Segregated configurations involving the square root of the laplacian and their free boundaries

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30 July 2021

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Segregated configurations involving the square root of the laplacian and their free boundaries

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DOI:10.1007/s00526-019-1529-9

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SEGREGATED CONFIGURATIONS INVOLVING THE SQUARE ROOT OF THE

LAPLACIAN AND THEIR FREE BOUNDARIES

DANIELA DE SILVA AND SUSANNA TERRACINI

Abstract. We study the local structure and the regularity of free boundaries of segre-gated minimal configurations involving the square root of the laplacian. We develop an improvement of flatness theory and, as a consequence of this and Almgren’s monotonic-ity formula, we obtain partial regularmonotonic-ity (up to a small dimensional set) of the nodal set, thus extending the known results in [4, 21] for the standard diffusion to some anomalous case.

1. Introduction

The analysis of the nodal sets of segregated stationary configurations for systems of

elliptic equations has been the subject of an intense study in the last decade, starting

from the works [9, 10, 11, 6, 4, 3, 17, 12, 13, 14]. The present paper is concerned with the

geometric structure of the nodal set, when the creation of a free boundary is triggered

by the interplay between

fractional diffusion and competitive interaction. A prototypical

example comes the following system of fractional Gross-Pitaevskii equations

(1.1)

(

(−∆ + m

2

i

)

1/2

u

i

+ V (x)u

i

= ω

i

u

3i

− βu

i

P

j6=i

a

ij

u

2j

u

i

∈ H

1/2

(R

N

),

with

a

_ij

= a

_ji

> 0, which is the relativistic version of the Hartree-Fock

approxima-tion theory for mixtures of Bose-Einstein condensates in different hyperfine states which

overlap in space.

The sign of

ω

_i

reflects the type of interaction of the particles within

each single state. If

ω

_i

is positive, the self interaction is attractive (focusing problems

de-focusing otherwise).

V represents an external potential. The sign of β, on the other hand,

accounts for the interaction of particles in different states. This interaction is attractive

when negative and repulsive otherwise. If the condensates repel, and the competition rate

tends to infinity, the densities eventually separate spatially, giving rise to a free boundary:

the common nodal set of the components

u

_i

’s. This phenomenon is called phase

separa-tion and has been described in the recent literature, both physical and mathematical, in

the case of standard diffusion. It is by now a well-established fact that in the case of elliptic

systems with standard diffusion this nodal set is comparable, as regards to the qualitative

properties, to that of the scalar solutions. The main reason can be attributed to the validity

of a weak reflection law (see [21]) which constitutes the condition of extremality at the

Date: November 28, 2018.

2010_{Mathematics Subject Classification. 35J70, 35J75, 35R11, 35B40, 35B44, 35B53,}

Key words and phrases. Free boundary problems, nonlocal diffusion, improvement of flatness, monotonicity formulas, blow-up classification.

Work partially supported by the ERC Advanced Grant 2013 n. 339958 Complex Patterns for Strongly Inter-acting Dynamical Systems - COMPAT .

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common interface.

Relevant connections have been established with optimal partition

problems involving spectral functionals (cfr [21, 18]).

We consider the following model with fractional diffusion: according to [5], the

n-dimensional half laplacian can be interpreted as a (nonlinear) Dirichlet-to-Neumann

op-erator.

For this reason we shall state all our results for harmonic functions with

non-linear Neumann boundary conditions involving strong competition terms. Precisely, the

following uniform-in-

β estimates have been derived in [23]. Here we denote by B

_r+

:=

B

r

∩ {z > 0} ⊂ R

n+1

and by

B

_r

:= B

_r

∩ {z = 0} (where z is the (n + 1)-th coordinate).

Theorem 1.1 (Local uniform H¨older bounds, [23]). Let the functions f

i,β

be continuous

and uniformly bounded (w.r.t. β) on bounded sets, and let {u

β

= (u

i,β

)

1≤i≤k

}

β

be a family

of H

1

_(B

+

1

)

solutions to the problems

(

P

_β

)

(

−∆u

i

= 0

in B

1+

∂

ν

u

i

= f

i,β

(u

i

) − βu

i

P

_j6=i

u

2j

on B

1

.

Let us assume that

ku

β

k

L∞_(B+ 1)

≤ M,

for a constant M independent of β. Then for every α ∈ (0, 1/2) there exists a constant

C = C(M, α), not depending on β, such that

ku

β

k

_C0,α_B+ 1/2

≤ C(M, α).

Furthermore, {u

β

}

β

is relatively compact in H

1

(B

_1/2+

) ∩ C

0,α

B

+_1/2

for every α < 1/2.

As a byproduct, up to subsequences, we have convergence of the solutions to (

P

_β

)

to some limiting profile, whose components are segregated on the boundary

B

₁

. If

fur-thermore

f

_i,β

→ f

_i

, uniformly on compact sets, we can prove that this limiting profile

satisfies

(1.2)

(

−∆u

i

= 0

in

B

+ 1

u

i

∂

ν

u

i

= f

i

(u

i

)u

i

on

B

₁

.

One can see that, for solutions of this type of equation, the highest possible regularity

corresponds indeed to the H ¨

older exponent

α = 1/2. As a matter of fact, it has been

proved that the limiting profiles do enjoy such optimal regularity.

Theorem 1.2 (Optimal regularity of limiting profiles, [23]). Under the assumptions above,

assume moreover that the locally Lipschitz continuous functions f

i

satisfy f

i

(s) = f

i0

(0)s +

O(|s|

1+ε

)

as s → 0, for some ε > 0. Then u ∈ C

0,1/2

B

_1/2+

.

It is worthwhile noticing that these result apply to the (local) minimizers of the

func-tionals

(1.3)

J

_β

(U ) =

k

X

i=1

Z

B+ 1

1

2 ∇u

i

(x, z)|

2

_{dx dz + β}

X

1≤i<j≤k

Z

B1

u

2_i

(x, 0)u

2_j

(x, 0) dx

in the set of all configurations with fixed boundary data. Taking the singular limit as

β →

+∞ we are naturally lead to consider the energy minimizing profiles which segregate

only at the characteristic hyperplane

{z = 0}.

Our main goal is to describe, from differential and geometric measure theoretical points

of view, the structure of the trace on the characteristic hyperplane

{z = 0} of the

com-mon nodal set of these limiting profiles. From now on, for the sake of simplicity we shall

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assume that the reactions

f

_i

’s are identically zero and we reflect the components

u

_i

’s

through the hyperplane

{z = 0}. It is worthwhile noticing that we cannot deduce from

the system (1.2) alone any regularity property of the common nodal set

N (u) = {x ∈

Ω ∩ R

n

_{× {0} : u(x, 0) = 0}, as the equations can be independently solved for arbitrary,}

though mutually disjoint, nodal sets

N (u

_i

) = {(x, 0) ∈ Ω ∩ R

n

× {0} : u

_i

(x, 0) = 0}

on the characteristic hyperplane

{z = 0}.

