Extremal functions for Morrey's inequality in convex domains

For a bounded domain $\Omega\subset \mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\|u\|^p_{\infty}\le \int_\Omega|Du|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(\Omega)$. We show that the ratio of any two extremal functions is constant provided that $\Omega$ is convex. We also explain why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this property. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the $p$-Laplacian.


Introduction
Suppose Ω ⊂ R n is a bounded domain and p > n. Morrey's inequality for W 1,p 0 (Ω) functions u may be expressed as where c > 0 is a constant that is independent of u. In particular, Let us define Observe that (1.2) and that c = λ p is the largest constant such that (1.1) is valid. Furthermore, if there is a function u ∈ W 1,p 0 (Ω) \ {0} such that equality holds in (1.1), then c = λ p .
It is plain to see that any multiple of an extremal is also an extremal. Using routine compactness arguments, it is not difficult to verify that extremal functions exist. We will argue below that any extremal u satisfies the boundary value problem which was derived by Ercole and Pereira in [7]. Here ∆ p ψ := div(|Dψ| p−2 Dψ) is the p-Laplacian, and x 0 is the unique point for which |u| is maximized in Ω. Moreover, using (1.3) we will be able to conclude that any extremal has a definite sign in Ω. And as the PDE in (1.3) is homogeneous, the optimal constant λ p can be interpreted as being an eigenvalue. The primary goal of this work is to address the extent to which extremal functions can be different. In particular, we would like to know if any two extremal functions are necessarily multiples of one another. If they are, we consider the set of extremals to be uniquely determined. For once one extremal is found, all others can be obtained by scaling. We will argue that annuli never have this uniqueness property. We will also exhibit starshaped domains for which this uniqueness property fails. However, we will see that if a planar domain has enough symmetry, then its extremals are one dimensional.
Our main result is that convex domains always have the aforementioned uniqueness property.
Theorem 1.1. Assume that Ω ⊂ R n is open, convex and bounded. If u and v are extremal, then u/v is constant throughout Ω.
We will also explain how Theorem 1.1 implies the following corollary involving the Green's function of the p-Laplacian in Ω. Corollary 1.2. Assume that Ω ⊂ R n is open, convex and bounded. Suppose that G(·, y) is the Green's function of the p-Laplacian in Ω with pole y ∈ Ω; that is, G(·, y) satisfies −∆ p w = δ y x ∈ Ω w = 0 x ∈ ∂Ω. Part of our motivation was to extend a previous result of Talenti. He considered extremal functions for the following inequality, which is also due to Morrey. For each weakly differentiable function u : R n → R, Here C(n, p) is an explicit constant depending only on p and n, and |supp(u)| is the Lebesgue measure of the support of u. Employing Schwarz symmetrization, Talenti showed in [21] that if equality holds in (1.5) there are a ∈ R, r > 0 and x 0 ∈ R n such that We also would like to mention the recent work of Cianchi [4], in which a quantitative version of Talenti's result was established. Noteworthy are also the papers [9] and [11], where uniqueness and nonuniqueness for eigenvalue problems of the type Unfortunately, R n and balls are the only known domains for which the extremals have such convenient characterizations. Nevertheless, in this paper, we believe that we have taken significant steps in understanding precisely which domains have a one dimensional collection of extremals. In Section 2, we will derive basic properties of extremal functions, and in Section 3, we consider the support function of an extremal. In Section 4, we will provide examples of domains for which uniqueness fails; these include annuli, bow tie and dumbbell shaped planar domains. In Section 5, we verify Theorem 1.1, and in Section 6, we use Steiner symmetrization to exhibit some nonconvex planar domains that have unique extremals.

Properties of extremals
We now proceed to deriving some properties of extremal functions. These properties will be crucial to our uniqueness study. First, we verify that extremal functions satisfy the boundary value problem (1.3). Then we will study the behavior of extremal functions on convex domains near their global maximum or minimum points and consider their support functions. We also refer the reader the recent paper [7] by Ercole and Pereira, where they studied properties of extremal functions in a wide class of inequalities that include (1.1).
