In this section, we are going to prove Theorem 1.3, that is, to identify the equality cases in Theorems1.1and1.2. As we already pointed out after the statement of Theorem1.2, we may extend (WSI) fromC0∞(Rn) to ˙Wσ1,p(Rn), that is larger space in order to search for a suitable candidate as an extremal function. To do this extension, a careful approximation argument is needed which is similar to the one carried out in [11, Lemma 7] for the unweighted case, and that was adapted to equal monomial weights in [25]. In fact, the idea to do this is to extend the integration by parts formula (2.7) to functions uin ˙Wσ1,p(Rn), a technical issue discussed in detail in [11, 25]. Since the same technique can be adapted also to our setting, we thus omit the details.
In order to prove Theorem 1.3, we shall need some preliminary results. First, we have the following characterization of concavity.
Lemma 3.1. LetE ⊆Rn be an open convex set and h:E→R be a continuous function which is a.e. differentiable in E. Then the following statements are equivalent.
(a) his concave inE.
(b) For a.e. x∈E and ally∈E, one hash(y)−h(x)∇h(x)·(y−x).
Proof. Although standard, we provide the proof since we did not find it in the literature. The implication ‘(a)⇒(b)’ is trivial. For ‘(b)⇒(a)’, letE0⊂E be the set wherehis differentiable;
clearly,Ln(E\E0) = 0. Letx0, y0∈E, 0< t <1, andz0= (1−t)x0+ty0. Ifz0∈E0, then by our assumption, we have thath(x0)−h(z0)∇h(z0)·(x0−z0) andh(y0)−h(z0)∇h(z0)· (y0−z0). Multiplying the first inequality by (1−t), the second byt, and adding them up yields (1−t)h(x0) +t h(y0)−h(z0)0. On the other hand, ifz0∈/ E0, pick a sequencezk∈E0such that zk →z0. Since E is open, we can pick sequences xk, yk∈E such that xk →x0, yk → y0, with zk = (1−t)xk+tyk. In particular, we have that h(xk)−h(zk)∇h(zk)·(xk−zk) and h(yk)−h(zk)∇h(zk)·(yk−zk). Multiplying the latter inequality byt and the former
by (1−t) yields (1−t)h(xk) +t h(yk)−h(zk)0. Since h is continuous, lettingk→ ∞ we
obtain the concavity ofh.
We are ready to prove a rigidity result based on the validity of condition C-0.
Proposition 3.1. Let E⊆Rn be an open convex cone and weights ω, σ:E→(0,∞) satisfying relation (1.1) withα >0, τ ∈R, continuous inE, differentiable a.e. in E. Assume in addition that at least one of the weights ω orσ is locally Lipschitz in E. Ifna <+∞, we have:
(i) if condition C-0 holds withC0>0 andτα, thenC0 na1−n; (ii) the following statements are equivalent.
(a) Condition C-0holds forC0=n 1
a−n andτα.
(b) ω=cσ for somec >0 (thusα=τ)andσ1/α is concave inE.
Proof. (i) From Euler’s theorem for homogeneous functions,∇ω(x)·x=τ ω(x) and∇σ(x)· x=ασ(x) for all a.e.x∈E. Pickingy=x∈E in C-0 yields 1C0(pτ +αp) . Using (1.4) and (1.5), we get that na =pτ−α+p(τ+n), and
na−n=p τ+n(α−τ)
τ−α+p τ+n
p(α−τ)τ+1
p(α−τ) = τ p +α
p,
where in the last estimate we used the assumption τ α. The lower estimate for C0 then follows.
(ii) ‘(b)⇒(a)’ On one hand, by Lemma3.1, we note that the concavity ofσ1/αinE implies σ(y)1/α−σ(x)1/α∇σ1/α(x)·(y−x) for a.e. x∈E and ally∈E.
By the 1-homogeneity of σ1/α and Euler’s theorem, it turns out thatσ(x)1/α=∇σ1/α(x)·x for a.e. x∈E, thus the last inequality is equivalent to
σ(y)1/α∇σ1/α(x)·y= 1
ασ(x)1/α−1∇σ(x)·y for a.e.x∈E and ally∈E. (3.1) On the other hand, since by assumptionω=cσ(for somec >0), one hasτ=αandna =n+α.
