where
(i) |E ∩ H+| = |E ∩ H−|, and
(ii) there exist R > 0, and functions u, v : BR → [0, ∞), with u, v ∈ C1(BR)∩ C(BR), u = v = 0 on ∂BR, and such that
∂E∩ H+={(z, x3)∈ H+:|z| < R, x3= u(z)} and ∂E∩ H−={(z, x3)∈ H− :|z| < R, x3=−v(z)}.
Here, and throughout this section, we write BR= B((0, 0), R) for the ball of radius R inR2.
Note that the upper and lower portions of a set E ∈ E can be described by possibly different C1 graphs, and that, besides C1 smoothness, and the fact that their common domain is a metric ball, no additional assumptions are made on the functions u and v. The main result in this section characterizes the isoperimetric sets within the class E.
Theorem 8.11 (Danielli–Garofalo–Nhieu). Let V > 0, and define R > 0 so that V =|B(o, R)| (see 8.4). Then the variational problem
min
E∈E:|E|=VPH(E)
has a unique solution in E given by the bubble set B(o, R).
We now present a step-by-step sketch of the proof of Theorem 8.11.
Step 1. First, we state without proof some invariance and symmetry properties of the horizontal perimeter. Consider the mapO : H → H defined by
O(x1, x2, x3) = (x2, x1,−x3). (8.29) It is easy to see that O preserves the Lebesgue measure. Noting that the map O is an isometry of (H, d), it follows that it also preserves the horizontal perimeter:
PH(O(E)) = PH(E) for every piecewise C1 domain E ⊂ H with finite horizontal perimeter. (Note that the reflection alone (x1, x2, x3) → (x1, x2,−x3) is not an isometry of H and does not preserve the horizontal perimeter.)
Using a standard contradiction argument, we can establish the following sym-metry result for isoperimetric sets whose intersection with the hyperplane{x3= 0} is a 2-dimensional ball.
Theorem 8.12. Let E⊂ H be a bounded open set such that ∂E ∩ H+ and ∂E∩ H− are C1 hypersurfaces, and assume that E satisfies the following condition:
E∩ {x3= 0} = BR (8.30)
for some R > 0. Suppose E is an isoperimetric set satisfying|E∩H+| = |E∩H−| =
1
2|E|. Then
PH(E;H+) = PH(E;H−).
164 Chapter 8. The Isoperimetric Profile ofH
Step 2. We now consider a domain Ω⊂ R2and a C1 function u : Ω→ [0, ∞). We assume that E⊂ H is a C1 domain enclosed by a t-graph, i.e., for which
E∩ H+={(z, x3)∈ H : z ∈ Ω, 0 < x3< u(z)}.
For z = (x1, x2)∈ R2, we set z⊥= (x2,−x1). Indicating by φ(z, x3) = x3− u(z) the defining function of E∩ H+, a simple computation gives
|∇0φ| =
Invoking the representation formula (5.2) for the horizontal perimeter yields PH(E,H+) = Next, we introduce the relevant functional class for our problem. The class of competing functions is defined as follows.
Definition 8.13. We let D denote the set of functions u ∈ Cloc1,1(BR)∩ W1,1(BR) for which there exists R > 0 so that u≥ 0 in BR,
BR=:
{BR+ρ: supp(u)⊂ BR+ρ}.
We note explicitly that, if u∈ D and R is as in Definition 8.13, then u = 0 on ∂BR. Furthermore, functions inD may have large zero sets, e.g., the graph of such a function may touch the hyperplane x3 = 0 in sets of large measure. We remark that D is not a vector space, nor is it a convex subset of V. We mention that the requirement u ∈ Cloc1,1(BR) in the definition of the class D, is justified by the following considerations. When we compute the Euler–Lagrange equation of the functional (8.32) we need to know that, with Ω = BR, the singular set Su = {z = (x1, x2) ∈ Ω ⊂ R2 : |∇zu(z) + z2⊥| = 0}, which is the projection of the characteristic set of the graph of u (see (4.27)), has vanishing 2-dimensional Lebesgue measure. This is guaranteed by Theorem 4.48.
