Electronic Journal of Differential Equations, Vol. 2010(2010), No. 02, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OPTIMIZATION IN PROBLEMS INVOLVING THE
P-LAPLACIAN
MONICA MARRAS
R
Abstract. We minimize the energy integral Ω |∇u|p dx, where g is a bounded
positive function that varies in a class of rearrangements, p > 1, and u is a
solution of
−∆p u = g
u=0
in Ω
on ∂Ω .
Also we maximize the first eigenvalue λ = λg , where
−∆p u = λgup−1
in Ω .
For both problems, we prove existence, uniqueness, and representation of the
optimizers.
1. Introduction
Let Ω be a bounded smooth domain in RN , and let g0 be a measurable function
satisfying 0 < g0 ≤ H in Ω for a positive constant H. Define G as the family of
measurable functions which are rearrangements of g0 . In Section 2 of this article,
we consider the problem
−∆p u = g in Ω,
(1.1)
u = 0 on ∂Ω,
0
where p > 1, g ∈ G. The operator ∆p : W01,p (Ω) → W −1,p (Ω), p0 = p/(p − 1),
stands for the usual p-Laplacian defined as
Z
h−∆p u, vi =
|∇u|p−2 ∇u · ∇v dx.
Ω
It is well known that (1.1) has a unique solution u ∈ W01,p (Ω). Corresponding to
g, we consider the so called energy integral
Z
I(g) =
|∇u|p dx.
Ω
It is useful to investigate the maximum or the minimum of I(g) when g varies in G.
Actually, the maximum of I(g) has been discussed in the paper [7]. In the present
paper we investigate the minimum of I(g) for g ∈ G, proving results of existence
2000 Mathematics Subject Classification. 35J25, 49K20, 47A75.
Key words and phrases. p-Laplacian; energy integral; eigenvalues; rearrangements;
shape optimization.
c
2010
Texas State University - San Marcos.
Submitted November 2, 2009. Published January 5, 2010.
1
2
M. MARRAS
EJDE-2010/02
and uniqueness of the minimizer ǧ. We also find a formula of representation for ǧ
in terms of the corresponding solution uǧ .
In Section 3, we consider the eigenvalue problem
−∆p u = λgup−1 ,
u(x) > 0
in Ω,
u = 0 on ∂Ω,
(1.2)
where λ is the first eigenvalue. It is well known that problem (1.2) has a first
positive eigenvalue λ = λg and a corresponding positive eigenfunction u = ug .
It is interesting to investigate the maximum and the minimum of λg for g ∈ G.
Actually, the minimum of λg has been discussed in the paper [8]. In the present
paper we investigate the maximum of λg for g ∈ G, proving results of existence and
uniqueness. We also find a formula of representation for the maximizer ĝ in terms
of the corresponding eigenfunction uĝ . We emphasize that the methods developed
here are different from those used in the papers [7] and [8].
Since we use the notion of rearrangements, let us recall the definition. Denote
with |E| the Lebesgue measure of the (measurable) set E. Given a function g0 (x)
defined in Ω and satisfying 0 < g0 (x) ≤ H for a constant H. We say that g(x)
belongs to the class of rearrangements G = G(g0 ) if
|{g(x) ≥ β}| = |{g0 (x) ≥ β}|
∀β ∈ (0, H).
Here we write {g(x) ≥ β} instead of {x ∈ Ω : g(x) ≥ β}. In what follows, we shall
use the following results
Lemma 1.1. Let g ∈ Lr (Ω), r > 1, and let u ∈ Ls (Ω), s = r/(r − 1). Suppose
that every level set of u has measure zero. Then there exists an increasing function
φ such that φ(u) is a rearrangement of g. Furthermore, there exists a decreasing
function ψ such that ψ(u) is a rearrangement of g.
Proof. The first assertion follows from [4, Lemma 2.9]. The second assertion follows
applying the first one to −u.
Denote with G the closure of G with respect to the weak* topology in L∞ (Ω).