Now using (2.1) we see that condition C-0 means σ(x)
σ(y) σ(x)
1/α
C0∇σ(x)·y for a.e.x∈E and ally∈E.
On account of (3.1), condition C-0 holds forC0=α1 = n1
a−n.
‘(a)⇒(b)’ This is the trickiest part of the proof and at the same time is the most important result to use later in the description of equality in (WSI).
Since by assumption, condition C-0 holds withC0=n 1
a−n, it follows from (2.1) that σ(y)
σ(x) τ+n
ω(x) ω(y)
α+n−pn(α−τ)+pτ1
1
na−n 1
p
∇ω(x) ω(x) +1
p
∇σ(x) σ(x)
·y for a.e.x∈E and ally∈E. (3.2) Choosing y=xin (3.2) yields
1 1
na−n τ
p +α p
. (3.3)
Let us recall from the proof of Part (i) that
na−n= p τ+n(α−τ) τ−α+p . This inserted into (3.3) yields
p τ+n(α−τ)
In this way, we obtain f(x) x∈Eand ally∈E. Furthermore, sincef is homogeneous of degree zero and differentiable a.e., one has that∇f(x)·x= 0, and thusgx(x) = 0 for a.e.x∈E. In particular, for a.e.x∈E, the
We are going to prove that f is locally Lipschitz in E; once we do that, by (3.6) we may conclude thatf is constant. To see this, leth:E→(0,∞) be the continuous, a.e. differentiable function given by
h(x) =
σ(x)α+n ω(x)α+n−p
pα1
, x∈E. (3.7)
From (3.6), it follows that ∇ω(x)ω(x) = ∇σ(x)σ(x) for a.e. x∈E. A simple computation then shows that ∇h(x) = α1h(x)∇ω(x)ω(x) for a.e.x∈E. Therefore, relation (3.4) reduces to
h(y)∇h(x)·y for a.e.x∈E and ally∈E. (3.8) Sincehis homogeneous of degree one, it follows by (3.8) that
h(y)−h(x)∇h(x)·(y−x) for a.e. x∈E and ally∈E. (3.9) Now, Lemma 3.1implies thathis concave inE, thus locally Lipschitz in E. By assumption, one of the weights is locally Lipschitz, thus the other one is locally Lipschitz too. In particular, f is also locally Lipschitz in E, and so from (3.6) we conclude the proof thatf is constant.
Henceω=cσinE for somec >0, and soh(x) =cp−α−npα σ(x)α1 for everyx∈E. Thereforeσα1
is concave inE concluding the proof.
Proof of Theorem1.3. Let us assume that equality holds in (WSI) for someu∈W˙σ1,p(Rn)\ {0}, and without loss of generality, assume thatuis nonnegative with
ˆ
E
u(x)qω(x)dx= 1.
A similar argument as in [11, Proposition 6] implies that ΔDφ is absolutely continuous on E0, where E0 denotes the interior of the set {x∈Rn:φ(x)<+∞}. We note that U∩E= supp(u)∩E⊂E0.
To prove Theorem 1.3, we discuss separately the equality cases forp >1 (see Theorem 1.1)
andp= 1 (see Theorem1.2), respectively.
3.1. Casep >1
We split the proof into several cases.
Case1: condition C-0 holds,p >1 andna<+∞.
Since u gives equality in (WSI), we must have equality in each step in the proof of Lemma2.1(i), Case 1. In particular, we have equality in the AM-GM inequality detD2Aφ(x) (ΔAnφ(x))n for μ-a.e. x∈E (recall that μ=uqω dx), thus identifying D2Aφ with DD2φ, it turns out thatD2Aφ(x) =λIn for a.e.x∈E, whereλ >0 andIn is then×n-identity matrix.
Therefore, for some x0∈Rn, one has
∇φ(x) =λx+x0 for a.e. x∈E∩E0. (3.10) Since∇φ(E)⊆E and 0∈E, we necessarily have thatx0∈E.
The equality in the second AM-GM inequality in the proof of Lemma 2.1(i) yields ω(∇φ(x))−1/qσ(∇φ(x))1/p
ω(x)−n/qnaσ(x)n/pna
na/(na−n)
=ΔAφ(x)
n ω(x)−1/qσ(x)1/p for a.e. x∈E∩E0. By rearranging the last equation, combined with ΔAφ(x) =λnfor a.e.x∈E∩E0 and (3.10), it follows that
ω(λx+x0)−1/qσ(λx+x0)1/p=λ(na−n)/naω(x)−1/qσ(x)1/p for a.e. x∈E∩E0. (3.11)
When we apply condition C-0 in the proof of Lemma2.1(i), the equality means that for a.e.
