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Algebra i analiz St. Petersburg Math. J.

Tom. 32 (2020), }4 S 1061-0022(XX)0000-0

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES

(SCALAR AND SYSTEM CASES)

L. EL HAJJ AND H. SHAHGHOLIAN

Dedicated to Nina Nikolaevna Ural ′tseva on the occasion of her 85th birthday

Abstract. Convexity is discussed for several free boundary value problems in exte-

rior domains that are generally formulated as

∆u=f(u) in Ω \D, |∇u|=gon ∂Ω, u ≥0 in Rn,

where uis assumed to be continuous in Rn, Ω = {u > 0}(is unknown), u= 1 on

∂D, and Dis a bounded domain in Rn(n≥2). Here g=g(x) is a given smooth

function that is either strictly positive (Bernoulli-type) or identically zero (obstacle

type). Properties for fwill be spelled out in exact terms in the text.

The interest is in the particular case where Dis star-shaped or convex. The focus

is on the case where f(u) lacks monotonicity, so that the recently developed tool of

quasiconvex rearrangement is not applicable directly. Nevertheless, such quasicon-

vexity is used in a slightly diﬀerent manner, and in combination with scaling and

asymptotic expansion of solutions at regular points. The latter heavily relies on the

regularity theory of free boundaries.

Also, convexity for several systems of equations in a general framework is dis-

cussed, and some ideas along with several open problems are presented.

§1. Introduction

We discuss several free boundary value problems and their geometric properties such

as that of being star-shaped or convex for (continuous) solutions in the exterior of a given

domain. Starting with a bounded domain D⊂Rn(n≥2) and a given function fon R

(with properties to be speciﬁed) we consider the following semilinear partial diﬀerential

equation

(1.1)

∆u=f(u) in Rn\D,

u= 1 on ∂D,

u≥0 in Rn,

and ask whether the super-level sets of positive (continuous) solutions to this equation

inherit certain geometric properties of D. We also discuss the particular case of the so-

called Bernoulli problem, where the bulk equation in (1.1) is satisﬁed in Ω := {u > 0},

and an extra boundary condition |∇u|=gis imposed on ∂Ω. We will be more exact on

these points later.

In order to give a fair chance to the problem for positive results, we need, to a certain

extent, restrict the problem to a reasonable class of right-hand sides f(u). The class we

shall consider is still very general and contains many classical free boundary problems as

well as some new ones.

2010 Mathematics Subject Classiﬁcation. Primary 35R35.

Key words and phrases. Convexity, starshapedness, uniqueness, system of equations.

H. Shahgholian was supported by Swedish Research Council.

c

2004 American Mathematical Society

1

2 L. EL HAJJ AND H. SHAHGHOLIAN

For the system case we consider equations of the type

(1.2)

∆u=f(x, u) in Dc,

u=kon D,

u≥0in Rn,

where u= (u1, u2,·, um) is continuous,

k= (k1,...,km) and D= (D1, D2,·, Dm),

with Dj⊂Rn. We use the componentwise notation throughout the paper; in particular

u≥0means that ui≥0 for i= 1,···m.

When f=∇uFwith F“reasonably” smooth, one has a variational formulation of the

problem. The reason for introducing the system of equations is to initiate and promote a

possible development of qualitative theory for free boundary value problems for systems,

which scarcely exists now. Our discussions for systems are at a heuristic level and we

present some ideas and an example with hand-waving arguments. We also state and give

a heuristic proof of the existence of starshaped solutions whenever the given domains

and ingredients have this property. We hope that mathematical tools will come in the

future, and that the current discussion will tease the appetite of many enough for such

a development.

What concerns the scalar case, our approach combines tools from quasiconvexiﬁcation

[7, 13] and those of free boundary regularity of both Bernoulli type and obstacle type

problems. For the system case we also resort to recent results of regularity theory for

the corresponding problems. In simple terms our approach can be explained as follows.

Given a solution to the above problem, we take the quasiconvex rearrangement u∗that is

larger than u. It is known that u∗satisﬁes the inequality ∆u∗≥f(u∗), provided f(u)≥0

and |∇u|>0 inside the set {u > 0}. Now if fis also monotone, one can conclude by the

comparison principle that u∗=u, and consequently the super-level sets of uare convex.

Since we lack this monotonicity, we can circumvent the diﬃculty by rescaling the solution

ut0(x) = u(t0x) such that ut0(x)≥u∗(x) and t0<1 is the largest such value. From

this, it must follow that the surfaces of u∗and ut0touch at some point(s) x0. If the

touch is in the set where u∗(x0)>0, then we use a standard argument combined with

the comparison principle between the two functions and reach a contradiction. If it is on

the boundary of the support, the matter is subtler, and we have to resort to regularity

theory and asymptotic expansion of the solutions, and ﬁnally comparison between them

to reach a contradiction.

The method in this paper can be generalized to other operators (such as fully nonlinear

operators) and a general right-hand side (depending on xas well as on ∇u) whenever

the corresponding quasiconvexiﬁcation and regularity theory of the free boundaries are

available.