Lemma 1.2. Let G be the set of rearrangements of a fixed function g0 ∈ Lr (Ω),
r > 1, and let u ∈ Ls (Ω), s = r/(r − 1). If there is an increasing function φ such
that φ(u) ∈ G then
Z
Z
gu dx ≤
φ(u)u dx ∀g ∈ G,
Ω
Ω
and the function φ(u) is the unique maximizer relative to G. Furthermore, if there
is a decreasing function ψ such that ψ(u) ∈ G then
Z
Z
gu dx ≥
ψ(u)u dx ∀g ∈ G,
Ω
Ω
and the function ψ(u) is the unique minimizer relative to G.
Proof. The first assertion follows from [4, Lemma 2.4]. To prove the second assertion we put φ(t) = ψ(−t). Since φ is increasing, applying the previous result we
have
Z
Z
g (−u) dx ≤
Ω
φ(−u) (−u) dx ∀g ∈ G,
Ω
EJDE-2010/02
PROBLEMS INVOLVING THE P-LAPLACIAN
3
and φ(−u) = ψ(u) is the unique function satisfying the inequality. Equivalently,
we have
Z
Z
gu dx ≥
ψ(u)u dx ∀g ∈ G.
Ω
Ω
The proof is complete.
2. Energy integral
Let Ω ⊂ RN be a bounded smooth domain and let p > 1, H > 0 be two real
numbers. Let G be the family of all rearrangements of a given function g0 with
0 < g0 (x) ≤ H. It is convenient to introduce G, the closure of G with respect to the
weak* topology in L∞ (Ω). By [3] or [4], we know that G is weakly compact and
convex. Moreover, each g ∈ G satisfies 0 < g(x) ≤ H a.e. in Ω.
For g ∈ G, we consider problem (1.1). It is a classical result that such a problem
has a unique positive solution u ∈ W01,p (Ω) which satisfies
Z
Z
Z
p
p
sup
(pgv − |∇v| )dx = (pgu − |∇u| )dx = (p − 1)
|∇u|p dx. (2.1)
v∈W01,p (Ω)
Ω
Ω
Ω
p
R
It is also known that the functional Ω (pgv − |∇v| )dx has a unique maximizer u in
W01,p (Ω), and this maximizer is a solution to problem (1.1). By regularity results
(see for example [17]), the solution u belongs to W 2,1 (Ω), and equation (1.1) holds
a.e. in Ω.
R
Lemma 2.1. For g ∈ G, let I(g) = Ω |∇u|p dx, where u is the solution to (1.1).
(a) The functional g 7→ I(g) is continuous with respect to the weak* topology
in L∞ (Ω).
(b) The functional g 7→ I(g) is strictly convex in G.
p
(c) The functional g 7→ I(g) is Gâteaux differentiable with derivative p−1
ug .
Proof. Part (a). Let gn * g, and let ug , ugn be the corresponding solutions to (1.1)
with g, gn respectively. Using (2.1) we have
Z
Z
(p − 1)I(g) +
p(gn − g)ug dx = (pgn ug − |∇ug |p ) dx ≤ (p − 1)I(gn )
Ω
ZΩ
Z
= (pgugn − |∇ugn |p ) dx +
p(gn − g)ugn dx
Ω
Ω
Z
≤ (p − 1)I(g) +
p(gn − g)ugn dx.
Ω
(2.2)
By assumption, we have
Z
(gn − g)ug dx = 0.
(2.3)
(gn − g)ugn dx = 0.
(2.4)
lim
n→∞
Ω
Let us prove that
Z
lim
n→∞
Ω
Using (1.1) with g = gn , Poincaré Theorem and Hölder inequality we have
Z
Z
Z
Z
1/p
p
|∇ugn | dx =
gn ugn dx ≤ H
ugn dx ≤ C
|∇ugn |p dx
|Ω|(p−1)/p .