Definition 1.3 (Segregated minimal configurations, class M(Ω)). For an open,

z-symme-tric

Ω ⊂ R

n+1

, we define the class

M(Ω) of the

segregated minimal configurations as the

set of all the even-in-

z vector valued functions u = (u

₁

, . . . , u

_k

) ∈ (H

1

(Ω))

k

∩C

0,1/2

(Ω),

whose components are all nonnegative and achieving the minimal the energy among

configurations segregating only at the characteristic hyperplane

{z = 0}, that is solutions

to

min

(

_k

X

i=1

Z

Ω

|∇u

i

|

2

:

u

i

(x, 0) · u

j

(x, 0) ≡ 0 R

n

-a.e.

for i 6= j,

u

i

= ϕ

i

, on ∂Ω for i = 1, . . . , k,

)

where the

ϕ

_i

’s are nonnegative

H

1/2

-boundary data which are even-in-

z and segregated

on the hyperplane

{z = 0}.

For such class of solutions, we are going to prove a theorem on the structure of the

nodal set

N (u), which is the perfect counterpart of the results in [4, 21].

Theorem 1.4 (Structure of the nodal set of segregated minimal configurations). Let Ω ⊂

R

n+1

, with n ≥ 2, u be a segregated minimal configuration and let N (u) = {x ∈ ˜

Ω :

u(x, 0) = 0}. Then, N (u) is the union of a relatively open regular part Σ

u

and a relatively

closed singular part N (u) \ Σ

u

with the following properties:

(i)

Σ

_u

is a locally finite collection of hyper-surfaces of class C

1,α

(for some 0 < α < 1).

(ii)

H

_dim

(N (u) \ Σ

_u

) ≤ n − 2

for any n ≥ 2. Moreover, for n = 2, N (u) \ Σ

_u

is a

locally finite set.

Remark 1.5. (a) In the light of the extension facts related to the half-laplacian, our

the-ory applies, among others, to segregated minimizing configurations involving non local

energies, like, for instance the solutions to the following problem (when

s = 1/2):

min

(

_k

X

i=1

Z

R2n

|u

i

(x) − u

i

(y)|

2

|x − y|

n+2s

:

u

i

(x, 0) · u

j

(x, 0) ≡ 0 a.e. in R

n

for i 6= j,

u

i

≡ ϕ

i

, on R

n

\ ˜

Ω for i = 1, . . . , k,

)

where

ϕ

_i

are nonnegative

H

1/2

(R

n

) data which are segregated themselves. In this

re-gard, our results extend those of [4] to the fractional case, or, equivalently, to the case

when the phase segregation takes place only on the characteristic hyperplane.

(b) It is worthwhile noticing that, in case of the standard diffusion, the nodal set of the

segregated minimal configurations shares the same measure theoretical features with the

nodal set of harmonic functions; this is not the case of the fractional diffusion; indeed, as

shown in [20], the stratified structure of the nodal set of

s-harmonic functions is far more

complex than that of the segregated minimal configurations. The asymptotics and

prop-erties of limiting profiles of competition diffusion systems with quadratic (Lotka-Volterra)

mutual interactions have been investigated in [25]; as discussed there, the free boundary,

in the Lotka-Volterra case, resembles the nodal set of

s-harmonic functions with some

important differences however, enlightened in that paper.

(c)

The theory developed in this paper is suitable to extend in order to cover the limiting

cases, as

β → ∞, of the variational problems associated with (1.1).

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In order to prove our main result, we will actually consider two other different notions

of solution. The first is the class of solutions in the variational sense - concept that we have

expressed through the validity of the domain variation formula in Definition 6.1 - and a

second notion of solution, expressed in terms of viscosity solutions in Definitions 2.3 (for

two components) and 7.1 (for

k components). As we will see in Theorem 7.3, the latter

notion is weaker (although they are probably equivalent) but carries precious information

on the regularity of the nodal set.

In particular, both notions encode a reflection rule

about the free boundary (see e.g. Proposition 6.3), which will be the ultimate reason for

the regularity of the nodal set.

A major achievement toward the proof of our structure theorem will be an

improve-ment of flatness arguimprove-ment for the case of two components, which was inspired by the

work in [15]. As a byproduct, it yields the following local regularity result: in two

dimen-sions, let

U (t, z) be defined in polar coordinates as

(1.4)

U (t, z) = r

1/2

cos

θ

2 ,

t = r cos θ,

z = r sin θ,

r ≥ 0,

−π ≤ θ ≤ π,

and let

¯

U (t, z) := U (−t, z).

By abuse of notation we denote,

U (x, z) := U (x

n

, z),

U (x, z) := ¯

¯

U (x

n

, z).

Theorem 1.6 (Local regularity of the free boundary). There exists ¯

ε > 0

small depending

only on n, such that if u = (u

1

, u

2

)

is a viscosity solution in B

1

in the sense of Definition

2.3 satisfying

(1.5)

ku

₁

− U k

_∞

≤ ¯

ε,

ku

₂

− ¯

U k

_∞

≤ ¯

ε

then N (u) is a C

1,α

_{graph in B}

1 2

∩ R

n

_{× {0} for every α ∈ (0, 1) with C}

1,α

_{norm bounded}

by a constant depending on α and n.

The paper is organized as follows. Section 2 contains the definition of viscosity solution

for

2-components systems, and the basic related facts. Section 3 is devoted to the study

of the linearized problem associated with

ε-domain variations around the fundamental

solution of the system defined in (1.4).

Next, Section 4 contains Harnack estimates for

viscosity solutions to the free boundary problem for two components.

Such estimates

will be crucial tools in proving the improvement of flatness result, in Section 5, which

concludes the analysis of the regular part of the free boundary of

2-systems and proves

Theorem 1.6.

The rest of the paper concerns

k-vector systems: Section 6 is devoted to

the study of

k-vector solutions in the variational sense (class G) and the consequences

of the associated Almgren’s monotonicity formula, with a focus on the existence and

classification of blow-ups and conic entire solutions. In Section 7 we introduce the notion

of

k-vector solutions in the viscosity sense and we connect it to the variational one, in

order to prove Theorem 1.4. The two appendices contain some ancillary known results.

1. Introduction

1

2. Viscosity solutions for two component systems

5

2.1. Notations and definitions.

5

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2.3. Renormalization.

8

2.4. Domain Variations

8

3. The linearized problem.

11

4. Harnack Inequality

19

5. Improvement of Flatness

25

6. Segregated critical profiles

29

6.1. Compactness and convergence of blowup sequences

32

6.2. Segregated entire homogenous profiles

33

7. Viscosity solutions to

k-component systems

35

7.1. Structure of the nodal set

35 Appendix A.

37 Appendix B.

38 References

39

2. Viscosity solutions for two component systems

2.1. Notations and definitions. First, we introduce some notations.

A point

X ∈ R

n+1

will be denoted by

X = (x, z) ∈ R

n

× R, and sometimes x =

(x

0

, x

n

) with x

0

= (x

1

, . . . , x

n−1

).

A ball in R

n+1

with radius

r and center X is denoted by B

_r

(X) and for simplicity B

_r

=

B

r

(0). Also, for brevity, B

r

denotes the

n-dimensional ball B

_r

∩ {z = 0} (previously

denoted with

∂

0

B

_r+

).

Let

v(X) be a continuous non-negative function in B

₁

. We associate with

v the

fol-lowing sets:

B

₁+

(v) := B

1

\ N (v),

N (v) := {(x, 0) : v(x, 0) = 0};

B

+₁

(v) := B

₁+

(v) ∩ B

1

;

F (v) := ∂

_Rn

N

o

(v) ∩ B

₁

,

N

o

(v) := Int

_Rn

(N (v)).

We now introduce the definition of viscosity solutions for a problem with two

com-ponents.

Let

u

₁

(x, z), u

₂

(x, z) be non-negative continuous functions in the ball B

₁

⊂

R

n+1

= R

n

× R, which vanish on complementary subsets of R

n

× {0} and are even in

the

z variable. We consider the following free boundary problem

(2.1)











∆u

i

= 0,

in

B

+ 1

(u

i

), i = 1, 2,

B

1

= N (u

1

) ∪ N (u

2

),

N

o

(u

1

) ∩ N

o

(u

2

) = ∅,

∂u

1

∂

√

t

=

∂u

2

∂

√

t

,

on

F (u

₁

, u

₂

) := ∂

Rn

N

o

_(u

1

) ∩ B

1

= ∂

Rn

N

o

_(u

2

) ∩ B

1

,

where

(2.2)

∂u

i

∂

√

t

(x

0

) := lim

t→0+

u

i

(x

0

+ tν

i

(x

0

), 0)

√

t

,

x

0

∈ F (u

1

, u

2

)

with

ν

_i

(x

₀

) the normal to F (u

₁

, u

₂

) at x

₀

pointing toward

{u

_i

(x, 0) > 0}.

Definition 2.1. Given g, v continuous in B

1

, we say that

v touches g by below (resp.

above) at

X

₀

∈ B

₁

if

g(X

₀

) = v(X

₀

), and

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If this inequality is strict in

O \ {X

₀

}, we say that v touches g strictly by below (resp.

above).

Definition 2.2. We say that the ordered pair (v

1

, v

2

) is a (strict) comparison subsolution

to (2.1) if the

v

_i

∈ C(B

₁

) are non-negative functions even-in-z that satisfy

(i)

v

₁

is

C

2

and

∆v

₁

≥ 0, in B

+

1

(v

1

),

(ii)

v

₂

is

C

2

and

∆v

₂

≤ 0, in B

+

1

(v

2

),

(iii)

B

₁

= N (v

₁

) ∪ N (v

₂

),

N

o

(v

₁

) ∩ N

o

(v

₂

) = ∅,

N

o

(v

_i

) 6= ∅;

(iv)

F (v

₁

, v

₂

) := F (v

₁

) = F (v

₂

) is C

2

and if

x

₀

∈ F (v

₁

, v

₂

) we have

v

i

(x

0

+ tν

i

(x

0

), 0) = α

i

(x

0

)

√

t + o(

√

t),

as

t → 0

+

,

with

α

1

(x

0

) ≥ α

2

(x

0

),

where

ν

_i

(x

₀

) denotes the unit normal at x

₀

to

F (v

₁

, v

₂

) pointing toward {v

_i

(x, 0) >

0};

(v)

Either the

v

_i

are both not harmonic in

B

+

1

(v

i

) or α

1

(x

0

) > α

2

(x

0

) at all x

0

∈

F (v

1

, v

2

).

Similarly the ordered pair

(w

₁

, w

₂

) is a (strict) comparison supersolution if (w

₂

, w

₁

)

is a (strict) comparison subsolution.

Definition 2.3. We say that (u

1

, u

2

) is a viscosity supersolution to (2.1) if u

i

≥ 0 is a

continuous function in

B

₁

which is even-in-

z and it satisfies

(i)

∆u

_i

= 0

in

B

₁+

(u

_i

);

(ii)

B

₁

= N (u

₁

) ∪ N (u

₂

),

N

o

(u

₁

) ∩ N

o

(u

₂

) = ∅;

(iii)

If

(v

₁

, v

₂

) is a (strict) comparison subsolution then v

₁

and

v

₂

cannot touch

u

₁

and

u

₂

respectively by below and above at a point

X

₀

= (x

₀

, 0) ∈ F (u

₁

, u

₂

) :=

∂N

o

(u

1

) ∩ B

1

.

Respectively, we say that

(u

₁

, u

₂

) is a viscosity subsolution to (2.1) if the

con-ditions above hold with (iii) replaced by

(iv)

If

(w

₁

, w

₂

) is a (strict) comparison supersolution then w

₁

and

w

₂

cannot touch

u

1

and

u

₂

respectively by above and below at a point

X

₀

= (x

₀

, 0) ∈ F (u

₁

, u

₂

).

We say that

(u

₁

, u

₂

) is a viscosity solution if it is both a super and a subsolution.

2.2. Comparison principle. We now derive a basic comparison principle.

Definition 2.4. Let (v

1

, v

2

) be a comparison subsolution to (2.1). We say that (v

1

, v

2

) is

monotone in the

e

_n

direction whenever

v

₁

is monotone increasing and

v

₂

is monotone

decreasing in the

e

_n

direction.

Lemma 2.5 (Comparison principle). Let (u

1

, u

2

), (v

1t

, v

t2

) ∈ C(B

1

)

be respectively a

so-lution and a family of comparison subsoso-lutions to (2.1), t ∈ [0, 1]. Assume that

(i)

v

₁0

≤ u

₁

,

v

₂0

≥ u

₂

in B

₁

;

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(iii)

v

₁t

< u

₁

on F (v

₁t

)

which is the boundary in ∂B

₁

of the set ∂B

+

1

(v

t

1

) ∩ ∂B

1

, for all

t ∈ [0, 1];

(iv)

v

₂t

> u

₂

on F (u

₂

)

which is the boundary in ∂B

₁

of the set ∂B

+

1

(u

2

) ∩ ∂B

1

, for

all t ∈ [0, 1];

(v)

v

t

i

is continuous in (x, t) ∈ B

1

× [0, 1] and B

+1

(v

it

)

is continuous in the Hausdorff

metric.

Then

v

t₁

≤ u

1

,

v

2t

≥ u

2

,

in B

1

, for all t ∈ [0, 1].

Proof. Let

A := {t ∈ [0, 1] : v

t₁

≤ u

1

, v

2t

≥ u

2

on

B

₁

}.

In view of (i) and (v)

A is closed and non-empty. Our claim will follow if we show that

A is open. Let t

0

∈ A, then v

t10

≤ u

1

, v

2t0

≥ u

2

on

B

₁

and by the definition of viscosity

solution

F (v

t0

i

) ∩ F (u

i

) = ∅,

i = 1, 2.

For

i = 1 we argue as follows. Together with (iii) the identity above implies that

B

+ 1

(v

t0 1

) ⊂ B

+ 1

(u

1

),

F (v

t10

) ∪ F (v

t0 1

) ⊂ {x ∈ B

1

: u

1

(x, 0) > 0}.

By (v) this gives that for

t close to t

₀

(2.3)

B

+ 1

(v

t 1

) ⊂ B

+ 1

(u

1

),

F (v

1t

) ∪ F (v

t 1

) ⊂ {x ∈

B

1

: u

1

(x, 0) > 0}.

Call

D := B

₁

\ (B

0₁

(v

₁t

) ∪ F (v

t₁

)). Combining (2.3) with assumption (ii) we obtain

v

t₁

≤ u

1

on

∂D,

and by the maximum principle the inequality holds also in

D. Hence

v

t₁

≤ u

1

in

B

₁

.

Similarly for

i = 2, combining (iv) and (v) we get that for t close to t

₀

,

v

t₂

≥ u

2

in

B

₁

.

Hence

t ∈ A which shows that A is open.

The corollary below is now a straightforward consequence of Lemma 2.5.

Corollary 2.6. Let (u

1

, u

2

)

be a solution to (2.1) and let (v

1

, v

2

)

be a comparison

subsolu-tion to (2.1) in B

2

which is strictly monotone in the e

n

direction in B

2+

(v

i

). Call

v

_it

(X) := v

i

(X + te

n

),

X ∈ B

1

.

Assume that for −1 ≤ t

0

< t

1

≤ 1

v

t0 1

≤ u

1

,

v

2t0

≥ u

2

in B

1

,

v

t1 1

≤ u

1

on ∂B

1

,

v

1t1

< u

1

on F (v

t11

),

and

v

t1 2

≥ u

2

on ∂B

1

,

v

t21

> u

2

on F (u

2

).

Then

v

t1 1

≤ u

1

,

v

2t1

≥ u

2

in B

1

.

(9)

2.3. Renormalization. The following result allows us to replace the flatness assumption

(1.5) with the property that

u

₁

and

u

₂

are trapped between two nearby translates of

U

and ¯

U respectively. Precisely, we have the following lemma.

Lemma 2.7. Let (u

1

, u

2

)

be non negative continuous functions in ¯

B

2

satisfying

∆u

i

= 0,

in B

2+

(u

i

),

(2.4)

B

₁

= N (u

₁

) ∪ N (u

₂

),

N

o

(u

₁

) ∩ N

o

(u

₂

) = ∅,

and the flatness assumption

(2.5)

ku

₁

− U k

_∞

≤ δ,

ku

₂

− ¯

U k

_∞

≤ δ

with δ > 0 small universal. Then in B

1

,

(2.6)

U (X − εe

_n

) ≤ u

₁

(X) ≤ U (X + εe

_n

),

U (X + εe

¯

_n

) ≤ u

₂

(X) ≤ ¯

U (X − εe

_n

),

for some ε = Kδ, K universal.

Lemma 2.7 follows immediately from Lemma A.3 in the Appendix. Indeed, (2.4)-(2.5)

guarantee that

u

_i

satisfies the assumption (A.5).

2.4. Domain Variations. We recall the definition of ε-domain variation corresponding

to

U and some basic lemmas from [15]. We also introduce in a similar fashion the

ε-domain variation corresponding to ¯

U and deduce its properties.

Let

ε > 0 and let g be a continuous non-negative function in B

_ρ

. Here and henceforth

we denote by

P

±

the half-hyperplanes

P

+

_{:= {X ∈ R}

n+1

: x

n

≥ 0, z = 0},

P

−

_{:= {X ∈ R}

n+1

: x

n

≤ 0, z = 0},

and by

L := {X ∈ R

n+1

: x

n

= 0, z = 0}.

To each

X ∈ R

n+1

\ P

−

we associate

g

˜

_ε

(X) ⊂ R via the formula

(2.7)

U (X) = g(X − εwe

_n

),

∀w ∈ ˜

g

_ε

(X).

Similarly, to each

X ∈ R

n+1

\ P

+

we associate

g

˜

¯

_ε

(X) ⊂ R via the formula

(2.8)

U (X) = g(X − εwe

¯

_n

),

∀w ∈ ˜

¯

g

_ε

(X).

By abuse of notation, we write

g

˜

_ε

(X), ˜

¯

g

_ε

(X) to denote any of the values in this set.

As observed in [15], if g satisfies

(2.9)

U (X − εe

_n

) ≤ g(X) ≤ U (X + εe

_n

)

in

B

_ρ

,

then for all

ε > 0 we can associate with g a possibly multi-valued function ˜

g

_ε

defined at

least on

B

_ρ−ε

\ P

−

and taking values in

[−1, 1] which satisfies

(2.10)

U (X) = g(X − ε˜

g

_ε

(X)e

_n

).

Moreover if

g is strictly monotone in the e

_n

direction in

B

+

ρ

(g), then ˜

g

ε

is single-valued.

A similar statement holds for

˜

¯

g

_ε

, when

g satisfies the flatness assumption

(2.11)

U (X + εe

¯

_n

) ≤ g(X) ≤ ¯

U (X − εe

_n

)

in

B

_ρ

.

The following elementary lemmas hold. The proof of the first one can be found in [15].

The second one can be obtained similarly.

(10)

Lemma 2.8. Let g, v be non-negative continuous functions in B

ρ

. Assume that g satisfies

the flatness condition (2.9) in B

ρ

and that v is strictly increasing in the e

n

direction in B

ρ+

(v).

Then if

v ≤ g

in B

ρ

,

and ˜

v

ε

is defined on B

ρ−ε

\ P

−

we have that

˜

v

ε

≤ ˜

g

ε

on B

ρ−ε

\ P

−

.

Viceversa, if ˜

v

ε

is defined on B

s

\ P

−

and

˜

v

ε

≤ ˜

g

ε

on B

s

\ P

−

,

then

v ≤ g

on B

s−ε

.

Lemma 2.9. Let g, v be non-negative continuous functions in B

ρ

. Assume that g satisfies

the flatness condition (2.11) in B

ρ

and that v is strictly decreasing in the e

n

direction in

B

+

ρ

(v).

Then if

v ≥ g

in B

ρ

,

and ˜

v

¯

ε

is defined on B

ρ−ε

\ P

+

we have that

˜

¯

v

ε

≤ ˜

¯

g

ε

on B

ρ−ε

\ P

+

.

Viceversa, if ˜

v

¯

ε

is defined on B

s

\ P

+

and

˜

¯

v

ε

≤ ˜

¯

g

ε

on B

s

\ P

+

,

then

v ≥ g

on B

s−ε

.

We now state and prove a key comparison principle, which will follow immediately

from the lemmas above and Corollary 2.6.

Lemma 2.10. Let (u

1

, u

2

), (v

1

, v

2

)

be respectively a solution and a comparison subsolution

to (2.1) in B

2

, with (v

1

, v

2

)

strictly monotone in the e

n

direction in B

2+

(v

i

).

Assume that u

1

and u

2

satisfy respectively the flatness assumptions (2.9)-(2.11) in B

2

for ε > 0 small and

that g

(v

1

)

ε

, g

(v

2

)

ε

are defined in B

2−ε

\ P

∓

and satisfy

| g

(v

1

)

ε

| ≤ C,

| g

(v

2

)

ε

| ≤ C.