§2. Scalar equations

In this section, we make the following general assumptions on f(see [10] for similar

type of problems):

(2.1)

f(t)>0 for t > 0,

f(t) = 0 for t≤0,

f(t) = btα+o(tα) for t < t1,some t1>0 and −1< α < 1,

where b≥0. We also assume that

fis either left- or right-continuous and its discontinuities are isolated.(2.2)

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES 3

We can relax the third condition to allow bbe a function of x, with certain assumptions

to ﬁt into the context of quasiconvex rearrangement, see [7]. For clarity, we leave this

generalization to readers to ﬁgure out.

Property (2.2) includes examples such as the singular perturbation problem ∆uǫ=

1

ǫχ{0<uǫ<ǫ}, with the above boundary value on D. It is well known that solutions to this

problem converge to the Bernoulli free boundary.

Theorem 1. Let Dbe a bounded convex domain in Rn(n≥2), and let fhave properties

(2.1)–(2.2). Let g(x)be a Cβ-function in the entire space satisfying max(b, g(x)) ≥b0

for some b0, and let g(x)be either identically zero or strictly positive. When g > 0,we

assume 1/g to be a concave function.

Then there exists a nonnegative function uwith Ω := {u > 0}solving1the free bound-

ary problem

(2.3)

∆u=f(u)in Ω\D,

u= 1 on ∂D,

|∇u|=gon ∂Ω,

with convex super-level sets,i.e.,{u > l}is convex for all 0≤l≤1.

If in addition there is a point z∈Dsuch that

(2.4) tg(t(x−z) + z)is monotone nondecreasing in tfor all x∈Rn,

then uis a unique solution.

Remark 1.The theorem can be generalized in two directions.

i) One may allow f=f(x, u, ∇u), with f(x, s, t)/t2convex in both xand tfor all s;

see [7].

ii) One may also allow f(t) to take zero values for t > 0 as well as being in L1, through

approximation of fby smooth functions, and then passing to the limit, because the

convexity of the super-level sets will be preserved. A technical issue is that one needs to

make sure that the supports of the approximating solutions will stay uniformly compact.

In this case the uniqueness proof of the theorem will however not work anymore.

Proof. The existence of solutions follows in a standard way as in the literature, by using

the corresponding functional: we vaguely approximate the function fwith c0uαfrom

below, replace Dwith a larger ball containing D, and then use standard comparisons

with the corresponding symmetric functional; see for example [12] for similar arguments2.

Therefore we focus on the convexity of solutions only.

We let u∗be the quasiconcave envelope of the function u(see [7, 13]). By deﬁnition,

u∗is the function whose super-level sets are the closed convex hulls of the corresponding

super-level sets of u. If we take the support of uto be Ω = {u > 0}and denote by Ω∗

the convex hull of Ω, then we have (see [7])

Ω∗={x:u∗>0}.

From [7, 13] we have

(2.5) ∆u∗≥f(u∗) in Ω∗,

1The notion of a solution here can be taken in the weak sense for the PDE, and in the continuous

sense for all boundary conditions including the supplementary gradient condition. We also remark that

for linear PDEs, weak and viscosity solutions are the same. See [14, Page 103].

2One can actually use the so-called nondegeneracy argument to show that any (local/global) mini-

mizer ushould satisfy supBR(z)u≥c0cR2/(1−α)+u(z), as long as BR(z)⊂Dcand u(z)>0. Since

u < 1 we have a uniform bound on Rand hence on the support of u. See, e.g., [2].

4 L. EL HAJJ AND H. SHAHGHOLIAN

in the sense of viscosity, and hence in the weak sense (since the Laplacian is linear), see

[14, page 103] for the deﬁnition of viscosity solutions and the equivalence of weak and

viscosity. We remark further that (see [4, Proposition 3.7])

(2.6) u∗∈C1,β because u∈C1,β .

Our aim is to prove that u∗≡u, using the so-called Lavrentiev method of scaling and

comparing. Hence we deﬁne

ut(x) = u(tx),Ωt:= {x:tx ∈Ω}={x:ut(x)>0},

and set

t0:= sup{t:ut(x)≥u∗(x), x ∈Ω∗}.

Clearly t0<1, otherwise u∗=uand we are done. Therefore there is x0with u∗(x0) =

ut0(x0) such that either

A: x0∈(Ω∗∩Ωt0)\D, or

B: x0∈∂Ω∗∩∂Ωt0.

In case A, we set s0:= u∗(x0) = ut0(x0)>0, and by the hypotheses (2.1)–(2.2) we

may consider two subcases:3

A1: fis continuous in a neighborhood of s0;

A2: fis continuous in a punctured neighborhood of s0.

For A1 we use the continuity of fto show that t2

0f(ut0(x)) ≤f(u∗(x)), in a small

neighborhood of x0. Indeed, for any xclose to x0we have

(2.7) f(ut0(x)) ≤f(ut0(x0)) + ω(r1),

where r1:= |ut0(x)−ut0(x0)|and ωis the modulus of continuity for f. Once again using

the continuity of fand (2.7) we have (with r2:= |ut0(x)−u∗(x0)|)

(2.8) f(ut0(x)) ≤f(u∗(x0)) + ω(r2)≤f(u∗(x)) + ω(r1) + ω(r2).

