Citation: Daniela De Silva, Giorgio Tortone. Improvement of flatness for vector valued free boundary problems[J]. Mathematics in Engineering, 2020, 2(4): 598-613. doi: 10.3934/mine.2020027
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Dedicato a Sandro Salsa, con amicizia ed ammirazione.
This note is concerned with the vector valued one-phase free boundary problem,
{ΔU=0in Ω(U):=Ω∩{|U|>0};|∇|U||=1on F(U):=Ω∩∂Ω(U). | (1.1) |
Here U(x):=(u1(x),…,um(x)),x∈Ω, with Ω a bounded domain in Rn. In the scalar case, m=1, (1.1) is the Euler-Lagrange equation associated to the classical one-phase Bernoulli energy functional (u≥0),
J(u,Ω):=∫Ω(|∇u|2+χ{u>0}) dx. | (1.2) |
Minimizers of J were first investigated systematically by Alt and Caffarelli. Two fundamental questions are answered in the pioneer article [1], that is the Lipschitz regularity of minimizers and the regularity of "flat" free boundaries, which in turns gives the almost-everywhere regularity of minimizing free boundaries. The viscosity approach to the associated free boundary problem was later developed by Caffarelli in [2,3,4]. In particular in [4] the regularity of flat free boundaries is obtained. There is a wide literature on this problem and the corresponding two-phase problem, and we refer the reader to the paper [9] for a comprehensive survey.
The system (1.1) can also be seen as the Euler-Lagrange equations associated to a vectorial Alt-Caffarelli type functional. Namely, given a regular open set Ω⊂Rn and Φ=(φ1,…,φm)∈H1/2(∂Ω,Rm), one can consider the vectorial free boundary problem
min{∫Ω|∇U|2dx+|Ω(U)|:U∈H1(Ω,Rm),U=Φ on ∂Ω}. | (1.3) |
In [6,14], the authors initiated the study of this problem where several flows are involved, and interact whenever there is a phase transition. In particular, they applied a reduction method to reduce the problem to its scalar counterpart by assuming nonnegativity of the components of U. More precisely, under this assumption, the components are weak solutions of
Δui=wiHn−1∟(Ω∩∂∗{|U|>0}),for i=1,…,m, |
with
wi(x)=limy∈{|U|>0},y→xui(y)|U|(y). |
Recently in [15], a different group of authors removed the sign assumption on the components. As expected, in this case the structure of the singular set changes and the set of branching points Sing2(F(U)) arises, as natural in two-phase problems (see also the recent work [7]). More precisely, the following theorem holds.
Theorem 1.1 ([15]). The problem (1.3) admits a solution U∈H1(Ω;Rm). Moreover, any solution is Lipschitz continuous in Ω⊂Rn and the set Ω(U) has a locally finite perimeter in Ω. More precisely, the free boundary F(U) is a disjoint union of a regular part Reg(F(U)), a (one-phase) singular set Sing1(F(U)) and a set of branching points Sing2(F(U)):
(1). Reg(F(U)) is an open subset of F(U) and is locally the graph of a smooth function.
(2). Sing1(F(U)) consists only of points in which the Lebesgue density of Ω(U) is strictly between 1/2 and 1. Moreover, there is n∗∈{5,6,7} such that:
● if n<n∗, then Sing1(F(U)) is empty;
● if n=n∗, then Sing1(F(U)) contains at most a finite number of isolated points;
● if n>n∗, then the (n−n∗)-dimensional Hausdorff measure of Sing1(F(U)) is locally finite in Ω.
(3). Sing2(F(U)) is a closed set of locally finite (n−1)-Hausdorff measure in Ω and consists of points in which the Lebesgue density of Ω(U) is 1 and the blow-up limits are linear functions.
As pointed out in [14], problem (1.3) is also related to a class of shape optimization problems involving the eigenvalues of the Dirichlet Laplacian. Precisely, if U∗ is the vector whose components are the Dirichlet eigenfunctions on the set Ω∗ which solves the shape optimization problem
min{m∑i=1λi(Ω):Ω⊂Rn open,|Ω|=1}, | (1.4) |
then U∗ can be seen as quasi-minimizers of (1.3). Indeed, [15] follows some of the main ideas developed in [14]. In [12,13], a different set of authors considered an even more general class of spectral functionals than (1.4) and used a viscosity approach based on an Harnack inequality and a linearization, in the same spirit of the method developed in [8] by the first author.