Ω
Ω
Ω
Ω
(2.5)
4
M. MARRAS
EJDE-2010/02
By (2.5) we infer that the norm k∇ugn kLp (Ω) is bounded by a constant independent
of n. Therefore, a sub-sequence of ugn (denoted again ugn ) converges weakly in
W 1,p (Ω) and strongly in Lp (Ω) to some function z ∈ W01,p (Ω). Since
Z
Z
Z
(gn − g) ugn dx = (gn − g)z dx + (gn − g)(ugn − z)dx,
Ω
and since
Ω
Ω
Z
(gn − g)(ugn − z)dx ≤ 2Hkugn − zkL1 (Ω) ,
Ω
Equality (2.4) follows. By (2.2), (2.3) and (2.4) we infer
lim I(gn ) = I(g).
n→∞
(2.6)
This yields the weak* continuity. We claim that the function z is actually the
solution of (1.1) corresponding to our function g. Indeed, from
Z
(p − 1)I(gn ) = (pgn ugn − |∇ugn |p ) dx,
Ω
Z
Z
gn ugn dx =
gz dx,
lim
n→∞
Ω
Ω
and the classical result
Z
|∇ugn |p dx ≥
lim inf
n→∞
Ω
Z
|∇z|p dx,
Ω
using (2.6) and (2.1) we get
Z
(pgz − |∇z|p ) dx ≤ (p − 1)I(g).
R
By the uniqueness of the maximizer of Ω (pgv − |∇v|p ) dx we must have z = ug , as
claimed.
Proof of (b). Let f, g ∈ G, let 0 < t < 1 and let v ∈ W01,p (Ω). We have
Z
p(tf + (1 − t)g)v − |∇v|p dx
Ω
Z
Z
p
= t (pf v − |∇v| ) dx + (1 − t) (pgv − |∇v|p ) dx.
(p − 1)I(g) ≤
Ω
Ω
Ω
By taking the superior of both sides relative to v ∈ W01,p (Ω), we get
I(tf + (1 − t)g)) ≤ tI(f ) + (1 − t)I(g),
that is, the convexity. Now, suppose equality holds in the above inequality for some
t ∈ (0, 1). Then, if ut is the solution corresponding to tf + (1 − t)g we have
Z
Z
t
pf ut − |∇ut |p dx + (1 − t)
pgut − |∇ut |p dx
Ω
Ω
Z
Z
p
= t (pf uf − |∇uf | ) dx + (1 − t) (pgug − |∇ug |p ) dx.
Ω
Ω
It follows that
Z
ZΩ
Ω
pf ut − |∇ut |p dx =
pgut − |∇ut |p dx =
Z
ZΩ
Ω
pf uf − |∇uf |p dx
pgug − |∇ug |p dx.
EJDE-2010/02
PROBLEMS INVOLVING THE P-LAPLACIAN
5
By the uniqueness of the maximizer, we must have ut = uf = ug . Moreover, since
−∆p uf = f,
−∆p ug = g,
a.e. in Ω,
a.e. in Ω,
if uf = ug , we must have f (x) = g(x) a.e. in Ω, and the strict convexity is proved.
Proof of (c). Let tn > 0 be a sequence such that tn → 0 as n → ∞. Let f, g ∈ R,
and let gn = g + tn (f − g). Then, by (2.2) we find
Z
p
I(g) + tn (f − g)
ug dx ≤ I g + tn (f − g)
p−1
Ω
Z
p
≤ I(g) + tn (f − g)
ug dx,
p
−
1 n
Ω
and
Z
(f − g)
Ω
I(g + tn (f − g)) − I(g)
p
ug dx ≤
≤
p−1
tn
Z
(f − g)
Ω
p
ug dx.
p−1 n
As already observed, the sequence ugn converges to ug in the norm of Lp (Ω). Therefore,
Z
Z
(f − g)ugn dx = (f − g)ug dx.
lim
n→∞
Ω
Ω
Hence, since the sequence tn is arbitrary, we have
Z
I(g + t(f − g)) − I(g)
p
= (f − g)
ug dx.
lim
t→0
t
p
−
1
Ω
It follows that I(g) is Gâteaux differentiable with derivative
complete.
p
p−1 ug .
The proof is
Theorem 2.2. Let 0 < g0 (x) ≤ H, and let G be the class of all rearrangements of
g0 . There exists a unique ǧ ∈ G such that
I(ǧ) = inf I(g).
g∈G
Furthermore, we have ǧ = ψ(uǧ ) for some decreasing function ψ.