Now by (2.8) and (2.5) it follows that

(2.9) ∆ut0=t2

0f(ut0(x)) ≤t2

0f(u∗(x)) + t2

0(ω(r1) + ω(r2)) ≤f(u∗(x)) ≤∆u∗,

provided r1, r2are chosen suﬃciently small such that

ω(r1) + ω(r2)<1−t2

0

t2

0

f(u∗(x)),

for xclose to x0. Hence ∆ut0≤∆u∗in Br(x0). Since also ut0≥u∗in a neighborhood

of x0and ut0(x0) = u∗(x0), we reach a contradiction to the strong comparison principle.

For the second subcase A2, we may assume for deﬁniteness that fis left-continuous.

The right-continuous case can be treated in exactly the same way as the left-continuous

case, and hence is left to the reader. Now take x∈Br(x0) with ut0(x)< ut0(x0) (since

we look at the left side of s0), and deﬁne K:= Br(x0)∩ {ut0(x)< s0}. Recall also that

s0:= u∗(x0) = ut0(x0)>0. Using a continuity argument as above, we easily arrive at

(2.8), and hence we deduce (2.9) for such x. From here we conclude, as argued above,

that

(2.10) ut0≥u∗, ut0(x0) = u∗(x0),∆ut0≤∆u∗in K.

Next, we observe that ∆ut0=t2

0f(ut0) is bounded and hence ut0is a C1,β function

locally near x0(for all 0 < β < 1). Recall (2.6) that u∗∈C1,β .

3We remark that case A2 covers also A1, but for clarity of the exposition we have considered the

cases separately.

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES 5

Along with (2.10), this implies

(2.11) |∇ut0(x0)|=|∇u∗(x0)|.

To this end we want to invoke Hopf’s boundary point lemma (in K) to contradict (2.11),

and conclude that u∗=u. To see that Hopf’s lemma can be invoked we only need (2.10),

the C1-regularity of both u∗and u, along with the interior C1,Dini-condition for Kat x0.

To show the last property, we use the fact that for each point z∈∂Ω∩∂Ω∗the super-level

set Lz:= {u(x)> u(z)}has a support plane and hence the standard Hopf lemma for u

applies at zfrom the interior of the set Ω \Lz, and we can conclude that |∇u(z)|>0.

This together with the C1,β -regularity of umentioned above implies that ∂Lzis C1,β in

a neighborhood of z. By the construction of u∗, the boundary of the super-level sets of

u∗must be C1,β , and hence Hopf’s lemma can be applied. This concludes the proof for

this case.

Now, we turn our attention to Case B, i.e., x0∈∂Ω∗∩∂Ωt0. In this case we know,

by the regularity theory, that near the special points xithat span ∂Ω∗\∂Ω, the free

boundary is smooth.4Without loss of generality, assume that x0= 0 and the inward

normal to ∂Ω∗at the origin is e1= (1,0,...,0). In particular, this means that the

inward normal to ∂Ω at each extremal point yithat span the origin5is e1, and hence

(by the regularity theory for free boundaries) we have a local asymptotics for uat these

extremal points:

(2.12) u(x) = ax+

1+b(x+

1)κ+o(|x|κ),where a=g(x0), κ =2

1−α.

Now, we consider two subcases, which appear in the assumptions (2.1).

B1: g≡0, i.e., u(x) = b(x+

1)κ+o(|x|κ).

B2: g > 0, i.e., u(x) = ax+

1+o(|x|), with a > 0.

For Case B1 a straightforward calculation in the 1-dimensional case gives

b=1

α(α−1)1/α−1

.

Similarly, we have

(2.13) ut0(x) = b0(x+

1)κ+o(|x|κ),where b0=t2

0

α(α−1)1/α−1

< b,

because t0<1,and 1

1−α>0. For small δ > 0,for some xi(i= 1,...,k with k≤n) we

have

(2.14)

k

X

i=1

λixi=δe1and u∗(δe1) = u(xi),

where Pk

i=1 λi= 1, and all xidepend on δ. Observe that the ﬁrst equation in (2.14)

implies

(2.15)

k

X

i=1

λixi

1=δ.

4Observe that for any point x∈∂Ω∩∂Ω∗the free boundary has a support plane and hence the free

boundary is C1,β at such points where there is some a priori ﬂatness from the exterior of the domain.

Indeed, also by property (2.1) our problem falls into the class of free boundaries that are well studied and

the regularity of the free boundaries for these cases is well known; see [1, 2, 10], and also generalization

to equations with right-hand side in some subsequent work.

5This means that Pk

i=1 λiyi=0, with k≤n

6 L. EL HAJJ AND H. SHAHGHOLIAN

Using (2.12)–(2.14), and the fact that ut0≥u∗, we have

b0δκ+o(δκ) = ut0(δe1)≥u∗(δe1) = u(xi) = b|xi

1|κ+o(|xi

1|κ).