If

(2.12)

(v

g

1

)

ε

+ c ≤ g

(u

1

)

ε

in (B

3/2

\ B

1/2

) \ P

−

,

and

(2.13)

(v

g

2

)

_ε

+ c ≤ g

(u

2

)

_ε

in (B

3/2

\ B

1/2

) \ P

+

,

then

(2.14)

(v

g

₁

)

_ε

+ c ≤ g

(u

₁

)

_ε

in B

_3/2

\ P

−

,

and

(2.15)

(v

g

₂

)

_ε

+ c ≤ g

(u

₂

)

_ε

in B

_3/2

\ P

+

.

(11)

Proof. We wish to apply Corollary 2.6 to the functions (u

1

, u

2

) and

v

i,t

= v

i

(X + εte

n

).

We need to verify that for some

t

₀

< t

₁

= c

(2.16)

v

_1,t

0

≤ u

1

v

2,t0

≥ u

2

in

B

1

,

and for all

δ > 0 and small

(2.17)

v

_1,t

1−δ

≤ u

1

on

∂B

1

,

v

1,t1−δ

< u

1

on

F (v

1,t1−δ

),

(2.18)

v

_2,t

1−δ

≥ u

2

on

∂B

1

,

v

2,t1−δ

> u

2

on

F (u

2

).

Then our Corollary implies

v

1,t1−δ

≤ u

1

in

B

1

,

v

2,t1−δ

≥ u

2

in

B

1

.

By letting

δ go to 0, we obtain that

v

1,t1

≤ u

1

,

v

2,t1

≥ u

2

in

B

1

,

which in view of Lemma 2.8 gives

^

(v

1,t1

)

ε

≤ g

(u

1

)

ε

in

B

1−ε

\ P

−

_,

^

(v

2,t1

)

ε

≤ g

(u

2

)

ε

in

B

1−ε

\ P

+

,

assuming that the

ε-domain variations on the left hand side exist on B

_1−ε

\ P

∓

. On the

other hand, it is easy to verify that on such set

(2.19)

(v

]

_1,t

)

ε

(X) = (

v

e

1

)

ε

(X) + t,

(v

]

2,t

)

ε

(X) = g

(v

2

)

ε

(X) + t,

and hence we have for

t

₁

= c/2,

g

(v

1

)

ε

+ t

1

≤ g

(v

1

)

ε

+ c ≤ g

(u

1

)

ε

in

B

1−ε

\ P

−

,

g

(v

2

)

ε

+ t

1

≤ g

(v

2

)

ε

+ c ≤ g

(u

2

)

ε

in

B

1−ε

\ P

+

,

which gives the desired conclusion.

We are left with the proof of (2.16)-(2.17)-(2.18).

In view of Lemmas 2.8-2.9, in order to obtain (2.16) it suffices to show that

^

(v

1,t0

)

ε

≤ g

(u

1

)

ε

,

in

B

1+ε

\ P

−

,

^

(v

2,t0

)

ε

≤ g

(u

2

)

ε

,

in

B

1+ε

\ P

+

_,

which by (2.19) becomes

g

(v

1

)

_ε

+ t

0

≤ g

(u

1

)

_ε

,

in

B

_1+ε

\ P

−

,

g

(v

2

)

ε

+ t

0

≤ g

(u

2

)

ε

,

in

B

1+ε

\ P

+

.

These last inequalities hold trivially for an appropriate choice of

t

₀

since the functions

involved are bounded.

For (2.17)-(2.18), notice that the first inequality follows easily from our assumption

(2.12)-(2.13) together with (2.19) and Lemmas 2.8-2.9. More precisely we have that for all

δ ≤ c/2, the stronger statement

(2.20)

v

_1,t

1

≤ u

1

v

2,t1+δ

≥ u

2

in

B

3₂−ε

\ B

1 2+ε

,

holds.

(12)

In particular, from the strict monotonicity of

v

₁

in the

e

_n

-direction in

B

+

2

(v

1

) we have

that

v

1,t1

> 0

on

F (v

1,t1−δ

),

which combined with the previous inequality gives that

u

1

> 0

on

F (v

_1,t 1−δ

),

that is the second condition in (2.17).

Similarly the second inequality in (2.17) follows

from the second inequality in (2.20) and the strict monotonicity of

v

₂

in

B

₂+

(v

₂

).

Finally, given

ε > 0 small and Lipschitz functions ˜

ϕ, ˜

ϕ defined on B

¯

_ρ

( ¯

X), with values

in

[−1, 1], then there exists a unique pair (ϕ

_ε

, ¯

ϕ

_ε

) defined at least on B

_ρ−ε

( ¯

X) such that

(2.21)

U (X) = ϕ

_ε

(X − ε ˜

ϕ(X)e

_n

),

X ∈ B

_ρ

( ¯

X),

(2.22)

U (X) = ¯

¯

ϕ

_ε

(X − ε ˜

ϕ(X)e

¯

_n

),

X ∈ B

_ρ

( ¯

X).

It is readily seen that if

(u

₁

, u

₂

) satisfies the flatness assumption (2.6) in B

₁

then (say

ρ, ε < 1/4, ¯

X ∈ B

1/2

,)

(2.23)

ϕ ≤ (˜

˜

u

₁

)

_ε

in

B

_ρ

( ¯

X) \ P

−

⇒ ϕ

_ε

≤ u

₁

in

B

_ρ−ε

( ¯

X),

(2.24)

ϕ ≤ (˜

˜

¯

u

₂

)

_ε

in

B

_ρ

( ¯

X) \ P

+

⇒ ¯

ϕ

_ε

≥ u

₂

in

B

_ρ−ε

( ¯

X).

The following proposition is contained in [16]. An analogous statement can be clearly

obtained for

ϕ

¯

_ε

.

Proposition 2.11. Let ϕ be a smooth function in B

λ

( ¯

X) ⊂ R

n+1

\ P

−

. Define (for ε > 0

small) the function ϕ

ε

as above by

(2.25)

U (X) = ϕ

_ε

(X − εϕ(X)e

_n

).

Then,

(2.26)

∆ϕ

_ε

= ε∆(U

_n

ϕ) + O(ε

2

),

in B

_λ/2

( ¯

X)

with the function in O(ε

2

₎

_{depending on kϕk}

C5

and λ.

3. The linearized problem.

We introduce here the linearized problem associated with (2.1). Given

g ∈ C(B

₁

) and

X

0

= (x

00

, 0, 0) ∈ B

1

∩ L, we call

g

r

(X

0

) :=

lim

(xn,z)→(0,0)

g(x

00

, x

n

, z) − g(x

00

, 0, 0)

r

,

r

2

_{= x}

2 n

+ z

2

.