Proof. By the compactness of G and the weak continuity of I(g) (proved in Lemma
2.1), we know that a minimizer ǧ exists in G. Since by Lemma 2.1, I(g) is strictly
convex, the minimizer ǧ is unique. We have to show that ǧ ∈ G.
With 0 < t < 1 and g ∈ G, let gt = ǧ + t(g − ǧ). Since I(g) is Gâteaux
differentiable at ǧ, we have
Z
p
I(gt ) = I(ǧ) + t (g − ǧ)
uǧ dx + o(t).
p
−
1
Ω
Since I(gt ) ≥ I(ǧ), we find
Z
I(ǧ) ≤ I(ǧ) + t
(g − ǧ)
Ω
It follows that
Z
0≤
(g − ǧ)
Ω
As t → 0 we find that
p
uǧ dx + o(t).
p−1
p
o(t)
uǧ dx +
.
p−1
t
Z
0≤
(g − ǧ)uǧ dx,
Ω
6
M. MARRAS
and
Z
EJDE-2010/02
Z
guǧ dx ≥
∀g ∈ G.
ǧuǧ dx,
Ω
(2.7)
Ω
The function u = uǧ satisfies the equation −∆p u = ǧ > 0 a.e. in Ω, therefore uǧ
cannot have flat zones in Ω (see [12, Lemma 7.7]). By Lemma 1.1 and Lemma 1.2
we can find a decreasing function ψ such that ψ(uǧ ) is a rearrangement of g0 and
Z
Z
guǧ dx ≥
ψ(uǧ )uǧ dx, ∀g ∈ G.
Ω
Ω
Comparing the latter inequality with inequality (2.7) and using Lemma 1.2 again,
we must have ǧ = ψ(uǧ ) ∈ G. The proof is complete.
We remark that Theorem 2.2 gives some information on the shape of the minimizer ǧ. Indeed, since the associate solution uǧ is positive in Ω, vanishes on the
boundary ∂Ω, and ǧ = ψ(uǧ ) with ψ decreasing, ǧ has to be large where uǧ is small,
that is close to ∂Ω.
3. Principal eigenvalue
We use the same assumptions and notation as in the previous section. For g ∈ G,
we consider problem (1.2). It is known that such a problem has a principal positive eigenvalue λg to which corresponds a unique (up to a normalization) positive
eigenfunction ug [1]. We have [14]
R
pgv p dx
1
RΩ
.
(3.1)
=
sup
p
λg
v∈W 1,p (Ω) Ω |∇v| dx
0
Following Auchmuty [2], we can prove that
hZ
Z
2 i Z
Z
2
p2 1
p
p
p
p
pg|v|
dx
−
|∇v|
dx
=
pg|u|
dx
−
|∇u|
dx
=
sup
.
4 λ2g
Ω
Ω
Ω
v∈W 1,p (Ω) Ω
0
(3.2)
Since we know that the principal eigenfunction is positive, we can take v > 0 in
(3.2). Therefore, for v > 0, define
Z
2
Z
p
A(v) =
pgv dx −
|∇v|p dx .
Ω
Ω
With t > 0, we have
A(tv) = tp
Z
pgv p dx − t2p
Ω
Z
2
|∇v|p dx .
Ω
It is easy to see that, for v fixed, A(tv) ≤ A(t0 v) with
R
pgv p dx
p
t0 = R Ω
2
2 Ω |∇v|p dx
Therefore,
R
p2 Ω pgv p dx 2
R
A(tv) ≤
.
4
|∇v|p dx
Ω
It follows that
p2
sup A(v) =
4
v∈W 1,p (Ω)
0
R pgv p dx 2
RΩ
sup
.
|∇v|p dx
v∈W 1,p (Ω)
Ω
0
(3.3)
EJDE-2010/02
PROBLEMS INVOLVING THE P-LAPLACIAN
7
Equation (3.2) follows from the latter equation and (3.1).
Note that if v is a maximizer in (3.1) then also νv with ν 6= 0 is a maximizer.
A maximizer u in (3.1) is also a maximizer in (3.2) when u is normalized so that
t0 = 1 in (3.3), that is
Z
Z
2
pgup dx = 2
|∇u|p dx .
(3.4)
Ω
Ω
Therefore, the (positive) maximizer u = ug in (3.2) satisfies (3.4) and is unique.