Using the inequality b0< b, we can rephrase this as

|xi

1| ≤ (1 −2ρ)δ+o(δ)≤(1 −ρ)δ

for some ρ=ρ(b, b0, κ)>0, and provided δis small such that o(δ)≤ρδ. From this and

(2.15) we have

δ≤

k

X

i=1

λi|xi

1| ≤ (1 −ρ)δ

k

X

i=1

λi= (1 −ρ)δ,

which is a contradiction.

For Case B2 we have the Bernoulli boundary condition, with g > 0. Once again using

the C1,β -regularity of ∂Ω∗∩∂Ω (see the comment in footnote 4), we conclude that ∂Ω∗

is C1,β regular. Similarly, all super-level sets of u∗will have the same regularity. In

particular, we can conclude as in [16] that

h(x) := 1

|∇u∗|−1

g

is convex on any line segment of ∂Ω∗\∂Ω, with end-points on ∂Ω∗∩∂Ω; we call such

line segments maximal. Since u≤u∗, we conclude that |∇u| ≤ |∇u∗|on ∂Ω∗\∂Ω,

and in particular at the end-points of any maximal line segments and hence h(x)≤0 at

such end-points. Now hbeing convex implies h≤0 on such line segments and therefore

|∇u∗| ≥ gon these line segments. This in turn implies that u∗is a subsolution to the

Bernoulli problem.

Next we use Lavrentiev’s scaling and comparison. For this, suppose without loss of

generality that zin the statement of the theorem is the origin, and deﬁne ut0(x) = u(t0x),

as was done above. Comparing ut0and u∗, we conclude that

g(x0)≤ |∇u∗|(x0)≤ |∇ut0|(x0) = t0|∇u|(t0x0) = t0g(t0x0),

which contradicts the assumption (2.4) on g.

Uniqueness for both cases can be proved in a similar fashion, by using the function u

and scaling the next solution vt(x) = v(tx), after which the above Lavrentiev argument

is applied.

This theorem can be extended to an “exterior-like” ring-shaped domain, as done in

[8]. A consequence of this is that one can state (with a straightforward proof as in the

above case) a general uniqueness and convexity theory for elliptic PDEs in ring-shaped

domains, for f(t)>0 and left- or right- continuous with isolated discontinuities. At this

point, one can use approximation (of the function f) to strengthen the results to the

existence of solutions to elliptic PDEs in convex rings with convex super-level sets. Since

the positivity of fis a crucial part in our approach, we ﬁnd it not worthwhile to enter

into any technical details here, and we hope to be back to this in the near future using

more elaborated combinations of convexity theory for justifying the existence theory for

solutions with quasiconvex super-level sets.

§3. Systems (A discussion)

Problems involving weakly coupled systems have been on rise recently, in the free

boundary community. Currently, there are two main directions that are pursued by the

community, one being cooperative, and the other competitive systems. We shall discuss a

few examples of cooperative systems here below. To do so, we start introducing vectorial

notation for vector-functions and vector-domains as u= (u1, u2,··· , um), and D=

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES 7

(D1, D2,··· , Dm), with Dj⊂Rn. The term cooperative refers to the speciﬁc behavior

of the system, which forces the supports of all components of the vector-function coincide.

For the simplicity of presentation, we shall denote Ω = {|u|>0}, and k= (k1,··· , km),

with ki>0.

In general, one may consider equations of the type

(3.1)

∆u=f(x, u) in Ω \D,

u=kon D,

G(∇u) = g(x) on ∂Ω.

In case |∇u|= 0 on ∂Ω, the last equation is dropped out and the ﬁrst equation is replaced

by ∆u=f(x, u) in Dc, i.e.,

(3.2) (∆u=f(x, u) in Dc,

u=kon D.

When f=∇uFwith F“reasonably” smooth, one may ﬁnd solutions to (3.2) using

minimizers of the functional

(3.3) J(v) = ZRn

|∇v|2+F(x, v),

over {v∈W1,2

0(Rn) : v=kon D}. When F(x, ∇u) = g2(x)χ{|u|>0}, one obtains (3.1)

with G(∇u) = |∇u|, which is a Bernoulli type free boundary problem for systems that

were studied in [6]:

(3.4) (∆u= 0 in Ω \D,

|∇u|=g(x) on ∂Ω.

The particular case of F(u) = |u|, recently studied in [3], gives rise to obstacle type

problems for the systems

(3.5) ∆u=u

|u|χ{|u|>0}.

The next goal is to see what kind of domains Dcan be of interest to consider. This

has several possible scenarios; here are a few:

(i) Di=Dj,i, j = 1,2,...,m,

(ii) D1⊂D2⊂ · · · ⊂ Dm,

(iii) D1⊂D2⊂ · · · ⊂ Dm, and they are homothetic,

(iv) TDi6=∅.