Once the change of unknowns (2.7)-(2.8) has been done, the linearized problem associated

with (2.1) is

(3.1)











∆(U

n

g

1

) = 0,

in

B

₁

\ P

−

,

∆(− ¯

U

n

g

2

) = 0,

in

B

₁

\ P

+

,

g

1

= g

2

,

(g

1

)

r

+ (g

2

)

r

= 0,

on

B

₁

∩ L.

Definition 3.1. We say that (g

1

, g

2

) is a viscosity solution to (3.1) if g

i

∈ C(B

1

), g

i

is

even-in-

z and it satisfies

(i)

∆(U

_n

g

₁

) = 0

in

B

₁

\ P

−

;

(ii)

∆(− ¯

U

_n

g

₂

) = 0

in

B

₁

\ P

+

;

(13)

(iii)

g

₁

= g

₂

on

B

₁

∩ L;

(iv)

Let

φ

_i

be continuous around

X

₀

= (x

0₀

, 0, 0) ∈ B

₁

∩ L and satisfy

φ

1

(X) = l(x

0

) + b

1

(X

0

)r + O(|x

0

− x

00

|

2

_{+ r}

3/2

_),

φ

2

(X) = l(x

0

) + b

2

(X

0

)r + O(|x

0

− x

00

|

2

_{+ r}

3/2

_),

with

l(x

0

) = a

₀

+ a

₁

· (x

0

− x

0 0

).

If

b

₁

(X

₀

) + b

₂

(X

₀

) > 0 then (φ

₁

, φ

₂

) cannot touch (g

₁

, g

₂

) by below at X

₀

, and

if

b

₁

(X

₀

) + b

₂

(X

₀

) < 0 then (φ

₁

, φ

₂

) cannot touch (g

₁

, g

₂

) by above at X

₀

.

We wish to prove the following existence and regularity result.

Theorem 3.2. Given a boundary data (h

1

, h

2

)

with h

i

∈ C( ¯

B

1

), |h

i

| ≤ 1, h

i

even-in-z

and h

1

= h

2

on B

1

∩ L, there exists a unique classical solution (g

1

, g

2

)

to (3.1) such that

g

i

∈ C(B

1

), g

i

= h

i

on ∂B

1

, g

i

is even-in-z and it satisfies

(3.2)

|g

_i

(X) − a

₀

− a

0

· (x

0

− x

0₀

) − b

_i

r| ≤ C(|x

0

− x

0₀

|

2

+ r

3/2

),

X

₀

∈ B

_1/2

∩ L,

for a universal constants C, and a

0

∈ R

n−1

_{, a}

0

, b

i

∈ R depending on X

0

and satisfying

b

1

+ b

2

= 0.

As a corollary of the theorem above we obtain the following regularity result.

Theorem 3.3 (Improvement of flatness). There exists a universal constant C such that if

(g

1

, g

2

)

is a viscosity solution to (3.1) in B

1

with

−1 ≤ g

i

(X) ≤ 1

in B

1

,

then

(3.3)

a

₀

+ a

0

· x

0

+ b

_i

r − C|X|

3/2

≤ g

_i

(X) ≤ a

₀

+ a

0

· x

0

+ b

_i

r + C|X|

3/2

,

for some a

0

∈ R

n−1

_{, a}

0

, b

i

∈ R such that

b

1

+ b

2

= 0.

Proof. Let (w

1

, w

2

) be the unique classical solution to (3.1) in B

1/2

with boundary data

(g

1

, g

2

). We will prove that w

i

= g

i

in

B

_1/2

and hence it satisfies the desired estimate in

view of (3.2). Denote by

¯

w

εi

:= w

i

− ε + ε

2

r.

Then, for

ε small

¯

w

_εi

< g

i

on

∂B

_1/2

.

We wish to prove that

(3.4)

w

¯

i_ε

≤ g

_i

in

B

_1/2

.

Now, notice that

( ¯

w

1_ε

, ¯

w

_ε2

) (and all its translations) is a classical strict subsolution to (3.1)

that is

(3.5)











∆(U

n

w

¯

1ε

) = 0,

in

B

1/2

\ P

−

,

∆(− ¯

U

n

w

¯

ε2

) = 0,

in

B

1/2

\ P

+

,

¯

w

1 ε

= ¯

w

2ε

,

( ¯

w

1ε

)

r

+ ( ¯

w

ε2

)

r

> 0,

on

B

_1/2

∩ L.

Since

g

_i

is bounded, for

t large enough ¯

w

i_ε

− t lies strictly below g

_i

. We let

t → 0 and

show that the first contact point cannot occur for

t ≥ 0. Indeed since ¯

w

i_ε

− t is a strict

subsolution which is strictly below

g

_i

on

∂B

_1/2

then no touching can occur either in

(14)

B

1/2

\ P

∓

or on

B

_1/2

∩ L. We only need to check that no touching occurs on P

∓

\ L.

This follows from Lemma A.4 in the Appendix.

Thus (3.4) holds. Passing to the limit as

ε → 0 we deduce that

w

i

≤ g

i

in

B

_1/2

.

Similarly we also infer that

w

i

≥ g

i

in

B

_1/2

,

and the desired equality holds.

We remark that there is no theory readily available for this class of degenerate

prob-lems with a Neumann type boundary condition on the thin boundary

L, which in this

setting has positive capacity (see the classical work on degenerate elliptic problems [

?]).

Thus Theorem 3.3 calls for a direct proof.

The existence of the classical solution of Theorem 3.2 will be achieved via a variational

approach in the appropriate functional space.

Precisely, define the weighted Sobolev

space

H

_ω1

(B

1

) := H

1

(B

1

, U

n2

dX) × H

1

_(B

1

, ¯

U

n2

dX)

endowed with the norm

k(h

1

, h

2

)k :=

Z

B1

(h

2₁

+ |∇h

1

|

2

)U

n2

dX +

Z

B1

(h

2₂

+ |∇h

2

|

2

) ¯

U

n2

dX

and its subspace

V

0

(B

1

) := {(h

1

, h

2

) : h

i

∈ C

0∞

(B

1

), h

1

= h

2

on

B

₁

∩ L}.

Let

H

1_0,ω

(B

₁

) := V

₀

(B

₁

) be the completion of V

₀

(B

₁

) in H

_ω1

(B

₁

).

Consider the energy functional

J (h

1

, h

2

) :=

Z

B1

(U

_n2

|∇h

1

|

2

+ ¯

U

n2

|∇h

2

|

2

)dX,

(h

1

, h

2

) ∈ H

ω1

(B

1

).

Given a boundary data

(h

₁

, h

₂

) with h

_i

∈ C

∞

( ¯

B

₁

), and h

₁

= h

₂

on ¯

B

₁

∩L, we minimize

J (h

1

+ ϕ

1

, h

2

+ ϕ

2

)

among all

(ϕ

₁

, ϕ

₂

) ∈ H

1

0,ω

(B

1

). Equivalently, a minimizing pair (g

1

, g

2

) must satisfy

J (g

1

, g

2

) ≤ J (g

1

+ ϕ

1

, g

2

+ ϕ

2

),

∀ϕ

i

∈ C

0∞

(B

1

), i = 1, 2,

with

(3.6)

ϕ

₁

= ϕ

₂

on

B

₁

∩ L.