Thus, by (3.10) and (3.11), it turns out that λ=C0 By (1.1), the last relation is equivalent to
1 =C0
Finally, in the last estimate of the proof of Lemma 2.1/(i), the equality requires C0 similar reasoning as before using (3.13) together with (3.14) imply now the reverse conclusion, that is τα.
We also note thatα >0. Indeed, if we assume thatα0, we would haveτ α0 and by picking y=x∈E in C-0, it follows 1C0(pτ +αp)0; a contradiction.
Summing up, from the above arguments one concludes that condition C-0 holds with C0=
1
na−n andτ αwithα >0. But now from Proposition3.1(ii) it follows that there existsc >0 such that ω(x) =cσ(x) for every x∈E (thusα=τ andna =n+α) andσ1/α is concave in E.
Now, we are precisely in the setting of [10, Theorem A.1]. In particular, by the equality in the H¨older inequality, it follows that the extremal function satisfies|∇u(x)|pσ(x) =c0u(x)q|x+ x0|pω(x) for somec0>0 and everyx∈E. Thus,|∇u(x)|p=c0cu(x)q|x+x0|p, obtaining that the extremal function in (WSI) isu(x) :=uγ(x) = (γ+|x+x0|p)−n+α−pp ,γ >0. We note that (3.12) reduces toI0(x) = 0 for a.ex∈E, thusx0∈E verifies∇ω(x)·x0=∇σ(x)·x0= 0 for a.e. x∈E. In this way, (WSI) takes the more familiar form (with only one weight)
ˆ
where
K˜0= p(na−1) na(na−p)
´
Euγ(y)q|y|pσ(y)dy p1´
Euγ(y)qσ(y)dy1q
´
Euγ(y)q(1−na1 )σ(y)dy (3.16) is the best constant in (3.15) (not depending onγ >0).
Case2: condition C-0 holds,p >1 andna= +∞.
In order to have equality in (WSI), we must have equality in the proof of Lemma 2.1(i), Case 3. In particular, we have ΔAφ(x) = 0 for a.e.x∈E, which leads us to the degenerate case
∇φ(x) =x0 for a.e.x∈E, for somex0∈E, which is not compatible with the Monge–Amp`ere equation (2.5). Therefore, no equality can be obtained in (WSI).
Case3: condition C-1 holds andp >1.
Equality in (WSI) requires equality in each estimate in the proof of Lemma 2.1(ii), Case 1. First, as before, the equality in the AM-GM inequality detDA2φ(x)(ΔAnφ(x))n forμ-a.e.
x∈E yields
∇φ(x) =λx+x0 for a.e. x∈E∩E0 (3.17) for someλ >0 andx0∈E. The equality in the second estimate, where (2.4) is applied, together with the continuity of the weights σandω implies
ω(x)q1 =C1σ(x)1/p for allx∈E, (3.18) where C1>0 is the constant in condition C-1. Furthermore, the equality when we apply condition C-1 requires
1 p
∇ω(x) ω(x) +1
p
∇σ(x) σ(x)
·(λx+x0) = 0 for a.e. x∈E∩E0.
A similar argument as before shows that the latter relation can be transformed equivalently into
τ p +α
p+I0(x) = 0 for a.e. x∈E∩E0, (3.19) where
I0(x) =λ−1 1
p
∇ω(x) ω(x) +1
p
∇σ(x) σ(x)
·x0.
By condition C-1, it is clear that pτ +αp 0 (takingy=x) andI0(x)0 for a.e.x∈E(taking y:=yk where{yk}k⊂E converges tox0). Therefore, by (3.19), we have that pτ +αp = 0 and I0(x) = 0 for a.e.x∈E∩E0. Sincena =n, it follows that τq = αp; this relation combined with
τ
p +αp = 0 gives thatτ =α= 0.
Due to (3.18), condition C-1 implies
∇ω(x)·y0, ∇σ(x)·y0 for a.ex∈E and ally∈E. (3.20) Letx∈Ebe any differentiability point ofωand fixρ >0 small enough such thatx+Bρ⊂E.