Proof. 1. First let us establish the following identity for u, φ ∈ C(Ω). For any x 0 such that |u(x 0 )| = u ∞ , Therefore, and so " ≥ " holds in (2.2). Now choose a sequence of positive numbers (ǫ j ) j∈N tending to 0 such that lim sup and select a sequence (x j ) j∈N maximizing |u + ǫ j φ| that converges to a maximizer x 0 of |u|. Such sequences exist by the continuity of u and φ, the compactness of Ω, and the inequalities Therefore, by the convexity of the function |x| p lim sup We conclude (2.2). 2. Any extremal u ∈ W 1,p 0 (Ω) \ {0} satisfies for each φ ∈ W 1,p 0 (Ω) and ǫ > 0 sufficiently small. Exploiting (2.2) Canceling the factor 1/ u p ∞ and replacing φ with −φ gives (2.1). 3. Of course if (2.1) holds, we can choose φ = u to verify that u is extremal.
Corollary 2.2. Each extremal function is everywhere positive or everywhere negative in Ω.
Observe that the left hand side of (2.1) is linear in φ, while the right hand side appears to be nonlinear in φ. We will argue that this forces the set {x ∈ Ω : |u(x)| = u ∞ } to be a singleton for any extremal function. Proof. Without any loss of generality, we may assume u > 0 and u ∞ = 1. By (2.1), for Suppose that there are two distinct points x 1 and x 2 for which u(x 1 ) = u(x 2 ) = 1. In this case, there are balls B δ (x 1 ), B δ (x 2 ) ⊂ Ω that are disjoint for some δ > 0 small enough. We choose functions φ 1 ∈ C ∞ c (Ω), φ 2 ∈ C ∞ c (Ω) that are nonnegative, have maximum value 1, and are supported in B δ (x 1 ) and B δ (x 2 ). It follows that Corollary 2.4. Assume u is an extremal function. Then |u| attains its maximum value uniquely at some x 0 ∈ Ω. Moreover, for each φ ∈ W 1,p 0 (Ω). In particular, u is a weak solution of (1.3).
It now follows that any solution u of (1.3) is differentiable with a locally Hölder continuous gradient in Ω \ {x 0 }, see [8,15,22]. However, we show below that u is not differentiable at x 0 . Example 2.5. As we noted above, when Ω = B r (x 0 ), we have an explicit extremal function for each a ∈ R. Moreover, any extremal is of the form (2.4) for some a ∈ R; in particular, any ball has a one dimensional collection of extremal functions. The corresponding optimal constant in (1.2) is We can use the extremals for balls (2.4) to study the behavior of general extremals near the points which maximize their absolute values. Note in particular, that the family of extremals (2.4) are Hölder continuous with exponent p−n p−1 ∈ (0, 1], which is a slight improvement of the exponent p−n p one has from the Sobolev embedding W 1,p 0 (Ω) ⊂ C 1− n p (Ω). We will first argue that each extremal has exactly this type of continuity at their maximizing or minimizing points.
Proposition 2.6. Assume u is an extremal function and that |u| achieves its maximum Proof. Assume that u > 0 and u(x 0 ) = 1, and define v(x) : = 1 Observe that u and w are p-harmonic in Ω \ {x 0 }, u(x 0 ) = w(x 0 ) and u ≤ w on ∂Ω. By weak comparison, u ≤ w in Ω. That is Corollary 2.7. Suppose that u is an extremal function such that |u| attains its maximum value at x 0 ∈ Ω. Then u is not differentiable at x 0 .
We will now refine the above estimates to deduce the exact behavior of an extremal u and its derivative near the point at which |u| achieves its maximum. The following proposition relies on the results of Kichenassamy and Veron in [12].
Proposition 2.8. Assume that u is an extremal function for which |u| attains its maximum value at x 0 ∈ Ω. Then Here ω n is the Lebesgue measure of B 1 (0) ⊂ R n .
Proof. Without any loss of generality, we may assume that u is positive in Ω and that u(x 0 ) = 1. Recall that u is p-harmonic in Ω \ {x 0 }, and in view of the previous lemma, u in Ω for some constant C. By Theorem 1.1 and Remark 1.6 in [12], there is γ > 0 such that and lim x→x 0 We may integrate by parts and exploit (2.7) to get In view of (2.5), we actually have Now we can apply (2.6). This limit gives weakly is called a potential function. Observe that every extremal is a multiple of a potential function but not vice versa. The strong maximum principle for p-harmonic functions implies that 0 < w < 1 in Ω \ {x 0 }. In particular, w is uniquely maximized at x 0 . Using similar arguments as in the proof of Proposition 2.8, one can easily show that Moreover, we can repeat the proof of Proposition 2.8 to find (2.10) These facts will be used to verify Theorem 1.1.