In this note, we are also inspired by [8], and we use a vectorial viscosity approach which does not reduce the problem to the scalar one-phase problem, as done in [15]. Since we work directly on the problem (1.1), our proof (in particular the choice of barriers, e.g., Theorem 2.1) is more straightforward then the one in [12,13] as it takes advantage of the fact that the norm |U| is a viscosity subsolution to the scalar one-phase problem.
As in [8], the approach carries on in other settings, for example in the presence of a right hand side or even when all components satisfy a uniformly elliptic equation Lui=fi in the positivity set (for the same operator L). One can also allow a more general free boundary condition as in [8].
One of the objectives of this note is indeed to develop a method suitable for other vectorial problems, for example Bernoulli-type problems involving nonlocal diffusion. In particular, in [5,10,11] the authors studied the regularity of a one-phase scalar free boundary problem for the fractional Laplacian. While in [5] general properties like optimal regularity, nondegeneracy and classification of global solutions were proved, in [10,11] the authors developed a viscosity approach in order to prove that flat free boundaries are actually C1,α. In a forthcoming paper, we plan to extend these results to the vectorial case, following the approach developed in this paper.
We now state our main theorem. From now on, we denote by {ei}i=1,…,n, {fi}i=1,…,m canonical basis in Rn and Rm respectively. Unit directions in Rn and Rm will be typically denoted by e and f. The Euclidean norm in either space is denoted by |⋅|, while the dot product is denoted by ⟨⋅,⋅⟩.
Definition 1.2. We say that U∈C(Ω,Rm) is a viscosity solution to (1.1) in Ω if
Δui=0in Ω(U),∀i=1,…,m; |
and the free boundary condition is satisfied in the following sense. Given x0∈F(U), and a test function φ∈C2 in a neighborhood of x0, with |∇φ|(x0)≠0, then
(ⅰ) If |∇φ|(x0)>1, then for all unit directions f in Rm, ⟨U,f⟩ cannot be touched by below by φ at x0.
(ⅱ) If |∇φ|(x0)<1, then |U| cannot be touched by above by φ at x0.
Our main theorem reads as follows.
Theorem 1.3. Let U be a viscosity solution to (1.1) in B1. There exists a universal constant ˉε>0 such that if U is ˉε flat in B1, i.e., for some unit directions e∈Rn,f∈Rm
|U(x)−f⟨x,e⟩+|≤ˉε,in B1, | (1.5) |
and
|U|≡0in B1∩{⟨x,e⟩<−ˉε} | (1.6) |
then F(U)∈C1,α in B1/2.
We remark that condition (1.6) is satisfied by flat minimizing solutions in view of non-degeneracy [15,Section 2.1].
Notice that in [15] the authors used a smaller class of viscosity solutions in which property (i) is replaced by the following:
(i') If |∇φ|(x0)>1, then |U| cannot be touched by below by φ at x0. |
Indeed, in [15,Lemma 3.2.] they proved that if U is a minimizing free boundary, then for every x0∈Reg(F(U))∪Sing1(F(U)) there exists a small radius r>0 such that U is a viscosity solution of
{ΔU=0in Ω(U)∩Br(x0);|∇|U||=1on F(U)∩Br(x0), | (1.7) |
in the sense of (ⅰ')–(ⅱ). The larger class in Definition 1.2 is better suited for the strategy of our proof, which relies on a vectorial Harnack inequality and improvement of flatness technique. Details of the Harnack inequality are carried on in Section 2, while the improvement of flatness argument is presented in Section 3.
In this Section we will prove a Harnack type inequality for solutions to problem (1.1). Precisely, the following is our main theorem. Notice that our strategy consists in tracking the improvement of |U|, rather than working component-wise as in [12,13]. By working with |U|, we avoid some of the difficulties related to the no-sign assumption.
Theorem 2.1. There exists a universal constant ¯ε>0 such that, if U solves (1.1) in B1, and for some point x0∈B1(U)∪F(U),
xn+a0≤u1≤|U|≤(xn+b0)+in Br(x0)⊂B1, | (2.1) |
with
b0−a0≤ˉεr, |
and
|ui|≤r(b0−a0r)3/4in B1/2(x0),i=2,…,m, |
then
xn+a1≤u1≤|U|≤(xn+b1)+in Br/20(x0), | (2.2) |
with
a0≤a1≤b1≤b0,b1−a1=(1−c)(b0−a0), |
for 0<c<1 universal.
We briefly postpone the proof of Theorem 2.1, and obtain the key corollary which will be used in the improvement of flatness argument. First, the following lemma allows to translate the flatness assumption on the vector-valued function U into the property that one of its components is trapped between nearby translation of a one-plane solution, while the remaining ones are small.
Lemma 2.2. Let U be a solution to (1.1) in B1 such that for ε>0
|U−f1x+n|≤ε,in B1, | (2.3) |
and
|U|≡0in B1∩{xn<−ε}. | (2.4) |
Then
i. For i=2,…,m,
|ui|≤Cε(xn+ε)+in B3/4; | (2.5) |
ii.
(xn−ε)≤u1≤|U|≤(xn+2ε)+in B1. | (2.6) |
Proof. The bounds in (ii) are an immediate consequence of the assumptions. For (i), let v be the harmonic function in B1∩{xn>−ε} with smooth boundary data ˉv, 0≤ˉv≤1 such that
{ˉv=0on B1−ε∩{xn=−ε};ˉv=1on ∂B1∩{xn>−ε}. |
Since |ui| is subharmonic and by (2.3)–(2.4)
|ui|≤ε,ui≡0on {xn=−ε}, |
by comparison and boundary regularity we get
|ui|≤εv≤Cε(xn+ε)in B1/2∩{xn>−ε}. |
Now denote by,
˜u1:=u1−xnε,~|U|:=|U|−xnε,x∈B1(U)∪F(U). |
The following corollary is a consequence of the results above.
Corollary 2.3. Let U be a solution to (1.1) in B1 such that for ε>0
|U−f1x+n|≤ε,in B1, | (2.7) |
and
|U|≡0in B1∩{xn<−ε}. | (2.8) |
There exists ˉε>0 small universal, such that if ε≤ˉε, then ˜u1 and ~|U| have a universal Hölder modulus of continuity at x0∈B1/2 outside a ball of radius rε, with rε→0 as ε→0.
Proof. In view of Lemma 2.2, u1,U satisfy the assumptions of Theorem 2.1 (for ˉε possibly smaller than the one in Theorem 2.1), with
a0=−ε,b0=2ε,r=1/4,x0∈B1/2. |
Hence, by applying repeatedly Theorem 2.1,
xn+ak≤u1≤|U|≤(xn+bk)+in Brρk(x0),ρk=20−k, | (2.9) |
with
bk−ak=(1−c)k(b0−a0), |
for 0<c<1 universal as long as,
20k(1−c)kε≤ˉε,ε≤ˉC((1−c)320)k, |
with ˉC universal. This implies that for such cases, in (Ω(U)∪F(U))∩Brρk(x0) the oscillation of the functions ˜u1 and ~|U| are less or equal than (1−c)k=20−αk=ραk, as we claimed.
The next lemma is the main ingredient in the proof of Theorem 2.1. It uses the observation that |U| is subharmonic in Ω(U), as it can be easily verified with a straightforward computation.
Lemma 2.4. Let U be a solution to (1.1) in B1 such that for ε>0
p(x)≤u1≤|U|≤(p(x)+ε)+in B1,p(x):=xn+σ,|σ|<1/10, | (2.10) |
and
|ui|≤ε3/4in B1/2,i=2,…,m, | (2.11) |
with C>0 universal. There exists ˉε>0, such that if 0<ε≤ˉε, then at least one of the following holds true:
p(x)+cε≤u1≤|U|in B1/2, | (2.12) |
or
u1≤|U|≤(p(x)+(1−c)ε)+,in B1/2, |
for 0<c<0 small universal.
Proof. We distinguish two cases. If at ˉx=15en
u1(ˉx)≤p(ˉx)+ε2, | (2.13) |
then we will show that
|U|≤(p(x)+(1−c)ε)+in B1/2 . | (2.14) |
Similarly, if
u1(ˉx)≥p(ˉx)+ε2, | (2.15) |
we will show that
u1≥p(x)+cεin B1/2. |
In either case, we let A=B3/4(ˉx)∖B1/20(ˉx) and
w={1in B1/20(ˉx);|x−ˉx|γ−(3/4)γ(1/20)γ−(3/4)γin A;0on ∂B3/4(ˉx). | (2.16) |
with γ<0 be such that Δw>0 in A.
Case 1. If u1(ˉx)≥p(ˉx)+ε2, the argument in [8,Lemma 3.3] carries on, even if u1 may change sign. For completeness, we provide the details.
Since |σ|<1/10 and by the flatness assumption
u1−p≥0in B1, | (2.17) |
we immediately deduce that B1/10(ˉx)⊂B1(U). Notice that, by definition of ˉx, we have
B1/2⊂⊂B3/4(¯x)⊂⊂B1. | (2.18) |
Hence, in view of (2.17), by Harnack inequality applied to u1−p we get for c0>0 universal
u1(x)−p(x)≥c(u1(¯x)−p(¯x))≥c0εin B1/20(ˉx), | (2.19) |
where in the second inequality we used assumption (2.15).
Now, let us set
vt(x)=p(x)+c0ε(w(x)−1)+tin ¯B3/4(ˉx) | (2.20) |
for t≥0. Thus, we deduce that Δvt=Δp+c0εΔw>0 on A and, by (2.17), we get
v0≤p≤u1in ¯B3/4(¯x). |
Thus, let ¯t>0 be the largest t>0 such that vt≤u1 in ¯B3/4(¯x). We want to show that ¯t≥c0ε. Indeed, by the definition of vt, we will get
u1≥v¯t≥p+c0εwin ¯B3/4(¯x). |
In particular, by (2.18), since w≥c1 on ¯B1/2, we get
u1≥p+cεin ¯B1/2, |
as we claimed.
Suppose by contradiction that ˉt<c0ε. Let ˜x∈¯B3/4(ˉx) be the touching point between v¯t and u, i.e.,
u(˜x)=v¯t(˜x), |
we want to prove that it can only occur on ¯B1/20(¯x). Since w≡0 on ∂B3/4(¯x) and ¯t<c0ε we get
v¯t=p−c0ε+¯t<uon ∂B3/4(¯x), |
thus it is left to exclude that ˜x belongs to the annulus A. By the definition (2.20), we get
|∇v¯t|≥|vn|≥|1+c0εwn|in A. | (2.21) |
Since w is radially symmetric wn=|∇w|νx⋅en in A, where νx is the unit direction of x−¯x. On one side, from the definition of w, we get that |∇w|>c on A and on the other νx⋅en is bounded by below in the region {v¯t≤0}∩A, since ¯xn=1/5 and for ε small,
{v¯t≤0}∩A⊂{p−c0ε≤0}∩A={xn≤−σ+c0ε}∩A⊂{xn<3/20}. |
Hence, we infer that |∇v¯t|≥1+c2(γ)ε in {v¯t≤0}∩A and consequently
|∇v¯t|>1on F(v¯t)∩A. | (2.22) |
Finally, since we observed that Δv¯t>ε2 in A, and vˉt≤u1, we deduce that the touching cannot occur in A∩B1(U) where u1 is harmonic. In view of (2.22) and Definition 1.2, we conclude that the touching cannot occur on A∩F(U) as well. Therefore ˜x∈¯B1/20(¯x) and
u1(˜x)=v¯t(˜x)=p(˜x)+¯t<p(˜x)+c0ε, |
in contradiction with (2.19).
Case 2. If u1(ˉx)≤p(ˉx)+ε2, by the lower bound in (2.10), |u1|=u1 in B1/10(ˉx)⊂B1(U). Thus by Harnack inequality and assumption (2.13)
p+ε−|u1|≥2c0εin B1/20(ˉx). | (2.23) |
Since the desired bound clearly holds in {p≤−ε}, where all the ui≡0, it is enough to restrict to the region {p>−ε}. Below, the superscript ε denotes such restriction.
Now, let us consider for t≥0
vt(x)=p(x)+ε−c0ε(w(x)−1)−tin ¯Bε3/4(ˉx). | (2.24) |
Thus, we have Δvt=−c0εΔw<0 on Aε and
v0≥p+ε≥|U|in ¯Bε3/4(¯x). |
Now, let ˉt>0 be the largest t>0 such that |U|≤vt in ¯Bε3/4(¯x). We want to show that ¯t≥c0ε. Indeed, by the definition of vt, this would give
|U|≤vˉt≤p+ε−c0εwin ¯Bε3/4(¯x). |
In particular, by (2.18), since w≥c1 on ¯B1/2, we get
|U|≤p+(1−c)εin ¯Bε1/2, |
as we claimed.