We should also remark that for minimizers uof the functional Jwe always have

{ui>0}={uj>0}for all i, j, provided Di∩Dj6=∅. This follows from the fact that in

such cases we can make variations in both directions (upward and downward), ui∓ǫφi

and uj∓ǫφj. Hence we have the Euler–Lagrange equation for both uiand ujwhenever

one of them is nonzero.

The question we want to raise is under what conditions on D, and equations above

we may expect geometric inheritance for solutions. For instance, will the star-shapness

of Dwith respect to a point z∈Dimply the same for Ω? Will the convexity of all

components of Dimply the convexity of Ω?

We ﬁrst observe that in the most general case (iv), the convexity of the support of ui

is not trivial without further conditions on Di. Indeed, consider the following example

with m= 2.

8 L. EL HAJJ AND H. SHAHGHOLIAN

Example 1 (Negative result).Let

D1={x:−k < x1< k, −1< x2<1}

and

D2={x:−1< x1<1,−k < x2< k},

where kis very large. By nondegeneracy in both problems (see [3, 6]), we see that the

free boundary has to be in a uniform neighborhood of D1∪D2and hence it cannot be

convex, for klarge.

Therefore, in what follows we consider cases (i) and (ii).

It is also noteworthy that from Theorem 1 it follows easily that for small perturbation

of Di(i= 1,...,m) the solution-domains Ω remain convex, provided the free boundary

for the perturbed problem is uniformly C2. This can be seen in an obvious way from the

continuity of the curvature in the C2-case.

When the Disatisfy D1⊂D2⊂ · · · ⊂ Dm,and each Diis star-shaped with respect to

z∈D1, then one may obtain certain results, but still incomplete, because the regularity

of the free boundary is essential for our technique. Indeed, one may show through

a minimizing functional (whenever it applies) that the super-level sets of any global

minimizer are star-shaped with respect to z. If a solution is unique (this is true for the

cases when there is a minimizing functional that is strictly convex), then obviously it is

also star-shaped. This follows by standard star-shaped rearrangement arguments.

Uniqueness for the star-shaped case is subtler and many times requires smoothness of

the free boundary. However, we can still state and prove the following weaker version of

uniqueness.

Theorem 2. In equation (3.1), let Di⊆Dj(for i < j), and suppose all domains Di

are star-shaped with respect to the origin. Let further

f(x, u) = ∇uF(x, u),and G(∇u) = |∇u|,

and let each component of f= (f1,...,fm)satisfy the assumptions (2.1)–(2.2) in the u

variable;next,let gbe continuous and satisfy 6(for some a < 2)

tg(x)≥g(tx), taf(t|x|,p)≤f(|x|,p),∀t≤1,and p= (p1,...,pm).

We also assume that for some b0, b1≥0 (with max(b0, b1)>0) either of the following

holds7

g≥b0,|f(x, t)| ≥ b1|t|αfor some −1< α < 1,and 0< t < t1,

and that f(t) = 0if min(t1,··· , tm)<0,and f(t)≥0for t≥0. Then there is a

solution of (3.1), with compact support,satisfying

u(tx)≥tu(x)∀t≤1.

Moreover,among the solutions that satisfy the free boundary condition |∇u|=g > 0on

∂Ω,or those that have C1,β free boundary in the case where g= 0,there exists only one

solution to (3.1).

A proof of a general version of this theorem, and related results for nonlinear equations,

will appear later in a separate paper. Here we present a narrative sketch of the proof for

expert readers.

6The reason of taking a < 2 is of technical nature, but we believe that a= 2 should also work. This

appears in comparison when we use the scaling of utas was done in the proof of Theorem 1.

7Compare the assumptions in the scalar case.

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES 9

Proof. Starting with the corresponding functional

J(v) = ZDc

|∇v|2+F(x, v) + g2(x)χ{|v|>0},

we ﬁrst need to show that the conditions of the theorem reinforce the statement that the

support of a global minimizer should be compact. This can be done by using comparison

arguments for the functional and another one that gives larger solutions, and where

Diis replaced by a ball BR(0) ⊃Difor all i. See, e.g., [12] for such a comparison

argument. In other words, one can prove that replacing the functional with symmetric

and “competitive” integrands as well as replacing all Diwith a ball BR(along with

Schwarz symmetrization) will give a solution with larger support. From here one can

prove that the assumptions of the theorem will imply that we may take a minimizing

sequence of functions that all have uniformly bounded supports.

In a similar fashion as above, one can consider a decreasing star-shaped rearrange-

ment u∗of any global minimizier uthat reduces the functional value, unless u=u∗.

This would then imply that global minimizers are star-shaped. The assumptions in the

theorem are designed to allow the star-shaped rearrangement to work. This will show

the existence of star-shaped solutions as stated in the theorem.

The uniqueness follows with similar techniques as that in Theorem 1, with more care

as the system has several components to be treated. Indeed, having a solution proved

to be star-shaped as in the above paragraph, we may use scaling and comparison of

Lavrentiev in the case of g > 0, and use Hopf’s boundary principle for the case of g= 0.

