Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

Improvement of flatness for vector valued free boundary problems

  • For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies C1, α regularity, as well-known in the scalar case [1,4]. While in [15] the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of [8]. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in [10,11].

    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

    Related Papers:

    [1] Aleksandr Dzhugan, Fausto Ferrari . Domain variation solutions for degenerate two phase free boundary problems. Mathematics in Engineering, 2021, 3(6): 1-29. doi: 10.3934/mine.2021043
    [2] Sandro Salsa, Francesco Tulone, Gianmaria Verzini . Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources. Mathematics in Engineering, 2019, 1(1): 147-173. doi: 10.3934/Mine.2018.1.147
    [3] La-Su Mai, Suriguga . Local well-posedness of 1D degenerate drift diffusion equation. Mathematics in Engineering, 2024, 6(1): 155-172. doi: 10.3934/mine.2024007
    [4] Yoshihisa Kaga, Shinya Okabe . A remark on the first p-buckling eigenvalue with an adhesive constraint. Mathematics in Engineering, 2021, 3(4): 1-15. doi: 10.3934/mine.2021035
    [5] Donatella Danielli, Rohit Jain . Regularity results for a penalized boundary obstacle problem. Mathematics in Engineering, 2021, 3(1): 1-23. doi: 10.3934/mine.2021007
    [6] Giuseppe Maria Coclite, Lorenzo di Ruvo . On the initial-boundary value problem for a Kuramoto-Sinelshchikov type equation. Mathematics in Engineering, 2021, 3(4): 1-43. doi: 10.3934/mine.2021036
    [7] Camilla Nobili . The role of boundary conditions in scaling laws for turbulent heat transport. Mathematics in Engineering, 2023, 5(1): 1-41. doi: 10.3934/mine.2023013
    [8] Morteza Fotouhi, Andreas Minne, Henrik Shahgholian, Georg S. Weiss . Remarks on the decay/growth rate of solutions to elliptic free boundary problems of obstacle type. Mathematics in Engineering, 2020, 2(4): 698-708. doi: 10.3934/mine.2020032
    [9] Manuel Friedrich . Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials. Mathematics in Engineering, 2020, 2(1): 75-100. doi: 10.3934/mine.2020005
    [10] Virginia Agostiniani, Lorenzo Mazzieri, Francesca Oronzio . A geometric capacitary inequality for sub-static manifolds with harmonic potentials. Mathematics in Engineering, 2022, 4(2): 1-40. doi: 10.3934/mine.2022013
  • For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies C1, α regularity, as well-known in the scalar case [1,4]. While in [15] the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of [8]. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in [10,11].


    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):=ΩΩ(U). (1.1)

    Here U(x):=(u1(x),,um(x)),xΩ, 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 (u0),

    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 ΩRn and Φ=(φ1,,φm)H1/2(Ω,Rm), one can consider the vectorial free boundary problem

    min{Ω|U|2dx+|Ω(U)|:UH1(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=wiHn1(Ω{|U|>0}),for i=1,,m,

    with

    wi(x)=limy{|U|>0},yxui(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 UH1(Ω;Rm). Moreover, any solution is Lipschitz continuous in ΩRn 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 (nn)-dimensional Hausdorff measure of Sing1(F(U)) is locally finite in Ω.

    (3). Sing2(F(U)) is a closed set of locally finite (n1)-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{mi=1λi(Ω):ΩRn 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 UC(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 x0F(U), and a test function φC2 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 eRn,fRm

    |U(x)fx,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 x0Reg(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 x0B1(U)F(U),

    xn+a0u1|U|(xn+b0)+in Br(x0)B1, (2.1)

    with

    b0a0ˉεr,

    and

    |ui|r(b0a0r)3/4in B1/2(x0),i=2,,m,

    then

    xn+a1u1|U|(xn+b1)+in Br/20(x0), (2.2)

    with

    a0a1b1b0,b1a1=(1c)(b0a0),

    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

    |Uf1x+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.

    (xnε)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, 0ˉv1 such that

    {ˉv=0on B1ε{xn=ε};ˉv=1on B1{xn>ε}.

    Since |ui| is subharmonic and by (2.3)–(2.4)

    |ui|ε,ui0on {xn=ε},

    by comparison and boundary regularity we get

    |ui|εvCε(xn+ε)in B1/2{xn>ε}.

    Now denote by,

    ˜u1:=u1xnε,~|U|:=|U|xnε,xB1(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

    |Uf1x+n|ε,in B1, (2.7)

    and

    |U|0in B1{xn<ε}. (2.8)

    There exists ˉε>0 small universal, such that if εˉε, then ˜u1 and ~|U| have a universal Hölder modulus of continuity at x0B1/2 outside a ball of radius rε, with rε0 as ε0.

    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,x0B1/2.

    Hence, by applying repeatedly Theorem 2.1,

    xn+aku1|U|(xn+bk)+in Brρk(x0),ρk=20k, (2.9)

    with

    bkak=(1c)k(b0a0),

    for 0<c<1 universal as long as,

    20k(1c)kεˉε,εˉC((1c)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 (1c)k=20αk=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<εˉε, then at least one of the following holds true:

    p(x)+cεu1|U|in B1/2, (2.12)

    or

    u1|U|(p(x)+(1c)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)+(1c)ε)+in B1/2 . (2.14)

    Similarly, if

    u1(ˉx)p(ˉx)+ε2, (2.15)

    we will show that

    u1p(x)+cεin B1/2.