We are left with the proof that ¯t≥c0ε. Suppose by contradiction that ˉt<c0ε. Let ˜x∈¯Bε3/4(ˉx) be the first touching point between vˉt and |U| in ¯Bε3/4(ˉx), i.e.,
|U|(˜x)=vˉt(˜x). |
We prove that such touching point can only occur on ¯B1/20(¯x). Since w≡0 on ∂B3/4(¯x), |U|≡0 on {p=−ε} and ˉt<c0ε we get
vˉt>|U|on ∂Bε3/4(¯x), |
thus we need to exclude that ˜x belongs to Aε. By the definition (2.24), we get
|∇vˉt|2=1−2c0εwn+O(ε2)in Aε. | (2.25) |
On the other hand, it easily follows from the definition (2.24) that
{vˉt=0}⊂{p+(1−c0)ε<0}⊂{xn<110−(1−c0)ε}, |
thus we can estimate that
wn≥c3(γ)>0on Aε∩{vˉt=0}. |
Hence, we infer that for ε small,
0≠|∇vˉt|<1on F(vˉt)∩Aε. | (2.26) |
Finally, since we observed that Δvˉt<0 in Aε, and vˉt≥|U|, we deduce that ˜x∉Aε∩B1(U). Moreover, by (2.26) and Definition 1.2, we also conclude that ˜x∉Aε∩F(U).
Therefore, ˜x∈¯B1/20(ˉx) and
|U|(˜x)=vˉt(˜x)=p(˜x)+ε−ˉt>p(˜x)+ε−c0ε, |
that is
p(˜x)+ε−|U|(˜x)<c0ε. |
This implies, using (2.11) and the fact that |u1| is bounded,
p(˜x)+ε−|u1|(˜x)−Cε3/2<c0ε, |
and we contradict (2.23), for ε small and C universal constant.
We are now ready to prove Theorem 2.1.
Proof. Let us rescale,
uir(x):=1rui(rx+x0),x∈B1,i=1,…,m. |
Then,
p(x)≤u1r≤|Ur|≤(p(x)+ε)+in B1, | (2.27) |
with
ε:=r−1(b0−a0)≤ˉε,p(x)=xn+σ,σ=r−1a0 |
and
|uir|≤ε3/4in B1/2. | (2.28) |
If
|a0|≤r10, |
then we can apply Lemma 2.4 and reach the desired conclusion. If a0<−r/10 then for ε small,
|Ur|≡0in B1/20, |
and again we obtain the claim. We are left with the case a0>r/10. Then B1/10⊂B1(Ur) and u1r>0 and harmonic in B1/10. Hence by standard Harnack inequality, either
u1r≥p(x)+cεin B1/20, |
and we are done, or
u1r≤p(x)+(1−c)εin B1/20. |
Finally, by (2.28), for ε sufficiently small
|Ur|≤u1r+Cε3/2≤p(x)+(1−c2)εin B1/20, |
with C,c>0 universal.
In this section we prove our main result, an improvement of flatness lemma, from which the desired Theorem 1.3 follows by standard techniques.
First, we recall some known facts. Consider the following boundary value problem, which is the linearized problem arising from our improvement of flatness technique:
{Δ˜U=0in B1/2∩{xn>0},∂∂xn(˜u1)=0,˜ui=0i=2,…,mon B1/2∩{xn=0}, | (3.1) |
with ˜U=(˜u1,…,˜um)∈C(B1/2∩{xn≥0},Rm). The Neumann problem for ˜u1 is satisfied in the following viscosity sense.
Definition 3.1. If P(x) is a quadratic polynomial touching ˜u1 by below (resp. above) at ˉx∈B1/2∩{xn≥0}, then
(ⅰ) if ˉx∈B1/2∩{xn>0} then ΔP≤0, (resp. ΔP≥0) i.e ˜u1 is harmonic in the viscosity sense;
(ⅱ) if ˉx∈B1/2∩{xn=0} then Pn(ˉx)≤0 (resp. Pn(ˉx)≥0.)
As usual, in the definition above we can choose polynomials P that touch ˜u1 strictly by below/above. Also, it suffices to verify that (ⅱ) holds for polynomials ˜P with Δ˜P>0.
Since the linearized problem (3.1) is a system completely decoupled, the regularity of solutions follows immediately by standard theory (see also [8,Lemma 2.6].)
Lemma 3.2. Let ˜U be a viscosity solution to (3.1) in Ω. Then ˜U is a classical solution to (3.1) and ˜U∈C∞(B1/2∩{xn≥0};Rm).
We are now ready to state and prove our key lemma.
Lemma 3.3 (Improvement of Flatness). Let U be a viscosity solution to (1.1) in B1 satisfying the ε-flatness assumption in B1
|U−f1x+n|≤εin B1, | (3.2) |
and
|U|≡0in B1∩{xn<−ε}, | (3.3) |
with 0∈F(U). If 0<r≤r0 for a universal r0>0, and 0<ε≤ε0 for some ε0 depending on r, then
|U−ˉf⟨x,ν⟩+|≤εr2in Br, | (3.4) |
and
|U|≡0in Br∩{⟨x,ν⟩<−εr2}, | (3.5) |
with |ν−en|≤Cε,|ˉf−f1|≤Cε, for a universal constant C>0.