For the latter, since the regularity of the free boundary for systems is so far at a foster

stage, we have imposed the extra condition of C1,β-regularity to obtain uniqueness. The

drill is standard.

Corollary 1. Retain the hypotheses of Theorem 2, and assume that

Di=Bri(0) = {|x|< ri},

f=f(|x|,u),and g:= g(|x|). Then there is a unique spherically symmetric solution

of (3.1) with compact support.

Proof. By Theorem 2, there exists a solution uto our problem. In what follows all

operations are componentwise.

If this solution is now spherically symmetric, we can deﬁne

v= inf

σu(σx),

where σis the class of all possible spherical rotations. In particular, ∆v≤f(|x|,v).

Similarly, we take w= supσu(σx), which is a subsolution, ∆w≥f(|x|,w).

If we have two diﬀerent solutions, we may take the minimum of the two solutions

u3:= min(u1,u2)

and then a second inﬁmum of all possible rotations of this minimum

u4= inf

σu3(σx).

A similar argument applies in taking the maximum of the two and then the supremum

of all possible rotations of this maximum.

In this way we have created one supersolution vand one subsolution wsatisfying

v≤w. Now, we can apply a scaling and comparison argument as we have done earlier,

i.e., for some largest possible t0<1 with vt0≥w, satisfying

∆vt0≤t2f(t0|x|,v)≤t2−a

0f(|x|,v).

10 L. EL HAJJ AND H. SHAHGHOLIAN

From this an argument precisely as that in Theorem 1 works to deduce that ∆vt0≤w

in the vicinity of the touching point zas in Theorem 1. We leave out the obvious details.

Observe that the regularity of the boundary that was used in Theorem 1 is not needed

here, because the spheres are smooth and Hopf’s lemma applies.

3.1. Bernoulli type problems for systems. Convexity results for the Bernoulli type

problems, represented in the system of equations (3.4), seem for now out of reach, at least

with the methods we know of. It is also not straightforward what conditions we should

impose on the domains D(besides being convex and maybe Di’s being homothetic).

For F(x, v) = χ{|v|>0}in case (i), i.e., when D=D1=D2=..., one may reduce the

problem to the scalar case by letting vi=ui

λiand Ω = {|v|>0}, to arrive at

(3.6)

∆v= 0 in Ω \D,

v= 0 on ∂Ω,

v= 1 on ∂D,

Pλ2

i|∇vi|2=g2on ∂Ω,

where the equations and boundary values are componentwise. The ﬁrst three equations

imply that vi=vjfor all i, j, and hence the last condition turns into (Pλ2

i)|∇v1|2=g2.

Therefore this case reduces to the scalar case.

The main question concerning the Bernoulli problem is the following. Let D=

(D1,··· , Dm) with Di⊂Di+1 and each Diconvex. Is there a unique solution u, with

Ω = {|u|>0}, to the Bernoulli free boundary problem

(3.7)

∆u= 0 in Ω \D,

u= 0 on ∂Ω,

u= 1 on ∂D,

|∇u|= 1 on ∂Ω,

with the set {|u|>0}convex? Uniqueness for this problem follows as in the scalar case,

by using Lavrentiev’s principle.8

3.2. Obstacle type problems for systems. The system case of the free boundary for

general F(x, p), even if Fis independent of xand convex in the p-variables, is of course

a nontrivial problem. Let us consider the particular case as in (3.5), i.e., F(x, v) = |v|, in

the minimization problem (3.3). As in the Bernoulli problem, case (i) reduces to scalar

cases. In fact since Di=Dj, for vi=ui

λiwe have ∆v=v

|u|, and if we let hij =vi−vj,

then (∆hij =hij

|u|in Ω \D,

hij = 0 on ∂(Ω \D).

Hence by the maximum principle we have hij ≡0, i.e., vi=vjfor all i, j, and it follows

that ui=λi

λjuj.

The harder case of Di⊂Di+1 ,i= 1,··· , m −1, does not seem to admit an obvious

approach. For the scalar case, we know of two approaches, where the ﬁrst one is the

classical approach of using a quasiconvexity function (see [15, 17]) and the second and

more straightforward is the quasiconvex rearrangement as in [7, 13]. Both of these

approaches seems to be nonelementary to implement in this case.

8Observe that the boundary condition |∇u|= 1 tacitly assumes that the gradient is continuous up

to the boundary.

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES 11

There are other models of the system case, which we mention here. These are

a) ∆ui=1 + X

j6=i

ujχ{|u|>0},b) ∆ui=ui

Pm

j=1 ujχ{|u|>0},

where both of them can be reduced to the scalar case by deﬁning U=Pm

i=1 ui, which

gives us the following equations

c) ∆U= (m+mU)χ{U >0},d) ∆U=χ{U >0},

respectively, where we have used the relation χ{|u|>0}=χ{U>0}. Even such a reformula-

tion does not seem to work by earlier methods, unless Di=Djfor all i, j. Indeed in this

case the equation becomes scalar with a ﬁxed D=Di, with boundary values k:= Pki

on D. Since the right-hand side is monotone in both cases, we can invoke earlier results

of [13] to conclude that Uhas convex level sets. It is however not obvious whether each

component will be quasiconvex. This is left to readers to explore.