Support function of an extremal
Suppose now that Ω is convex and that u ∈ W 1,p 0 (Ω) is a positive extremal which achieves its maximum at x 0 ∈ Ω. The results of Lewis in [14] imply that {u > t} is convex for each t ∈ R, and {u = t} has positive Gaussian curvature for each t ∈ (0, u ∞ ). (3.1) By the implicit function theorem, it is also follows that the level sets of u are smooth.
We define the support function of u as is the usual support function of the convex set {u ≥ t}; if u(0) ≥ t and |ξ| = 1, h(ξ, t) represents the distance from the origin to the hyperplane that supports {u ≥ t} with outward normal ξ. It follows from (3.1) and Theorem 4 of [6], See [16]. In particular, since ξ is the outward unit normal to the hypersurface {u = t} at the point x, D ξ h(ξ, t) is the inverse image of the Gauss map at x. Moreover, as D 2 ξ h(ξ, t)ξ = 0, the restriction of the linear transformation D 2 ξ h(ξ, t) : R n → R n to ξ ⊥ := {z ∈ R n : z · ξ = 0} is the inverse of the second fundamental form of {u = t} at the point x (see section 2.5 of [20] for more on this point). In particular, D 2 ξ h(ξ, t)| ξ ⊥ : ξ ⊥ → ξ ⊥ is positive definite and its eigenvalues are the reciprocals of the principle curvatures of {u = t} at x.

Nonuniqueness
We now begin our study of the uniqueness of extremals for a given domain Ω. First, we first explain that uniqueness does not hold in general by providing explicit examples below. These instances include planar annuli, bow tie and dumbbell shaped domains. The perceptive reader will also see how to construct other examples from our remarks below.  Define Ω r 1 ,r 2 := {x ∈ R n : r 1 < |x| < r 2 } for r 1 , r 2 > 0 with r 1 < r 2 . As mentioned above, there is a positive extremal u that achieves is maximum at a single point x 0 ∈ Ω r 1 ,r 2 . Notice that for any n × n orthogonal matrix O, v := u • O is a positive extremal and v ∞ = u ∞ . Consequently, for each y 0 ∈ Ω r 1 ,r 2 with |y 0 | = |x 0 |, there is a distinct positive extremal with supremum norm equal to u ∞ . Thus, uniqueness of extremals does not hold for annuli as showed in Figure 2.
for ǫ > 0. Note, in particular, that Ω ǫ is star-shaped with respect to the origin; see Figure  3. Let u ǫ be a positive extremal for Ω ǫ with u ǫ ∞ = 1. If u ǫ is unique, then it must be that u ǫ (0, 0) = 1. This is due to the fact that the Ω ǫ and the p-Laplacian are invariant with respect to reflection about the x 1 and x 2 axes. Let us assume (4.1) holds for each ǫ > 0 and extend u ǫ to be 0 outside of Ω ǫ . Notice that the resulting function, which we also denote as u ǫ , belongs to W 1,p (R 2 ). Also note that since Ω 0 ⊂ Ω ǫ Consequently, there is a decreasing sequence of positive numbers (ǫ j ) j∈N tending to 0 and a continuous function u 0 : R 2 → [0, 1] for which u ǫ j → u 0 locally uniformly on R 2 . In view of (4.1), u 0 (0, 0) = 1.
On the other hand, u ǫ (0, 2ǫ) = 0 for all ǫ > 0. Thus which is a contradiction. As a result, we conclude that there is some ǫ > 0 such that u ǫ does not achieve its maximum value at (0, 0). For this value of ǫ, Ω ǫ will have a least two positive extremals with supremum norm equal to 1.