Step 3. Following classical ideas from the calculus of variations, we next introduce the admissible variations for the problem at hand, see [122] and [248].
Definition 8.14. Given u ∈ D, we say that φ ∈ V, with supp φ ⊆ supp u, is
8.4. Minimizers with symmetries 165
With (8.32) in mind, we define
J [u] =
supp(u)
|∇zu|2+1
4|z|2+∇zu, z⊥ dx1dx2 (8.34) for such u. Within the class of C1 graphs over R2, the isoperimetric problem consists in minimizing the functional J [u] subject to the constraint G[u] = V , where V > 0 is given and BR is replaced by an a priori unknown domain Ω. We emphasize that finding the section of the isoperimetric profile with the hyperplane {x3= 0}, i.e., finding the domain Ω, constitutes here part of the problem. Because of the lack of an obvious symmetrization procedure, this seems a difficult question.
To avoid this obstacle, we restrict the class of domains by requiring that their section with the hyperplane{x3= 0} be a ball, i.e., we assume that, given E ∈ E, there exists R = R(E) > 0 such that Ω = BR. Under this hypothesis, one can appeal to Theorem 8.12. The latter implies that it suffices to solve the following variational problem: given V > 0, find R0 > 0 and uo∈ D with supp(uo) = BR0 for which the following holds:
J [uo] = min
u∈DJ [u] and G[uo] = V
2. (8.35)
Step 4. Next, we reduce (8.35) to an unconstrained problem using an application of the following standard version of the Lagrange multiplier theorem (see, e.g., Proposition 2.3 in [248]).
Proposition 8.15. LetD be a subset of a normed vector space V, and consider func-tionals F, G1,G2,. . . ,Gk defined onD. Suppose there exist constants λ1, . . . , λk ∈ R, and uo∈ D, such that uominimizes
F + λ1G1+ λ2G2+· · · + λkGk (8.36) (uniquely) on D. Then uo minimizesF (uniquely) when restricted to the set
{u ∈ D : Gj[u] =Gj[uo], j = 1, . . . , k}.
The procedure of applying the above proposition when solving a problem of the type
min
u∈DF [u]
subject to the constraintsG1[u] = V1, . . . ,Gk[u] = Vk, consists of two steps. First, one shows that constants λ1, . . . , λk and a function uo ∈ D can be found so that uo solves the Euler–Lagrange equation of (8.36), and uo satisfies G1[uo] = V1, . . . ,Gk[uo] = Vk. Next, one proves that the solution uo of the Euler–Lagrange equation is indeed a minimizer of (8.36). If the functional involved possesses ap-propriate convexity properties, then one can show in addition that such minimizer uo is unique.
166 Chapter 8. The Isoperimetric Profile ofH
The constrained variational problem (8.35) is thus equivalent to the follow-ing one without constraint (provided the parameter λ is appropriately chosen):
minimize the functional
over the setD introduced in Definition 8.13. It is easily recognized that the Euler–
Lagrange equation of (8.37) is
divz
Step 5. As we pointed out above, solving (8.38) on an arbitrary domain of Ω⊂ R2 is a difficult task. However, when Ω is a ball in R2, the equation (8.38) admits a familiar class of spherically symmetric solutions. We note explicitly that for a graph x3= u(z) with spherical symmetry in z, the only characteristic points can occur at the intersection of the graph with the x3-axis.
Theorem 8.16. Given R > 0, for every λ∈ [−2/R, 0), equation (8.38) with Dirich-let condition u = 0 on ∂BR, admits the cylindrically symmetric solution uR,λ∈ D, where
Regarding the regularity of the functions uR,λ, it suffices to consider the upper half of the “normalized” candidate isoperimetric profile Eo⊂ H, where ∂Eo
is the graph of the function x3= uo(z), with uo= u1,−2. The characteristic locus of Eois{(0, 0, ±π8)}. Unlike its Euclidean counterpart, the hypersurface So= ∂Eo is not C∞at the characteristic points (0, 0,±π8). In fact, it is C2, but not C3, near its characteristic locus Σ. However, Sois C∞(in fact, real-analytic) away from Σ.