As a ﬁnal simple example we mention the following system:

∆ui=1 + (m−1)ui−X

j6=i

ujχ{|u|>0},

and with U=Puiwe have ∆U=mχ{U>0}, with Di=Djfor all i, j , and the

corresponding boundary value. This is an obstacle problem and it is well known that the

only solution to this problem is quasiconvex.

For the above problems, with Di⊂Di+1 , the question of quasiconvexity for each

component, or any combination of the components, or even the convexity of the support

Ω = {|u|>0}remains tantalizing.

We know of no quasiconvexity results (or even problems) for systems in existing liter-

ature. Even the simplest equation such as

∆u= (1 + 2v)χ{u>0}in Dc,

∆v= (1 + u)χ{v>0}in Dc,

u= 1 on ∂D,

v= 2 on ∂D,

does not seem to be easy to handle with existing methods for convexity. Here we have

chosen equations such that the system cannot be reduced to a scalar case. A ma jor

problem is that solutions in exterior domains become quasiconvex and not convex. This

means that any existing method would imply that the right-hand side of each equation

above is a quasiconvex function and this does not suﬃce to force through the existing

methods, as they require the right-hand sides be convex functions. This may also be an

indication that what we ask for is not true, but we have neither a counterexample.

Example 2. (A positive result) We shall now consider an ad hoc system, whose solution

can be shown to have convex level sets. Consider the equation

∆u= (10 −ev)χ{u>0}in Dc

∆v= (10 −eu)χ{v>0}in Dc,

u= 1 on ∂D,

v= 2 on ∂D,

where euand evare the concave envelops of u, respectively, v.9This system is specially

designed to create quasiconvex solutions, by using standard techniques.

9The reason we take the concave envelop is that it is nonzero. On the other hand, the convex envelop

of a positive function can be zero.

12 L. EL HAJJ AND H. SHAHGHOLIAN

To ﬁnd a solution we start with the obstacle problem

(∆u0= 10χ{u0>0}in Dc,

u0= 1 on ∂D,

which has a (unique) nonnegative solution u0with convex level sets, that is u0is quasi-

convex (see [7]). We also remark that u0≤1, by the maximum principle and hence the

concave envelop eu0does not exceed 1, because the constant linear function h≡1 is a

linear function above u0.

Next for this u0we ﬁnd a nonnegative solution v0to

(∆v0= (10 −eu0)χ{v0>0}in Dc,

v0= 2 on ∂D.

Let v∗

0≥v0be the quasiconvex solution of v0, see [7]. The right-hand side g(x, v0) :=

(10 −eu0)χ{v0>0}satisﬁes the hypothesis for the quasiconvexity theory (it is nonnegative,

monotone increasing in v0, and gis convex in the x-variables). Hence we can apply

Theorems 4.2 and 4.4 in [7] (see also [13]) to arrive at

∆v∗

0≥g(x, v∗

0)≥g(x, v0) = ∆v0.

Since both functions have the same boundary values on the set {v∗

0>0}\D, we conclude

by the comparison principle that v0=v∗

0.

We also have u0≤v0by the comparison principle for the obstacle-type problems (this

is straightforward and left to the readers) which gives eu0≤ev0, and hence solutions of

their corresponding obstacle problems must satisfy the reverse order v0≥u1.

Next we solve (∆u1= (10 −ev0)χ{u1>0}in Dc,

u1= 1 on ∂D,

and observe that due to the same convexity argument and the comparison principle as

mentioned above, we see that u1is quasiconvex and u1≥u0.

This u1is used once again as before to ﬁnd v1that solves

(∆v1= (10 −eu1)χ{v1>0}in Dc,

v1= 2 on ∂D.

Since the left-hand side of the equation is monotone increasing in v1and u1≥u0, and

hence eu1≥eu0, we deduce (by the comparison principle) that v1≤v0. Similarly, we

obtain v1≥u1.

Continuing in this vein we will have a sequence of pairs (ui, vi) solving the problem

∆ui= (10 −evi−1)χ{ui>0}in Dc,

∆vi= (10 −eui)χ{vi>0}in Dc,

u= 1 on ∂D,

v= 2 on ∂D,

for i= 1,2,..., and with (u0, v0) as obtained above.

We denote by

(u0, v0)≺(u1, v1)

the ordering

u0≤u1,and v0≥v1,

and this is repeated in the correct order (using the iteration above indeﬁnitely),

(ui, vi)≺(ui+1, vi+1 ).

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES 13

In other words uiis an increasing sequence and viis a decreasing sequence.

From here, with some footwork, one can obtain a convergence result and hence a

solution to our system above.

3.3. Singular perturbation for systems. In the scalar case, the Bernoulli problem

discussed earlier can also be obtained as a limit problem of the singular perturbation; the

literature is vast for this problem, so none mentioned none forgotten. A similar theory,

yet to be developed, seems plausible for the system case, which seems to have much more

possibilities and variations than its scalar counterpart.