Convex domains
Throughout this section, we will assume that Ω ⊂ R n is a bounded convex domain. We will also suppose that u 0 , u 1 ∈ W 1,p 0 (Ω) are positive extremal functions that satisfy for some x 0 , x 1 ∈ Ω. We aim to show that It is easy to see that Theorem 1.1 follows from (5.1). For each ρ ∈ (0, 1), we define the Minkowski combination of u 0 and u 1 u ρ (z) := sup min{u 0 (x), u 1 (y)} : z = (1 − ρ)x + ρy, x, y ∈ Ω z ∈ Ω. We recall that u 0 and u 1 are quasiconcave. Using the definition above, it is straightforward to verify that for each t ∈ R Here the addition is the usual Minkowski addition of convex sets. In particular, u ρ itself is quasiconcave. The Minkowski combination was introduced in work of Borell in [1] when he studied capacitary functions; although his work was motivated by the previous papers of Lewis [14] and Gabriel [10]. We also were particularly inspired to utilize the Minkowski combination after we became aware of the work of Colesanti and Salani in [6], who verified a Brunn-Minkowski inequality for p-capacitary functions (1 < p < n), and the work of Cardaliaguet and Tahraoui in [3] on the strict concavity of the harmonic radius.
Along the way to proving (5.1), we will need some other useful properties of u ρ .
Proposition 5.1. Define Then the following hold: We omit the proof of the above proposition. However, we remark that (i) and (ii) are elementary; Theorem 4 of [6] and Theorem 1 of [14] together imply (iii); and (iv) follows from Section 2 of [3] or Section 7 of [16]. Using these properties we will verify that u ρ itself is an extremal for each ρ ∈ (0, 1).
Proof. We first show that u ρ is p-subharmonic and integrate by parts to derive an upper bound on the integral Ω |Du ρ | p dz. Then we show that u ρ satisfies the limits in Proposition 2.8 (that are also satisfied by every extremal function). Finally, we combine the upper bound and the limits to arrive at the desired conclusion.
1. Let z ∈ Ω \ {x ρ }, and select x ∈ Ω \ {x 0 } and y ∈ Ω \ {x 1 } such that Recall that such x, y exist by Proposition 5.1. We have Note that min{1, p−1} > 0 is a lower bound on the eigenvalues of the matrix I n +(p−2)e⊗e. Therefore,

The divergence theorem gives
On the other hand, since u ρ is a positive p-subharmonic function in Ω \ {x ρ } As u ρ ≤ 1, 3. Let w ρ be a solution of the PDE (2.9) with x ρ replacing x 0 . As u ρ is p-subharmonic, u ρ (x ρ ) = 1 and u ρ | ∂Ω = 0, weak comparison implies u ρ ≤ w ρ . This is a version of Borell's inequality , see [1,3]. In particular, Setting λ := Ω |Dw ρ | p dz, we have from Proposition 2.8, (2.10) and (5.4) that It follows that λ = λ p . In view of (5.4), and since the sequence z k was arbitrary, (5.5) 4. Again let let z k → x ρ with z k = x ρ for all k ∈ N sufficiently large. By Proposition 5.1, Since u ρ (z k ) → 1, (5.6) implies that x k → x 0 and y k → x 1 as u 0 and u 1 are uniquely maximized as these points, respectively. Combining this fact with Proposition 2.8, (5.5) and again with (5.6) also gives We can now employ the gradient limit in Proposition 2.8 and (5.8) to obtain And since z k was arbitrary, (5.9) 5. Using the upper bound (5.3) and the limits (5.5) and (5.9), we can proceed with the same arguments as in the proof of Proposition 2.8 to conclude In order to verify (5.1), we will employ the respective support functions h 0 , h 1 , and h ρ of u 0 , u 1 and u ρ ; recall the support function of an extremal was defined in (3.2). In particular, we note that the identity (5.2) implies Using this identity with the fact that h 0 , h 1 and h ρ all satisfy equation (3.3), Colesanti and Salani showed for each t ∈ (0, 1) there is C(t) > 0 such that and for all |ξ| = 1 (see the proof of Theorem 1 in [6]). As D 2 ξ h 0 (ξ, t)ξ = D 2 ξ h 0 (ξ, t)ξ = 0 ∈ R n , it follows from (5.10) and the homogeneity of h 0 and h 1 that for some a(t) ∈ R n and b(t) ∈ R. Since h 0 and h 1 are homogeneous of degree one, it must be that b(t) = 0 for each t ∈ (0, 1). Thus, Taking the time derivative of both sides of this equation gives Comparing with (5.11) leads us to Suppose that C ′ (t 0 ) = 0 for some t 0 ∈ (0, 1). Then This would imply the level set {u 1 = t 0 } is the singleton {−a ′ (t 0 )/C ′ (t 0 )}, which is not possible. Therefore, C ′ (t) = 0 for t ∈ (0, 1) and thus a ′ (t) · ξ = 0 for all ξ. Consequently, a ′ (t) = 0 for t ∈ (0, 1). Since h 0 and h 1 coincide at t = 0, C(t) = 1 and a(t) = 0 for all t ∈ [0, 1]. As a result, h 0 ≡ h 1 and so {u 0 ≥ t} = {u 1 ≥ t} for each t ∈ [0, 1] (Theorem 8.24 in [19]). This verifies (5.1).