Since

J is strictly convex, we conclude that (g

₁

, g

₂

) is a minimizer if and only if

lim

ε→0

J (g

1

, g

2

) − J (g

1

+ εϕ

1

, g

2

+ εϕ

2

)

ε

= 0,

∀ϕ

i

∈ C

∞ 0

(B

1

), i = 1, 2,

satisfying (3.6), or equivalently,

(3.7)

Z

B1

(U

_n2

∇g

1

· ∇ϕ

1

+ ¯

U

n2

∇g

2

· ∇ϕ

2

)dx = 0,

∀ϕ

i

∈ C

0∞

(B

1

), i = 1, 2,

satisfying (3.6).

Lemma 3.4. Let (g

1

, g

2

)

be a minimizer to J in B

1

, then

∆(U

n

g

1

) = 0

in B

1

\ P

−

,

(15)

Proof. By (3.7)

Z

B1

(U

_n2

∇g

1

· ∇ϕ

−

+ ¯

U

n2

∇g

2

· ∇ϕ

+

)dx = 0,

∀ϕ

∓

∈ C

0∞

(B

1

\ P

∓

).

After integration by parts we obtain,

div

(U

_n2

∇g

₁

) = 0

in

B

₁

\ P

−

,

div

( ¯

U

_n2

∇g

₂

) = 0

in

B

₁

\ P

+

.

Since

U

_n

, ¯

U

_n

are bounded in

B

₁

\ P

−

, B

₁

\ P

+

respectively, we conclude by elliptic

regularity that

g

1

∈ C

∞

(B

1

\ P

−

), g

2

∈ C

∞

(B

1

\ P

+

).

Now a simple computation concludes the proof. Precisely,

div

(U

2 n

∇g

1

) = U

n2

∆g

1

+ 2

n+1

X

i=1

U

n

U

ni

(g

1

)

i

= 0

in

B

₁

\ P

−

.

Since

U

_n

> 0 and ∆U = 0 in B

₁

\ P

−

the identity above is equivalent to

∆(U

n

g

1

) = U

n

∆g

1

+ 2∇U

n

· ∇g

1

= 0

in

B

₁

\ P

−

,

as desired. The computation for

g

₂

is obtained analogously.

Lemma 3.5. Let (g

1

, g

2

) ∈ C(B

1

)

be a solution to

∆(U

n

g

1

) = 0

in B

1

\ P

−

,

∆(− ¯

U

n

g

2

) = 0

in B

1

\ P

+

,

with g

1

= g

2

on B

1

∩ L and assume that

lim

r→0

(g

i

)

r

(x

0

_{, x}

n

, z) = b

i

(x

0

),

(g

i

)

r

(X) =

x

n

r

g

xn

(X) +

z

r

g

z

(X);

with b

i

(x

0

)

a continuous function. Then (g

1

, g

2

)

is a minimizer to J in B

1

if and only if

b

1

+ b

2

≡ 0.

Proof. The pair (g

1

, g

2

) minimizes J if and only if

Z

B1

(U

_n2

∇g

1

· ∇ϕ

1

+ ¯

U

n2

∇g

2

· ∇ϕ

2

)dx = 0,

for all

ϕ

_i

∈ C

₀∞

(B

₁

) which coincide on L.

We compute that

(3.8)

Z

B1

U

n2

∇g

1

· ∇ϕ

1

dx =

Z

B1\P−

div

(U

_n2

g

₁

)ϕ

₁

dx + lim

δ→0

Z

∂Cδ∩B1

U

n2

ϕ

1

∇g

1

· νdσ

where

C

_δ

= {r ≤ δ} and ν is the inward normal to C

_δ

, and similarly

(3.9)

Z

B1

¯

U

_n2

∇g

2

· ∇ϕ

2

dx =

Z

B1\P+

div

( ¯

U

2 n

g

2

)ϕ

2

dx + lim

δ→0

Z

∂Cδ∩B1

¯

U

_n2

ϕ

2

∇g

2

· νdσ.

Indeed, say for

g

₁

, given δ > 0, for P

_ε

a strip of width

ε around P

−

Z

B1\Cδ

U

n2

∇g

1

· ∇ϕ

1

dx = lim

ε→0

Z

(B1\Cδ)∩(B1\Pε)

U

n2

∇g

1

· ∇ϕ

1

dx

and integrating by parts

Z

B1\Cδ

U

_n2

∇g

1

· ∇ϕ

1

dx =

Z

(B1\Cδ)\P−

div

(U

_n2

g

₁

)ϕ

₁

dx +

Z

∂Cδ∩B1

U

_n2

ϕ

1

∇g

1

· νdσ

(16)

+ lim

ε→0

Z

∂Pε∩(B1\Cδ)

U

_n2

ϕ

1

∇g

1

· νdσ.

It is easily seen that the last limit goes to zero, hence our claim is proved. In fact, in the

region of integration we have

U

n

≤ Cε,

and

|∇(U

n

g

1

)|, |∇U

n

| ≤ C,

from which it follows that

U

n

|∇g

1

| ≤ C.

Finally using the formula for

U we can compute that

(3.10)

lim

δ→0

Z

∂Cδ∩B1

U

_n2

ϕ

1

∇g

1

· νdσ = π

Z

L

b

1

(x

0

)ϕ

1

(x

0

, 0, 0)dx

0

,

(3.11)

lim

δ→0

Z

∂Cδ∩B1

¯

U

_n2

ϕ

2

∇g

2

· νdσ = π

Z

L

b

2

(x

0

)ϕ

1

(x

0

, 0, 0)dx

0

.

Precisely, say for

g

₁

,

Z

∂Cδ∩B1

U

_n2

ϕ

1

∇g

1

· νdσ =

1 δ

Z

∂Cδ∩B1

cos

2

(

θ

2 )(g

1

)

r

ϕ

1

dσ

=

Z

∂C1∩B1

cos

2

(

θ

2 )((g

1

)

r

ϕ

1

)(X

0

_{, δ cos θ, δ sin θ)dx}

0

_dθ,

and the desired equality follows.

Combining (3.8)-(3.9)-(3.10)-(3.11) with the fact that

ϕ

₂

= ϕ

₂

on

L, and the

computa-tion at the end of Lemma 3.4, we conclude the proof.

The key ingredient in showing the regularity of the minimizing pair is the following

Harnack inequality.

Lemma 3.6 (Harnack inequality). Let (g

1

, g

2

)

be a minimizer to J in B

1

among

competi-tors which are even-in-z, and assume |g

i

| ≤ 1. Then g

i

∈ C

α

(B

1/2

)

and

[g

i

]

C0,α_(B

1/2)

≤ C,

with C universal.

Indeed, since our linear problem is invariant under translations in the

x

0

-direction, we

see that discrete differences of the form

g

i

(X + τ ) − g

i

(X),

with

τ in the x

0

-direction are also minimizers. Thus by standard arguments (see [2]) we

obtain the following corollary.