2
Lemma 3.1. For g ∈ G, let J(g) = p4 λ12 , where λg is the principal eigenvalue of
g
problem (1.2).
(a) The functional g 7→ J(g) is continuous with respect to the weak* topology
in L∞ (Ω).
(b) The functional g 7→ J(g) is strictly convex in G.
(c) The functional g 7→ J(g) is Gâteaux differentiable with derivative pupg .
Proof. Parts (a) and (b) of this lemma are essentially proved in [8]; however we
give here a slightly different proof.
Proof of (a). Let gn * g, and let ug , ugn be the corresponding maximizers of
(3.2) (eigenfunctions) with g, gn respectively. Using (3.2) we have
Z
Z
Z
2
|∇ug |p dx ≤ J(gn )
J(g) +
p(gn − g)upg dx =
pgn upg dx −
Ω
Ω
ZΩ
Z
2
p
=
pgugn dx − |∇ugn |p dx +
p(gn − g)upgn dx
Ω
Ω
Z
≤ J(g) +
p(gn − g)upgn dx.
Ω
(3.5)
By assumption, we have
Z
(gn − g)upg dx = 0.
(3.6)
(gn − g)upgn dx = 0.
(3.7)
lim
n→∞
Ω
Let us prove that
Z
lim
n→∞
Ω
Using (3.4) with g = gn and Poincaré Theorem we have
Z
Z
Z
2 Z
2
|∇ugn |p dx =
pgn upgn dx ≤ pH
upgn dx ≤ C
|∇ugn |p dx.
Ω
Ω
Ω
(3.8)
Ω
By (3.8) we infer that the norm k∇ugn kLp (Ω) is bounded by a constant independent
of n. Therefore, a sub-sequence of ugn (denoted again ugn ) converges weakly in
W 1,p (Ω) and strongly in Lp (Ω) to some function z ∈ W01,p (Ω). Since
Z
Z
Z
p
p
(gn − g) ugn dx = (gn − g) z dx + (gn − g)(upgn − z p )dx,
Ω
Ω
Ω
and since
Z
Z
p
p
|ugn − z|(ugn + z)p−1 dx
(gn − g)(ugn − z )dx ≤ 2HCp
Ω
Ω
Z
(p−1)/p
≤ 2HCp kugn − zkLp (Ω)
(ugn + z)p dx
,
Ω
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M. MARRAS
EJDE-2010/02
Equality (3.7) follows. By (3.5), (3.6) and (3.7) we infer
lim J(gn ) = J(g).
(3.9)
n→∞
This yields the weak* continuity. We claim that the function z is actually the
eigenfunction corresponding to g. Indeed, from
Z
Z
2
p
J(gn ) =
pgn ugn dx −
|∇ugn |p dx) ,
Ω
Z
ZΩ
p
lim
gn ugn dx =
gz p dx,
n→∞ Ω
Z
ZΩ
p
lim inf
|∇ugn | dx ≥
|∇z|p dx,
n→∞
Ω
Ω
using (3.9) and (3.2), we obtain
Z
2
Z
|∇z|p dx ≤ J(g).
J(g) ≤
pgz p dx −
Ω
Ω
R
2
By the uniqueness of the maximizer of Ω pgv p dx − Ω |∇v|p dx we must have
z = ug , as claimed.
Proof of (b). Let f, g ∈ G, let 0 < t < 1 and let v ∈ W01,p (Ω). We have
Z
Z
2
p(tf + (1 − t)g)v p dx −
|∇v|p dx
Ω
Ω
Z
Z
Z
2
Z
2
p
p
p
pgv dx −
|∇v|p dx .
=t
pf v dx −
|∇v| dx + (1 − t)
R
Ω
Ω
Ω
By taking the superior of both sides relative to v ∈
Ω
W01,p (Ω),
we get
J(tf + (1 − t)g)) ≤ tJ(f ) + (1 − t)J(g),
that is, the convexity. To prove strict convexity, suppose equality holds in the
above inequality for some t ∈ (0, 1). Then, if ut is the eigenfunction corresponding
to tf + (1 − t)g we have
hZ
Z
2 i
hZ
Z
2 i
p
p
p
t
pf ut dx −
|∇ut | dx
+ (1 − t)
pgut dx −
|∇ut |p dx
Ω
Ω
Ω
Ω
hZ
Z
2 i
hZ
Z
2 i
p
p
p
=t
pf uf dx −
|∇uf | dx
+ (1 − t)
pgug dx −
|∇ug |p dx
.
