotonicity, that uu(x)ˆ in an open neighborhood of xˆ+φ(x)eˆ n. Since v(x)u(x+φ(x)en)we conclude that vv(x)ˆ in an open neighborhood ofx.ˆ In case (b) it follows from Definition 1.4 applied tou, and the Hopf boundary maximum principle, thatu≡0 inB(¯ x, φ(ˆ x)). Hence, once againˆ using-monotonicity we have thatv0=v(x)ˆ in an open neighborhood ofx.ˆ Thusvis monotone inDwith respect to the directions inΓ (en, θ).Lipschitzness ofF (v)∩Dfollows from an easy geometric argument using monotonicity ofvand the definition ofF (v).The proof of Lemma 3.6 is now complete. 2
Finally, we will use the following set of functions{φt}, 0t1, to construct appropriatep-subharmonic functions to be used in the proof of Theorems 1 and 2.
Lemma 3.7. LetΛ= {(x, xn)∈Rn: xn=λ(x)}whereλ:Rn−1→R,λ(0)=0, andλLip(Rn−1)M <∞for someM1.Givenh,0< h <10−3,letΛ(h)= {(x, xn): |xn−λ(x)|< h} ∩ ¯Q2,8M(0).Ifβ∈(0,1),then there exists a family of functions{φt},0t1,C2-regular inΛ(h), andc=c(p, n, M, β)1,h0=h0(p, n, M, β) >0, such that the following holds. There existsμ=μ(p, n),0< μ2, such that forx∈Λ(h), t∈ [0,1],andh∈(0, h0], we have
(i) 1φt(x)1+μt, (ii) ∇φt(x) ct hβ−1, (iii) φt(x)P
∇2φt(x)
50pn ∇φt(x) 2,
(iv) φt(x)≡1 wheneverx∈Λ(h)\Q1−2h1−β,4M(0),
(v) 1+μt−φt(x)ct hβ wheneverx∈Λ(h)∩Q1−100h1−β,4M(0).
Proof. We could prove Lemma 3.7 by arguing as in [10] however we prefer to use a covering argument and the construction in [19]. This construction in turn was based on a construction in [22] (the construction in [23] appears incorrect). To begin the argument let {aij} ∈Ap and letL be the operator n
i,j=1
∂
∂xi(aij∂x∂
j).Givenxˆ∈Rn, ρ∈ (0,10−2),we claim there exists 0fˆ∈C2(Rn\ ¯B(x, ρ/2))ˆ satisfying
Lfˆ0 inRn\B(x, ρ)ˆ and fˆ≡0 inRn\B(x,ˆ 10). (3.8) Also, for somek+=k+(ρ, p, n)1,we have
(i) k+−1min(f , Lˆ f )ˆ onB(x,ˆ 6)\ ¯B(x, ρ),ˆ
(ii) |∇ ˆf|Lfˆk+ inRn\B(x, ρ).ˆ (3.9)
For example, letN be a non-negative integer and let f (x)˜ = | ˆx−x|−2N wheneverx∈Rn\B(x, ρ/2).ˆ
It is easily checked thatf˜satisfies (3.8), (3.9) onB(x,10)\B(x, ρ), forN=N (ρ, p, n) >0 large enough, except that f˜does not have support inB(x,ˆ 10).To remedy this letfˆ= [max(f˜−10−2N,0)]4.ThenfˆisC2onRn\ ¯B(x, ρ/2)ˆ andfˆ≡0 inRn\B(x,ˆ 10).From the definition ofLand (3.9) (ii) we find that
Lfˆ4f˜3Lf˜4f˜3|∇ ˜f| = |∇ ˆf| wheneverρ|x− ˆx|<10.
Thus (3.8), (3.9) are valid.
Next we chooseρ=(100M)−1and use a well-known covering lemma to get{B(wi, ρh)}with{B(wi, ρh/10)} pairwise disjoint, and
(a) wi∈Λ(3h)\Λ(2h), i=1,2, . . . .
(b)
B(wi, ρh)∩Λ(h)= ∅. (c) Λ(h)∩Q1−50h1−β,4M(0)⊂
B(wi,6h).
(d)
Λ(h)\Q1−2h1−β,4M(0)
∩
B(wi,10h)= ∅. (3.10)
Existence of{wi}satisfying (a), (c), (d) is easily seen. Note that (b) follows from (a) our choice ofρ,and the fact that λhas Lipschitz normM.Next givenwi we takexˆ=wiin the definition offˆand set
fˆi(x)= ˆf
wi+x−wi h
, x∈Rn\B(wi, ρh), i=1,2, . . . . Let
ψ (x)=hβ ˆ
fi(x), whenx∈Rn\
B(wi, ρh).
Observe from (3.8) thatfˆi has support inB(w¯ i,10h).Using this fact, as well as the disjointness of{B(wi, ρh/10)}, we deduce that ifx ∈Rn\
B(wi, ρh),then{i: fˆi(x)=0}has cardinality at mostc˜wherec˜= ˜c(p, n, M)1.
Using these facts (3.8)–(3.10) we see for somec−=c−(p, n, M, β)1 that (α) Lψ (x)c−−1hβ−2 whenx∈Λ(h)∩Q1−50h1−β,4M(0), (β) Lψh−1|∇ψ| onΛ(h),
(γ ) max
ψ, h|∇ψ|
c−hβ onΛ(h),
(δ) ψ≡0 onΛ(h)\Q1−2h1−β,4M(0). (3.11)
Letθ∈C0∞(Rn−1)with 0θ1 and (α) θ≡1 on
x∈Rn−1: x 1−100h1−β , (β) θ≡0 on
x∈Rn−1: x 1−50h1−β , (γ ) h1−β
n−1
i,j=1
∂2θ
∂xi∂xj +
n−1
i=1
∂θ
∂xi
chβ−1 onRn−1, (3.12)