Corollary 3.7. Let (g

1

, g

2

)

be as in Lemma 3.6. Then D

β

x0

g

i

∈ C

0,α

(B

1/2

)

and

[D

β_x0

g

i

]

Cα_(B

1/2)

≤ C,

with C depending on β.

The proof of Harnack inequality relies on the following comparison principle for

min-imizers.

(17)

Lemma 3.8 (Comparison Principle). Let (g

1

, g

2

)

and (h

1

, h

2

)

be minimizers to J in B

1

.

If

g

i

≥ h

i

a.e in B

1

\ B

ρ

,

i = 1, 2,

then

g

i

≥ h

i

a.e. in B

1

,

i = 1, 2.

Proof. Call

M

i

:= max{g

i

, h

i

},

m

i

:= min{g

i

, h

i

},

i = 1, 2.

We have that

(3.12)

J (M

₁

, M

₂

) + J (m

₁

, m

₂

) = J (g

₁

, g

₂

) + J (h

₁

, h

₂

).

On the other hand, since

(g

₁

, g

₂

) and (h

₁

, h

₂

) are minimizing pairs, and under our

as-sumptions

(M

₁

, M

₂

), (m

₁

, m

₂

) are admissible competitors,

J (M

1

, M

2

) ≥ J (g

1

, g

2

),

J (m

1

, m

2

) ≥ J (h

1

, h

2

).

We conclude that

J (M

1

, M

2

) = J (g

1

, g

2

),

J (m

1

, m

2

) = J (h

1

, h

2

),

and by uniqueness,

M

i

= g

i

,

m

i

= h

i

,

i = 1, 2.

Our desired result follows.

Proof of Lemma 3.6. The key step consists in proving the following claim. The

remain-ing remain-ingredients are the standard Harnack inequality and Boundary Harnack inequality

for harmonic functions.

Claim: There exist universal constants η, c such that if g

i

≥ 0 a.e. in B

1

and

g

1

(

1

4 e

n

) ≥ 1,

then

g

i

≥ c

a.e. in

B

_η

.

By boundary Harnack inequality, since

U

_n

g

₁

and

U

_n

are non-negative and harmonic in

B

1

\ P

−

and vanish on

P

−

we conclude that

g

1

≥ c

a.e. on

C := {|x

0

| ≤ 3/4, 1/8 ≤ r ≤ 1/2}

. Let

v

1

(X) := −

|x

0

_|

2

n − 1

+ 2(x

n

+ 1)r,

v

2

(X) := −

|x

0

_|

2

n − 1

− 2(x

n

+ 1)r.

We show that

(v

₁

, v

₂

) is a minimizer to J in B

₁

.

To do so we prove that

(v

₁

, v

₂

)

satisfies Lemma 3.5.

To prove that

∆(U

n

v

1

) = 0

in

B

₁

\ P

−

,

we use that

2rU

_n

= U and that U, U

_n

are harmonic outside of

P

−

and do not depend on

x

0

. Thus

∆(U

n

v

1

) = −∆(

|x

0

_|

2

n − 1

U

n

) + ∆((x

n

+ 1)U ) = −2U

n

+ 2U

n

= 0.

We argue similarly for

v

₂

.

(18)

Finally the fact that

lim

r→0

(v

1

)

r

+ (v

2

)

r

= 0,

follows immediately from the definition of

v

_i

, i = 1, 2.

Now by choosing

δ small depending on c we can guarantee that

v

1

+ δ ≤ c ≤ g

1

,

v

2

+ δ ≤ 0 ≤ g

2

a.e. on

{|x

0

| ≤ 3/4, δ/2 ≤ r ≤ δ},

v

i

+ δ ≤ 0 ≤ g

i

,

i = 1, 2

on

{|x

0

| = 3/4, r ≤ δ}.

By our comparison principle, Lemma 3.8 we conclude that

v

i

+ δ ≤ g

i

a.e in

{|x

0

| ≤ 3/4, r ≤ δ.}

Since

v

_i

+ δ is a continuous function positive at zero, our claim follows immediately.

Proof of Theorem 3.2. Let (g

1

, g

2

) be a minimizer to J in which achieves the boundary

data

(h

₁

, h

₂

). By approximation, we can assume that h

_i

is smooth. Then by Lemma 3.4,

(g

1

, g

2

) satisfy

∆(U

n

g

1

) = 0

in

B

₁

\ P

−

,

∆(− ¯

U

n

g

2

) = 0

in

B

₁

\ P

+

,

and in view of Lemma 3.6 and Corollary 3.7,

∆

_x0

g

_i

∈ C

0,α

(B

₁

) with universally bounded

norm in say

B

_3/4

. Thus, it is shown in Theorem 8.1 in [15] that (after reflecting

g

₂

)

(3.13)

|g

_i

(x

0

, x

_n

, z) − g

_i

(x

0

, 0, 0) − b

_i

(x

0

)r| = O(r

3/2

),

(x

0

, 0, 0) ∈ B

_1/2

∩ L,

with

b

_i

Lipschitz and moreover,

(3.14)

|(g

_i

)

_r

(x

0

, x

_n

, z) − b

_i

(x

0

)| ≤ Cr

1/2

,

(x

0

, 0, 0) ∈ B

_1/2

∩ L,

with

C universal. Hence by Lemma 3.5,

(g

1

)

r

(x

0

) + (g

2

)

r

(x

0

) = b

1

(x

0

) + b

2

(x

0

) = 0

on

B

_1/2

∩ L.

For completeness, we sketch the argument to obtain (3.13)-(3.14), say for

g

₁

.

Since

g

₁

solves

∆(U

n

g

1

) = 0

in

B

₁

\ P

−

and

U

_n

is independent on

x

0

we can rewrite this equation as

(3.15)

∆

_x

n,z

(U

n

g

1

) = −U

n

∆

x0

g

1

,

and according to Corollary 3.7 we have that

∆

x0

g

₁

∈ C

α

(B

_1/2

),

with universal bound. Thus, for each fixed

x

0

, we need to investigate the 2-dimensional

problem

∆(U

t

g

1

) = U

t

f,

in

B

_1/2

\ {t ≤ 0, z = 0}

with

f ∈ C

α

(B

1/2

),

and

g

₁

bounded. Without loss of generality, for a fixed

x

0

we may assume

g

₁

(x

0

, 0, 0) = 0.

Let

H(t, z) be the solution to the problem

∆H = U

t

f,

in

B

_1/2

\ {t ≤ 0, z = 0},

such that

(19)

We wish to prove that

(3.16)

U

_t

h = H.

First notice that

∆(H − U

t

h) = 0

in

B

_1/2

\ {t ≤ 0, z = 0},

and

H = 0

on

∂B

_1/2

∪ (B

_1/2

∩ {t < 0, z = 0}).

We claim that

(3.17)

lim

(t,z)→(0,0)

H − U

t

h

U

t

=

lim

(t,z)→(0,0)

H

U

t

= 0.

If the claims holds, then given any

ε > 0

−εU

t

≤ H − U

t

h ≤ εU

t

,

in

B

_δ

with

δ = δ(ε). Then by the maximum principle the inequality above holds in the whole