Ω
Ω
Ω
Ω
It follows that
Z
Z
2 Z
Z
2
p
p
p
pf ut dx −
|∇ut | dx =
pf uf dx −
|∇uf |p dx ,
Ω
Ω
Ω
ZΩ
Z
2 Z
Z
2
p
p
p
pf ut dx −
|∇ut | dx =
pgug dx −
|∇ug |p dx .
Ω
Ω
Ω
Ω
By the uniqueness of the maximizer, we must have ut = uf = ug and λf = λg .
Moreover, since
−∆p uf = λf f up−1
,
f
a.e. inΩ,
−∆p ug = λg gup−1
,
g
a.e. inΩ,
if uf = ug and λf = λg , we must have f (x) = g(x) a.e. in Ω, and the strict
convexity is proved.
EJDE-2010/02
PROBLEMS INVOLVING THE P-LAPLACIAN
9
Proof of (c). Let tn > 0 be a sequence such that tn → 0 as n → ∞. Let f, g ∈ R,
and let gn = g + tn (f − g). Then, by (3.5) we find
Z
Z
J(g) + tn (f − g)pupg dx ≤ J g + tn (f − g) ≤ J(g) + tn (f − g)pupgn dx,
Ω
Ω
Z
Z
J(g + tn (f − g)) − J(g)
p
≤ (f − g)pupgn dx.
(f − g)pug dx ≤
t
n
Ω
Ω
As already observed, the sequence ugn converges to ug in the norm of Lp (Ω). Therefore,
Z
Z
lim
(f − g)upgn dx = (f − g)upg dx.
n→∞
Ω
Ω
Hence, since the sequence tn is arbitrary, we have
Z
J(g + t(f − g)) − J(g)
= (f − g)pupg dx.
lim
t→0
t
Ω
It follows that J(g) is Gâteaux differentiable with derivative pupg . The proof is
complete.
Theorem 3.2. Let 0 < g0 (x) ≤ H, and let G be the class of all rearrangements of
g0 . There exists a unique ĝ ∈ G such that
J(ĝ) = inf J(g).
g∈G
Furthermore, ĝ = ψ(uĝ ) for some decreasing function ψ.
Proof. By the compactness of G and the weak continuity of J(g) (proved in Lemma
3.1), we know that a minimizer ĝ exists in G. Since by Lemma 3.1 J(g) is strictly
convex, the minimizer ĝ is unique. We have to show that ĝ ∈ G.
With 0 < t < 1 and g ∈ G, let gt = ĝ + t(g − ĝ). Since J(g) is Gâteaux
differentiable at ĝ, we have
Z
J(gt ) = J(ĝ) + t (g − ĝ)pupĝ dx + o(t).
Ω
Since J(gt ) ≥ J(ĝ), we find
Z
J(ĝ) ≤ J(ĝ) + t
(g − ĝ)puĝ dx + o(t).
Ω
It follows that
Z
0≤
(g − ĝ)puĝ dx +
Ω
As t → 0 we find
o(t)
.
t
Z
0≤
(g − ĝ)uĝ dx,
Ω
and
Z
Ω
gupĝ
Z
dx ≥
Ω
ĝupĝ dx,
∀g ∈ G.
(3.10)
The function u = uĝ satisfies the equation −∆p u = λĝ ĝuĝp−1 > 0 a.e. in Ω;
therefore, upĝ cannot have flat zones in Ω. By Lemmas 1.1 and 1.2 we can find a
decreasing function ψ such that ψ(upĝ ) is a rearrangement of g0 and
Z
Z
p
guĝ dx ≥
ψ(uĝ )uĝ dx, ∀g ∈ G.