Therefore, G(y, y) ≤ λ 6 Steiner symmetric domains Theorem 1.1 implies that if a convex domain has some reflectional symmetry, then we have additional information on the location of the maximum points of positive extremals. More precisely, we can make the following observation. Corollary 6.1. Assume Ω ⊂ R n is a convex domain that is invariant with respect to reflection across the hyperplanes {x ∈ R n : x j = 0} for j = 1, . . . , n. Then any positive (negative) extremal achieves its maximum (minimum) value at 0 ∈ R n .
We now seek to extend this observation. We will show below that certain symmetric two dimensional domains have uniqueness of extremals without assuming the domains were convex to begin with. To this end, we employ Steiner symmetrization. In particular, we will make use of the results by Cianchi and Fusco in [5] on the equality condition in the Pólya-Szegö inequality associated with Steiner symmetrization. We also use special properties of the critical points of p-harmonic functions in two dimensions due to Manfredi in [17].
Let us first briefly recall the notion of the Steiner symmetrization of a subset of R 2 . For a given A ⊂ R 2 and a ∈ R, we will denote A ∩ {x 1 = a} as the intersection of A with the vertical line x 1 = a. We also will write L m for the outer Lebesgue measure defined on all subsets of R m (m = 1, 2).
The Steiner symmetrization of A with respect to the x 1 axis is A is said to be Steiner symmetric with respect to the x 1 axis if A * 1 = A. Now suppose u : R 2 → [0, ∞) is Lebesgue measurable. We can use the above definition to provide the following rearrangement of u This function is called the Steiner rearrangement of u with respect to the x 1 axis. Observe that for each t ≥ 0. Note that u * 1 (x 1 , ·) and u(x 1 , ·) have the same distribution for L 1 almost every x 1 ∈ R.
It is known that if p ∈ [1, ∞), Ω ⊂ R 2 is a bounded domain and u ∈ W 1,p 0 (Ω), then u * 1 ∈ W 1,p 0 (Ω). Moreover, if Ω * 1 is Lebesgue measurable, the Pólya-Szegö inequality holds, see [2,5]. Cianchi and Fusco showed that if Ω * 1 = Ω and equality holds in (6.2), then u * 1 = u provided L 2 ({u x 2 = 0}) = 0 (6.3) (Theorem 2.2 in [5]). All of the above definitions and facts regarding Steiner symmetrization and rearrangements with respect to the x 1 axis have obvious counterparts with respect to the x 2 axis. Our main assertion regarding the uniqueness of extremals on Steiner symmetric domains is as follows. Proposition 6.3. Assume Ω ⊂ R 2 is a bounded domain that is equal to its Steiner symmetrization about the x 1 and x 2 axes. Then any positive (negative) extremal achieves its maximum (minimum) value at 0 ∈ R 2 .
Let us now show that any positive extremal u ∈ W 1,p 0 (Ω) satisfies (6.3). Recall that u is p-harmonic in Ω \ {z} and therefore, u ∈ C 1 loc (Ω \ {z}). By the results of Manfredi in [17], we know the zeros of Du are isolated in Ω \ {z}. Consequently, u is locally real analytic in S := Ω \ ({z} ∪ {|Du| = 0}), which is an open set of full measure. In particular, u x 2 is also locally real analytic in S. Therefore, if L 2 ({x ∈ S : u x 2 (x) = 0}) > 0, then it must be that u x 2 ≡ 0 in S; see section 3.1 of [13]. Since u x 2 is continuous in Ω \ {z}, it would then follow that u x 2 ≡ 0 in Ω \ {z}, as well. However, this is clearly not possible as the function [0, ∞) ∋ t → u(z + te 2 ) is positive at t = 0 and vanishes for all t > 0 sufficiently large. As a result, (6.3) holds and the assertion follows.