Proof. Following the strategy of [8], we divide the proof in three different steps.
Step 1 - Compactness. Fix r≤r0 with r0 universal (the value of r0 will be given in Step 3), suppose by contradiction that there exists εk→0 and a sequence of solutions (Uk)k of (1.1) such that 0∈F(Uk) and (3.2) and (3.3) are satisfied for every k, i.e.,
|Uk−f1x+n|≤εk,in B1, | (3.6) |
and
|Uk|≡0in B1∩{xn<−εk}, | (3.7) |
but the conclusions (3.4) and (3.5) of the Lemma do not hold.
Let us set
˜Uk=Uk−f1xnεk,Vk=|Uk|−xnεkin Ω(Uk):=B1(Uk)∪F(Uk)⊂{xn≥−εk}. | (3.8) |
By the flatness assumptions (3.6)–(3.7), (Uk)k and (Vk)k are uniformly bounded in B1. Moreover, F(Uk) converges to B1∩{xn=0} in the Hausdorff distance. Now, by Corollary 2.3 and Ascoli-Arzelà, it follows that, up to a subsequence, the graphs of the components ˜uik of ˜Uk and of Vk over B1/2∩(B1(Uk)∪F(Uk)) converge in the Hausdorff distance to the graph of Holder continuous functions ˜ui∞,V∞ on B1/2∩{xn≥0}, for every i=1,…,m. Moreover, by Corollary 2.3,
V∞≡˜u1∞in B1/2∩{xn≥0}. | (3.9) |
Step 2 - Linearized problem. We show that ˜U∞ satisfies the following problem in the viscosity sense:
{Δ˜U∞=0in B1/2∩{xn>0},∂∂xn(˜u1∞)=0,˜ui∞=0i=2,…,mon B1/2∩{xn=0}. | (3.10) |
In view of Lemma 2.2, part (i), the conclusion for i=2,…,m is immediate. We are left with the case i=1.
First, let us consider the case a polynomial P touches ˜u1 at ¯x∈B1/2∩{xn≥0} strictly by below. Then the arguments of [8] apply. Indeed, we need to show that
ⅰ. if P touches ˜u1∞ at ¯x∈B1/2∩{xn>0}, then ΔP(¯x)≤0,
ⅱ. if P touches ˜u1∞ on {xn=0}, then Pn(¯x)≤0,
Since ˜u1k→˜u1∞ uniformly on compacts, there exists (xk)k⊂B1/2∩(B1(Uk)∪F(Uk)), with xk→¯x, and ck→0 such that
˜u1k(xk)=P(xk)+ck, |
and ˜u1k≥P+ck in a neighborhood of xk. From the definition of the sequence ˜u1k, we infer u1k(xk)=Q(xk) and u1≥Q in a neighborhood of xk, with
Q(x)=xn+εk(P(x)+ck) | (3.11) |
If ¯x∈B1/2∩{xn>0}, then xk∈B1/2(Uk), for k sufficiently large, and hence since Q touches u1k by below at xk
ΔQ(xk)=εkΔP(xk)≤0, |
which leads to ΔP(¯x)≤0 as k→∞.
Instead, if ¯x∈B1/2∩{xn=0}, then we can assume ΔP>0. It is not restrictive to suppose that, for k sufficiently large, xk∈F(Uk). Otherwise xkn∈B1/2(Ukn) for a subsequence kn→∞ and in that case ΔP(xkn)≤0, in contradiction with the strict subharmonicity of P.
Thus, for k large, xk∈F(Uk). Then noticed that ∇Q=en+ε∇P and |∇Q|>0 for k sufficiently large, since Q touches u1k by below, by Definition 1.2 we deduce that |∇Q|2(xk)≤1, i.e.,
εk|∇P|2(xk)+2Pn(xk)≤0. |
Passing to the limit as k→∞ we obtain the desired conclusion.
Consider now the case when P touches ˜u1∞ at ¯x∈B1/2∩{xn≥0} strictly by above. Since the case ¯x∈B1/2∩{xn>0} follows the same reasoning of the previous part, we move on to the case ¯x∈B1/2∩{xn=0} and assume that ΔP<0. We claim that
Pn(¯x)≥0. |
Since ˜u1∞=V∞, for k sufficiently large we get |Uk|(xk)=Q(xk) and |Uk|≤Q in a neighborhood of xk→ˉx, with
Q(x)=xn+εk(P−ck),ck→0. |
As before, since |Uk| is subharmonic, we can assume that xk∈F(Uk). By the definition of viscosity solution, we deduce that |∇Q|2(xk)≥1, i.e.