Ω
Ω
10
M. MARRAS
EJDE-2010/02
Comparing the latter inequality with inequality (3.10) and using Lemma 1.2 again,
we must have ĝ = ψ(upĝ ) ∈ G, and the statement of the theorem follows.
2
Remarks. Since J(g) = p4 λ12 , the minimization of J(g) corresponds to the
g
maximization of λg . Theorem 3.2 gives some information on the shape of the
maximizer of λg , ĝ. Indeed, since the associate eigenfunction uĝ is positive in Ω,
vanishes on the boundary ∂Ω, and ĝ = ψ(upǧ ) with ψ decreasing, ĝ has to be large
where uĝ is small, that is close to ∂Ω.
We underline that the maximization and the minimization of λg for g ∈ G in
case of p = 2 are discussed in [9]. However, the (interesting) method developed in
[9] for the investigation of the maximum of λg seems to not work in the nonlinear
case p 6= 2. Related problems are discussed in [6, 10, 11, 13, 15, 16].
References
[1] A. Anane; Simplicité et isolation de la première valeur propre du p-Laplacien avec poids. C.
R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725–728.
[2] G. Auchmuty; Dual variational principles for eigenvalue problems, in Nonlinear Analysis and
its applications (ed. F.E. Browder), Proc. Symposia in Pure Mathematics, Vol. 45, Part. 1,
AMS 1986, 55–72.
[3] G. R. Burton; Rearrangements of functions, maximization of convex functionals and vortex
rings. Math. Ann., 276 (1987), 225–253.
[4] G. R. Burton; Variational problems on classes of rearrangements and multiple configurations
for steady vortices. Ann. Inst. Henri Poincaré, 6(4) (1989), 295–319.
[5] G. R. Burton and J. B. McLeod; Maximisation and minimisation on classes of rearrangements.
Proc. Roy. Soc. Edinburgh Sect. A, 119 (3-4) (1991), 287–300.
[6] S. Chanillo, D. Grieser, M. Imai, K. Kurata, I. Ohnishi; Symmetry breaking and other
phenomena in the optimization of eigenvalues for composite membranes. Commun. Math.
Phys., 214 (2000), 315–337.
[7] F. Cuccu, B. Emamizadeh, G. Porru; Nonlinear elastic membrane involving the p-Laplacian
operator, Electronic Journal of Differential Equations. Vol 2006, no. 49 (2006), 1–10.
[8] F. Cuccu, B. Emamizadeh, G. Porru, G.; Optimization of the first eigenvalue in problems
involving the p-Laplacian, Proc. Amer. Math. Soc. 137 (2009), 1677–1687.
[9] S. J. Cox, J. R. McLaughlin; Extremal eigenvalue problems for composite membranes, I, II.
Appl. Math. Optim., 22 (1990),153–167; 169–187.
[10] F. Cuccu, G. Porcu; Existence of solutions in two optimization problems. Comp. Rend. de
l’Acad. Bulg. des Sciences, 54(9) (2001), 33–38.
[11] J. Garcı́a-Melián, J. Sabina de Lis; Maximum and comparison principles involving the pLaplacian. Journal Math. Anal. Appl., 218 (1998), 49–65.
[12] D. Gilbarg, N. S. Trudinger; Elliptic Partial Differential Equations of Second Order, Second
edition, Springer, Berlin, 1998.
[13] A. Henrot; Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics, Birkhuser Verlag, Basel, 2006.
[14] B. Kawohl, M. Lucia, S. Prashanth; Simplicity of the principal eigenvalue for indefinite
quasilinear problems. Adv. Differential Equations, 12 (2007), 407–434.
[15] J. Nycander, B. Emamizadeh; Variational problems for vortices attached to seamounts. Nonlinear Analysis, 55 (2003), 15–24.
[16] W. Pielichowski; The optimization of eigenvalue problems involving the p-Laplacian. Univ.
Iag. Acta Math., 42 (2004), 109–122.
[17] P. Tolksdorf; Regularity for a more general class of quasilinear elliptic equations. Journal of
Differential Equations, 51 (1984), 126–150.
Monica Marras
Dipartimento di Matematica e Informatica, Universitá di Cagliari, Via Ospedale 72,
09124 Cagliari, Italy
E-mail address: mmarras@unica.it