    In either case, we let A=B3/4(ˉx)B1/20(ˉx) and

    w={1in B1/20(ˉx);|xˉx|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

    u1p0in 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 u1p 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 t0. Thus, we deduce that Δvt=Δp+c0εΔw>0 on A and, by (2.17), we get

    v0pu1in ¯B3/4(¯x).

    Thus, let ¯t>0 be the largest t>0 such that vtu1 in ¯B3/4(¯x). We want to show that ¯tc0ε. Indeed, by the definition of vt, we will get

    u1v¯tp+c0εwin ¯B3/4(¯x).

    In particular, by (2.18), since wc1 on ¯B1/2, we get

    u1p+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 w0 on B3/4(¯x) and ¯t<c0ε we get

    v¯t=pc0ε+¯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|νxen 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 νxen is bounded by below in the region {v¯t0}A, since ¯xn=1/5 and for ε small,

    {v¯t0}A{pc0ε0}A={xnσ+c0ε}A{xn<3/20}.

    Hence, we infer that |v¯t|1+c2(γ)ε in {v¯t0}A and consequently

    |v¯t|>1on F(v¯t)A. (2.22)

    Finally, since we observed that Δv¯t>ε2 in A, and vˉtu1, we deduce that the touching cannot occur in AB1(U) where u1 is harmonic. In view of (2.22) and Definition 1.2, we conclude that the touching cannot occur on AF(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 {pε}, where all the ui0, it is enough to restrict to the region {p>ε}. Below, the superscript ε denotes such restriction.

    Now, let us consider for t0

    vt(x)=p(x)+εc0ε(w(x)1)tin ¯Bε3/4(ˉx). (2.24)

    Thus, we have Δvt=c0εΔw<0 on Aε and

    v0p+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 ¯tc0ε. Indeed, by the definition of vt, this would give

    |U|vˉtp+εc0εwin ¯Bε3/4(¯x).

    In particular, by (2.18), since wc1 on ¯B1/2, we get

    |U|p+(1c)εin ¯Bε1/2,

    as we claimed.

    We are left with the proof that ¯tc0ε. 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 w0 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=12c0εwn+O(ε2)in Aε. (2.25)

    On the other hand, it easily follows from the definition (2.24) that

    {vˉt=0}{p+(1c0)ε<0}{xn<110(1c0)ε},

    thus we can estimate that

    wnc3(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 ˜xAεB1(U). Moreover, by (2.26) and Definition 1.2, we also conclude that ˜xAεF(U).

    Therefore, ˜x¯B1/20(ˉx) and

    |U|(˜x)=vˉt(˜x)=p(˜x)+εˉt>p(˜x)+εc0ε,

    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),xB1,i=1,,m.

    Then,

    p(x)u1r|Ur|(p(x)+ε)+in B1, (2.27)

    with

    ε:=r1(b0a0)ˉε,p(x)=xn+σ,σ=r1a0

    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 B1/10B1(Ur) and u1r>0 and harmonic in B1/10. Hence by standard Harnack inequality, either

    u1rp(x)+cεin B1/20,

    and we are done, or

    u1rp(x)+(1c)εin B1/20.

    Finally, by (2.28), for ε sufficiently small

    |Ur|u1r+Cε3/2p(x)+(1c2)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{xn0},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 ˉxB1/2{xn0}, then

    (ⅰ) if ˉxB1/2{xn>0} then ΔP0, (resp. ΔP0) i.e ˜u1 is harmonic in the viscosity sense;

    (ⅱ) if ˉxB1/2{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 ˜UC(B1/2{xn0};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

    |Uf1x+n|εin B1, (3.2)

    and

    |U|0in B1{xn<ε}, (3.3)

    with 0F(U). If 0<rr0 for a universal r0>0, and 0<εε0 for some ε0 depending on r, then

    |Uˉfx,ν+|εr2in Br, (3.4)

    and

    |U|0in Br{x,ν<εr2}, (3.5)

    with |νen|Cε,|ˉff1|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 rr0 with r0 universal (the value of r0 will be given in Step 3), suppose by contradiction that there exists εk0 and a sequence of solutions (Uk)k of (1.1) such that 0F(Uk) and (3.2) and (3.3) are satisfied for every k, i.e.,

    |Ukf1x+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=Ukf1xnεk,Vk=|Uk|xnεkin Ω(Uk):=B1(Uk)F(Uk){xnεk}. (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{xn0}, for every i=1,,m. Moreover, by Corollary 2.3,

    V˜u1in B1/2{xn0}. (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 ¯xB1/2{xn0} strictly by below. Then the arguments of [8] apply. Indeed, we need to show that

    ⅰ. if P touches ˜u1 at ¯xB1/2{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)kB1/2(B1(Uk)F(Uk)), with xk¯x, and ck0 such that

    ˜u1k(xk)=P(xk)+ck,

    and ˜u1kP+ck in a neighborhood of xk. From the definition of the sequence ˜u1k, we infer u1k(xk)=Q(xk) and u1Q in a neighborhood of xk, with

    Q(x)=xn+εk(P(x)+ck) (3.11)

    If ¯xB1/2{xn>0}, then xkB1/2(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 ¯xB1/2{xn=0}, then we can assume ΔP>0. It is not restrictive to suppose that, for k sufficiently large, xkF(Uk). Otherwise xknB1/2(Ukn) for a subsequence kn and in that case ΔP(xkn)0, in contradiction with the strict subharmonicity of P.