Applying (3.20) for y:=x+z with arbitrarily z∈Bδ and using the fact that ∇ω(x)·x= 0 (sinceτ= 0), it follows that∇ω(x)·z0 for everyz∈Bδ. This holds in fact for everyz∈Rn, which implies ∇ω(x) = 0. Since ω is locally Lipschitz (thanks to our assumption an (3.18)), the latter relation impliesωis a constant,ω≡cω>0; in a similar way,σ≡cσ>0. By (3.18), one hasc
1q
ω =C1c
1p
σ. We also note thatx0 can be arbitrarily fixed inE.
A similar argument as in Case 1 shows that when we use H¨older inequality in the proof of Theorem 1.1(ii), the equality case implies that the extremal function verifies |∇u(x)|p= c1u(x)q|x+x0|p for somec1>0 and everyx∈E. The rest is the same as in (3.15) and (3.16),
where we may choose without loss of generality σ= 1; in fact, (3.15) is a Talenti-type sharp Sobolev inequality on convex cones.
3.2. Casep= 1
We now turn our attention to analyze the equality cases in Theorem 1.2. Since the proof is similar to the casep >1, we outline only the differences.
Case1: condition C-0 holds,p= 1 and na<+∞.
We follow the proof of Lemma2.1(i), Case 2. First, for someλ >0 andx0∈E, one has that
∇φ(x) =λx+x0 for a.e.x∈E∩E0. Similarly to (3.11), one necessarily has that ω(λx+x0)−1/qσ(λx+x0) =λ(na−n)/naω(x)−1/qσ(x) for a.e. x∈E∩E0. Furthermore, it follows that
λ=C0∇σ(x)
σ(x) ·(λx+x0) for a.e. x∈E∩E0, which can be written as
1 =C0(α+I0(x)) for a.e. x∈E∩E0,
where I0(x) =λ−1∇σ(x)σ(x) ·x0. Since I0(x)0 for a.e. x∈E (due to condition C-0 for p= 1), it follows that C0α1. Clearly, condition C-0 for p= 1 and y=x gives that 1C0α.
Thus C0α= 1. On the other hand, we must also have C0(1−nna) =n1
a, that is, C0=n1
a−n. Consequently, we obtain n1
a−n = α1, which is equivalent to (α−τ)(n+α−p) = 0. Due to (1.2), it follows that α=τ. We can apply again Proposition3.1(ii) to obtain the existence ofc >0 such that ω(x) =cσ(x) for every x∈E, and the σ1/α is concave in E. In this way, (WSI) reduces to
ˆ
E
|u(x)|na−1na σ(x)dx 1−na1
K˜0
ˆ
E
|∇u(x)|σ(x)dx for allu∈W˙σ1,1(Rn), (3.21) where
K˜0= 1 na
ˆ
B∩E
σ(y)dy −na1
. (3.22)
The constant ˜K0 in (3.21) is sharp. Indeed, according to [6, rel. (1.14), p. 2977], one has Pσ(B, E) = (n+α)´
B∩Eσ(x)dx. Sincena=n+α, by consideringu(x) :=1B∩E(x), it yields ˆ
E
|∇u(x)|σ(x)dx=Pσ(B, E) =na ˆ
B∩Eσ(x)dx= ˜K0−1 ˆ
B∩Eσ(x)dx 1−na1
= ˜K0−1 ˆ
E
|u(x)|na−1na σ(x)dx 1−na1
, which gives equality in (3.21).
Case2: condition C-0 holds,p= 1 and na= +∞.
We must have equality in the proof of Lemma2.1(i), Case 4. Thus we have ΔAφ(x) = 0 for a.e. x∈E, which contradicts again the Monge–Amp`ere equation (2.5). Thus, no equality can be obtained in (WSI).
Case3: condition C-1 holds andp= 1.
The discussion is similar to Case 3 with p >1, obtaining that equality in (WSI) implies that bothω andσare constant, ω≡cω>0,σ≡cσ>0, andc
1q
ω =C1cσ, where C1>0 is the constant in condition C-1. Therefore, (WSI) becomes the (usual) sharp isoperimetric inequality
on the coneE. This concludes the proof of Theorem1.3.