Here we shall give a few examples of such models that might be interesting for the

readers. The simplest equation is of the form

(3.8) ∆ui=1

ǫχ{0<|u|<ǫ}

in Dc, for a given (convex/star-shaped) D, and with u=kon D. This equation can be

reduced to the scalar case when k= (k,...,k), but stays a system with no possibility

of reduction when k= (k1,...,km), with k1< k2<··· < km. The more complicated

version would be to replace Dwith D= (D1,...,Dm) as earlier. This problem does not

seem to have a variational formulation, but one may study this through supersolution

techniques.

The following model equation

(3.9) ∆u=1

ǫ

u

|u|χ{0<|u|<ǫ}

has a variational formulation and is a direct generalization of the singular perturbation

problem to the system. This is also related to the obstacle problem for the systems as

we mentioned above.

3.4. Serrin type problem for systems. Contrary to convexity problems, the sym-

metry methods, such as the moving plane technique, work very well for free boundary

value problems for systems, when there is a symmetry in the given equation, the bulk

domain, and boundary values. Here, we present the simplest example, leaving several

obvious generalizations towards other problems to the reader. The original Saint Venant

problem, with an overdetermination of the boundary gradient condition (here expressed

as a system) is to show that whenever there is a solution vector uto the following problem

(3.10)

∆u=−kin Ω,

u=0on ∂Ω,

|∇u|= 1 on ∂Ω,

the domain Ω has to be a ball. For the scalar case, James Serrin [21] gave a very nice

proof of this based on the moving plane technique (of A. D. Alexandrov); see also [23]

for a diﬀerent proof.

For the system case, one may think of solutions being

u=k1−|x|2

2n,with Ω = n|x|<n

|k|o.

Same techniques carry over to the system case above, mutatis mutandis, by reﬂection

principle as long as the boundary is C2,β , to make Serrin’s argument work. Indeed, using

the moving plane technique one makes comparison componentwise along with Hopf’s

lemma and then adds up the gradients to achieve a contradiction. As this is a graduate

exercise, we leave out the details to the readers.

14 L. EL HAJJ AND H. SHAHGHOLIAN

It is however more interesting to consider the semilinear case of Serrin’s problem for

systems, that is to prove spherical symmetry for the same system as in (3.10) where one

replaces −kwith f(u), with mild assumptions on fsuch as Lipschitz regularity.

Another related problem is the discrete Bernoulli problem discussed in [22] (for the

scalar case), see also [11]. Let Ω be a domain (in Rn,n≥2) with C1boundary, and

r, l > 0 given constants, with l < maxΩu. Suppose the solution u= (u1,...,um) to the

problem

∆u=−k,in Ω,u= 0 on ∂Ω

has the property that, for all x∈∂Ω,

(3.11) dist(x, Γl) = r, Γl={|u|=l}.

Then, Ω is necessarily a ball.

See [18, 19, 20] for related problems in the scalar case.

§4. A few problems to ponder!

Several natural questions that arise from our analysis are the following.

Problems for Subsection 3.1–3.2. Both Bernoulli and obstacle type problems for

systems seem to be very hard to treat with existing methods and tools. All problems

discussed for both these are open and new methods are needed to treat them. The

quasiconvexiﬁcation as used in the above does not apply, because each right-hand side

of the equation contains other components of the solution and the requirement is that

these have to be convex functions in order to apply the method of quasiconvex envelope.

As is seen from Example (2), we force this by an ad hoc assumption in order to obtain a

positive result.

We strongly believe that these problems, as well as their interior counterparts will be

of interest for future study. All we need is some starting ideas and new tools to treat the

systems.

Problems for Subsection 3.3. In the scalar case, the ﬁrst author of this paper gave a

proof of convexity for the singular perturbation problem [8], with a slightly diﬀerent argu-

ment. Our proof in §2 provides a more general result and is probably simpler. However,

none of these methods seem to work for the system case for the singular perturbation

problem. It would be nice to see some new ideas for how the system case can work.

Problems for Subsection 3.4.

(1) How far can one stretch the conditions on f? Can we allow each component of f

to change sign, or be in L1? This question is yet not answered in the scalar case.

(2) Can we derive a stability theory for the system case, as this was done for the

scalar case [5]?

(3) Try to prove symmetry results for the discrete Bernoulli problem in the system

case.

Problems in unbounded domains. Most of the above questions and open problems

can naturally be stated for unbounded domains and half-spaces. Several authors have

considered the scalar case of such problems and the literature is vast. The reader may

consult the paper [9], and the references therein for these problems.

It would be interesting to see if methods can be invented to treat not only problems

in bounded domains, but also in unbounded ones as developed in the literature.

REMARKS ON THE CONVEXITY OF FREE BOUNDARIES 15

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American University in Dubai, Dubai, UAE

Email address:lhajj@aud.edu

KTH Royal institute of Technology, Stockholm, Sweden

Email address:henriksh@kth.se

Received 21/JULY/2019

Originally published in English