εk|∇P|2(xk)+2Pn(xk)≥0, |
which leads to the claimed result as k→∞..
Step 3 - Improvement of flatness. Since (˜Uk)k is uniformly bounded in B1, we get a uniform bound on |˜ui∞|, for every i=1,…,m. Furthermore, since 0∈F(˜U∞), by the regularity result in Lemma 3.2 we deduce that
|˜ui∞(x)−⟨∇˜ui∞(0),x⟩|≤C0r2in Br∩{xn≥0},i=1,…,m |
for a universal constant C0>0. On one side, since ∂xn(˜u1∞)=0 on B1/2∩{xn=0}, we infer
⟨x′,˜ν1⟩−C0r2≤˜u1∞(x)≤⟨x′,˜ν1⟩+C0r2in Br∩{xn≥0}, |
where ˜ν1=∇˜u1∞(0) is a vector in the variables x1,…,xn−1, with |˜ν1|≤M, for some M universal constant. Thus, fixed the notation x′=(x1,⋯,xn−1), for k sufficiently large there exists C1 such that
⟨x′,˜ν1⟩−C1r2≤˜u1k(x)≤⟨x′,˜ν1⟩+C1r2in Ω(Uk)∩Br |
and exploiting the definition of ˜u1k we read
xn+εk⟨x′,˜ν1⟩−εkC1r2≤u1k(x)≤xn+εk⟨x′,˜ν1⟩+εkC1r2in Ω(Uk)∩Br. |
Thus, called
ν=(εk˜ν1,1)√1+ε2k|˜ν1|2∈Sn, |
since for k sufficiently large 1≤√1+|˜ν1|2ε2k≤1+M2ε2k/2, we deduce that
⟨x,ν⟩−r2M2ε2k−εkC1r2≤u1k(x)≤⟨x,ν⟩+r2M2ε2k+εkC1r2in Ω(Uk)∩Br. | (3.12) |
It follows that, for r0≤1/(8C1) and k large,
F(Uk)∩Br⊂{|⟨x,ν⟩|≤εkr4} | (3.13) |
and since |Uk|≡0 in {xn<−εk} and |Uk|>0 in {xn>εk}, we conclude that
|Uk|≡0in Br∩{⟨x,ν⟩<−εkr4}, | (3.14) |
and
|Uk|>0in Br∩{⟨x,ν⟩>εkr4}. | (3.15) |
Also, for r0<1/(8C1) and k large,
⟨x,ν⟩−r8εk≤u1k(x)≤⟨x,ν⟩+r8εkin Ω(Uk)∩Br. | (3.16) |
On the other, since ˜ui∞=0 on B1/2∩{xn=0}, for i=2,…,m we get
⟨x,˜νi⟩−C0r2≤˜ui∞(x)≤⟨x,˜νi⟩+C0r2in Br∩{xn≥0}, |
where ˜νi=Mien,|Mi|≤M, for M universal constant, and for k sufficiently large
|uik(x)−Mixnεk|≤r8εkin Ω(Uk)∩Br. | (3.17) |
Finally, set
ˉf1k=1√1+ε2km∑i=2|Mi|2andˉfik=εkMi√1+ε2km∑i=2|Mi|2 |
for i=2,…,m. Thus, by (3.16) we get for k large,
|u1k−ˉf1k⟨x,ν⟩|≤|u1k−⟨x,ν⟩|+(m−1)M22ε2kr≤r4εk,in Ω(Uk)∩Br, | (3.18) |
and similarly, by (3.17), we obtain for the other components
|uik−ˉfik⟨x,ν⟩|≤|u1k−εkMixn|+εkMr|en−f1kν|≤r8εk+m2ε3kM3r≤r4εk,in Ω(Uk)∩Br. | (3.19) |
Summing (3.18) and (3.19) for all i=2,⋯,m we finally get
|Uk−fk⟨x,ν⟩|≤εkr4in Ω(Uk)∩Br. |
In view of (3.14), we only need to show that we can replace ⟨x,ν⟩ with its positive part, in the region
−r4εk<⟨x,ν⟩<0. |
Since in this region,
|Uk|≤|Uk−ˉfk⟨x,ν⟩|+|ˉfk⟨x,ν⟩|<εkr4, |
we obtain,
|Uk−ˉfk⟨x,ν⟩+|≤εkr2,in Ω(Uk)∩Br. |
In view of (3.15), this inequality holds in Br, which combined with (3.14) leads us to a contradiction.
Theorem 1.3 now follows from a standard iterative argument, which we briefly sketch below.
Choose ˉε=ε0, then by rescaling and iterating Lemma 3.3 we conclude that (say 0∈F(U))
|U−˜fk⟨x,νk⟩+|≤ε02−krk0,in Brk0, |
with
|νk−νk+1|≤Cε02−k,|˜fk−˜fk+1|≤Cε02−k. |
From this we deduce the existence of limiting ν∗0,˜f0 such that for γ=γ(r0),
|U−˜f0⟨x,ν∗0⟩+|≤Cε0r1+α,in Br,r≤1. |
By repeating the same argument at all x0∈F(U)∩B1/2, we conclude that
|U−˜fx0⟨x,ν∗x0⟩+|≤Cε0r1+α,in Br(x0),r≤1/2. |
Moreover,
|U|≡0,in Br(x0)∩{⟨x,ν∗x0⟩<−cε0r1+α}. |
Now a standard argument gives that |ν∗x0−ν∗y0|≤Cε0|x0−y0|α, with x0,y0∈F(U)∩B1/2 and the claim follows.
G. Tortone is partially supported by the ERC Advanced Grant 2013 n. 339958 Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT, held by Susanna Terracini.
The authors declare no conflict of interest.
[1] | Caffarelli LA, Alt HW (1981) Existence and regularity for a minimum problem with free boundary. J Reine Angew Math 325: 105-144. |
[2] | Caffarelli LA (1987) A Harnack inequality approach to the regularity of free boundaries. Part I. Lipschitz free boundaries are C1,α. Rev Mat Iberoamericana 3: 139-162. |
[3] | Caffarelli LA (1988) A Harnack inequality approach to the regularity of free boundaries. Part III. Existence theory, compactness, and dependence on x. Ann Scuola Norm Sci 15: 583-602. |
[4] | Caffarelli LA (1989) A Harnack inequality approach to the regularity of free boundaries. Part II. Flat free boundaries are Lipschitz. Commun Pure Appl Math 42: 55-78. |
[5] | Caffarelli LA, Roquejoffre JM, Sire Y (2010) Variational problems for free boundaries for the fractional Laplacian. J Eur Math Soc 12: 1151-1179. |
[6] |
Caffarelli LA, Shahgholian H, Yeressian K (2018) A minimization problem with free boundary related to a cooperative system. Duke Math J 167: 1825-1882. doi: 10.1215/00127094-2018-0007
![]() |
[7] | De Philippis G, Spolaor L, Velichkov B (2019) Regularity of the free boundary for the two-phase Bernoulli problem. arXiv:1911.02165. |
[8] | De Silva D (2011) Free boundary regularity for a problem with right hand side. Interface Free Bound 13: 223-238. |
[9] | De Silva D, Ferrari F, Salsa S (2014) On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete Contin Dyn Syst Ser S 7: 673-693. |
[10] |
De Silva D, Roquejoffre JM (2012) Regularity in a one-phase free boundary problem for the fractional Laplacian. Ann I H Poincare An 29: 335-367. doi: 10.1016/j.anihpc.2011.11.003
![]() |
[11] | De Silva D, Savin O, Sire Y (2014) A one-phase problem for the fractional Laplacian: Regularity of flat free boundaries. Bull Inst Math Acad Sin 9: 111-145. |
[12] |
Kriventsov D, Lin FH (2018) Regularity for shape optimizers: The nondegenerate case. Commun Pure Appl Math 71: 1535-1596. doi: 10.1002/cpa.21743
![]() |
[13] |
Kriventsov D, Lin FH (2019) Regularity for shape optimizers: The degenerate case. Commun Pure Appl Math 72: 1678-1721. doi: 10.1002/cpa.21810
![]() |
[14] |
Mazzoleni D, Terracini S, Velichkov B (2017) Regularity of the optimal sets for some spectral functionals. Geom Funct Anal 27: 373-426. doi: 10.1007/s00039-017-0402-2
![]() |
[15] |
Mazzoleni D, Terracini S, Velichkov B (2020) Regularity of the free boundary for the vectorial Bernoulli problem. Anal PDE 13: 741-763. doi: 10.2140/apde.2020.13.741
![]() |
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