One immediately obtains the following consequence.
Corollary 8.17. Let V > 0 be given, and define R > 0 so that V =|B(o, R)|. Let
8.4. Minimizers with symmetries 167
Furthermore,
Ω
uR(z) dx1dx2=1
2V. (8.42)
At this point, recalling that (8.38) is the Euler–Lagrange equation of the unconstrained functional (8.37), we deduce the following result.
Theorem 8.18. Let J and G be as in (8.34) and (8.33) respectively. Given V > 0 there exists R = R(V ) > 0 so that the function uo = uR in (8.41) is a critical point onD of the functional J[u] subject to the constraint G[u] = V/2.
Step 6. Our next objective is to prove that the function uo in (8.41) is, first, a global minimizer of the variational problem (8.35), and second, the unique global minimizer. We will need some basic facts from the calculus of variations, which we now recall.
Definition 8.19. Let V be a normed vector space, and D ⊂ V. Given a functional F : D → R, u ∈ D, and if φ is D-admissible at u, one calls
δF(u; φ) def= lim
→0
F[u + φ] − F[u]
the Gˆateaux derivative ofF at u in the direction φ if the limit exists.
Definition 8.20. LetV be a normed vector space, and D ⊂ V. Consider a functional F : D → ¯R. F is said to be convex over D if for every u ∈ D, and every φ ∈ V such that φ isD-admissible at u, and u + φ ∈ D, one has
F[u + φ] − F[u] ≥ δF(u; φ),
whenever the right-hand side is defined. We say that F is strictly convex if strict inequality holds in the above inequality except when φ≡ 0.
We then have the following result.
Theorem 8.21. Suppose F is convex and proper over a nonempty convex subset D∗⊂ V (i.e., F ≡ ∞ over D∗), and suppose that uo∈ D∗is such that δF(uo; φ) = 0 for all φ which are D∗-admissible at uo (that is, uo is a critical point of the functional F), then F has a global minimum in uo. If moreover F is strictly convex at uo, then uo is the unique element in D∗ satisfying
F[uo] = inf?
F[v] : v ∈ D∗@ .
Our next goal is to adapt the above results to (8.35). Given V > 0 consider the number R = R(V ) > 0 defined in Corollary 8.17, the corresponding fixed ball BR, and the normed vector spaceV(R) = {u ∈ C(BR) : u = 0 on ∂BR}. Let D(R) be the collection of functions u∈ V(R) with u ≥ 0, u ∈ C2(BR)∩ W1,1(BR), and
BR=:
{BR+ρ: supp(u)⊂ BR+ρ}}.
168 Chapter 8. The Isoperimetric Profile ofH
We note that D(R) is a nonempty convex subset of V(R), and that u = 0 on
∂BR for every u∈ D(R). Consider the functional (8.37). Given u ∈ D(R) and φ which isD(R)-admissible at u, in view of Theorem 4.48, we see that J is Gˆateaux differentiable at u in the direction of φ, and
δJ (u; φ) =
B((0,0),R)
hu(z, u(z),∇u(z)) φ(z) (8.43)
+ ∇ph(z, u(z),∇u(z)), ∇φ(z)
dx1dx2
=
BR
∇zu + z⊥/2,∇zφ
|∇zu + z⊥/2| + λ φ
dx1dx2.
At this point, using an algebraic agument, it is possible to show the existence of a global minimizer ofJ . Such global minimizer is indeed provided by the spherically symmetric function uR in (8.41).
Proposition 8.22. Given V > 0, let R = R(V ) > 0 be as in Corollary 8.17. The functional J in (8.37) is convex on D(R). As a consequence, the function uR in (8.41) is a global minimizer of J on D(R).
Finally, the proof of Theorem 8.11 is complete once one shows that uois the unique minimizer of the variational problem (8.35). This, in turn, follows from the fact that for every function φ, not identically zero, which is D(R)-admissible at uR, the strict inequality
J [uR+ φ] > J [uR] holds.