    Thus, for k large, xkF(Uk). Then noticed that Q=en+εP 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 ¯xB1/2{xn0} strictly by above. Since the case ¯xB1/2{xn>0} follows the same reasoning of the previous part, we move on to the case ¯xB1/2{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 xkˉx, with

    Q(x)=xn+εk(Pck),ck0.

    As before, since |Uk| is subharmonic, we can assume that xkF(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 0F(˜U), by the regularity result in Lemma 3.2 we deduce that

    |˜ui(x)˜ui(0),x|C0r2in Br{xn0},i=1,,m

    for a universal constant C0>0. On one side, since xn(˜u1)=0 on B1/2{xn=0}, we infer

    x,˜ν1C0r2˜u1(x)x,˜ν1+C0r2in Br{xn0},

    where ˜ν1=˜u1(0) is a vector in the variables x1,,xn1, with |˜ν1|M, for some M universal constant. Thus, fixed the notation x=(x1,,xn1), for k sufficiently large there exists C1 such that

    x,˜ν1C1r2˜u1k(x)x,˜ν1+C1r2in Ω(Uk)Br

    and exploiting the definition of ˜u1k we read

    xn+εkx,˜ν1εkC1r2u1k(x)xn+εkx,˜ν1+εkC1r2in Ω(Uk)Br.

    Thus, called

    ν=(εk˜ν1,1)1+ε2k|˜ν1|2Sn,

    since for k sufficiently large 11+|˜ν1|2ε2k1+M2ε2k/2, we deduce that

    x,νr2M2ε2kεkC1r2u1k(x)x,ν+r2M2ε2k+εkC1r2in Ω(Uk)Br. (3.12)

    It follows that, for r01/(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,νr8εku1k(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,˜νiC0r2˜ui(x)x,˜νi+C0r2in Br{xn0},

    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=11+ε2kmi=2|Mi|2andˉfik=εkMi1+ε2kmi=2|Mi|2

    for i=2,,m. Thus, by (3.16) we get for k large,

    |u1kˉf1kx,ν||u1kx,ν|+(m1)M22ε2krr4εk,in Ω(Uk)Br, (3.18)

    and similarly, by (3.17), we obtain for the other components

    |uikˉfikx,ν||u1kεkMixn|+εkMr|enf1kν|r8εk+m2ε3kM3rr4εk,in Ω(Uk)Br. (3.19)

    Summing (3.18) and (3.19) for all i=2,,m we finally get

    |Ukfkx,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||Ukˉfkx,ν|+|ˉfkx,ν|<εkr4,

    we obtain,

    |Ukˉfkx,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 0F(U))

    |U˜fkx,νk+|ε02krk0,in Brk0,

    with

    |νkνk+1|Cε02k,|˜fk˜fk+1|Cε02k.

    From this we deduce the existence of limiting ν0,˜f0 such that for γ=γ(r0),

    |U˜f0x,ν0+|Cε0r1+α,in Br,r1.

    By repeating the same argument at all x0F(U)B1/2, we conclude that

    |U˜fx0x,νx0+|Cε0r1+α,in Br(x0),r1/2.

    Moreover,

    |U|0,in Br(x0){x,νx0<cε0r1+α}.

    Now a standard argument gives that |νx0νy0|Cε0|x0y0|α, with x0,y0F(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
  • This article has been cited by:

    1. Giorgio Tortone, Regularity of shape optimizers for some spectral fractional problems, 2021, 281, 00221236, 109271, 10.1016/j.jfa.2021.109271
    2. Morteza Fotouhi, Henrik Shahgholian, A minimization problem with free boundary for p-Laplacian weakly coupled system, 2024, 13, 2191-950X, 10.1515/anona-2023-0138
    3. Masoud Bayrami, Morteza Fotouhi, Henrik Shahgholian, Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the p-Laplacian, 2024, 412, 00220396, 447, 10.1016/j.jde.2024.08.040
    4. D. De Silva, O. Savin, An energy model for harmonic functions with junctions, 2024, 447, 00018708, 109682, 10.1016/j.aim.2024.109682
    5. Daniela De Silva, Giorgio Tortone, A vectorial problem with thin free boundary, 2023, 62, 0944-2669, 10.1007/s00526-023-02561-z
    6. Lorenzo Ferreri, Bozhidar Velichkov, A one-sided two phase Bernoulli free boundary problem, 2025, 00217824, 103659, 10.1016/j.matpur.2025.103659
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3553) PDF downloads(266) Cited by(6)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog