# Extremal functions for Morrey’s inequality in convex domains

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## Abstract

*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 show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green’s function for the

*p*-Laplacian.

## Mathematics Subject Classification

35J60 35J70 35P30 39B62## 1 Introduction

*u*may be expressed as

*u*. In particular,

### Definition

A function \(u\in W^{1,p}_0(\Omega ){\setminus }\{0\}\) is an *extremal* if equality holds in (1.2).

*u*satisfies the boundary value problem

*p*-Laplacian, and \(x_0\) is the

*unique*point for which \(\left| u\right| \) is maximized in \(\Omega \). Moreover, using (1.3) we will be able to conclude that any extremal has a definite sign in \(\Omega \). And as the PDE in (1.3) is homogeneous, the optimal constant \(\lambda _p\) 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 star-shaped domains for which this uniqueness property fails. However, we will see that if a planar domain has certain 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 \(\Omega \subset \mathbb {R}^n\) is open, convex and bounded. If *u* and *v* are extremal, then *u* / *v* is constant throughout \(\Omega \).

We will also explain how Theorem 1.1 implies the following corollary involving the Green’s function of the *p*-Laplacian in \(\Omega \).

### Corollary 1.2

*p*-Laplacian in \(\Omega \) with pole \(y\in \Omega \); that is, \(G(\cdot ,y)\) satisfies

*c*(

*n*,

*p*) is an explicit constant depending only on

*p*and

*n*, and \(|\text {supp}(u)|\) is the Lebesgue measure of the support of

*u*. Employing Schwarz symmetrization, Talenti showed in [22] that if equality holds in (1.5) there are \(a\in \mathbb {R}\), \(r>0\) and \(x_0\in \mathbb {R}^n\) such that

Unfortunately, \(\mathbb {R}^n\) 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 Sect. 2, we will derive basic properties of solutions of (1.3), and in Sect. 3, we consider the support function of an extremal. In Sect. 5, we will provide examples of domains for which uniqueness fails; these include annuli, bow tie and dumbbell shaped planar domains. In Sect. 4, we verify Theorem 1.1, and in Sect. 6, we use Steiner symmetrization to exhibit some nonconvex planar domains that have unique extremals.

## 2 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 solutions of (1.3) near their global maximum or minimum points. We also refer the reader to the recent paper [9] by Ercole and Pereira, where they studied properties of extremal functions in a wide class of inequalities that include (1.1). In particular, they obtained analogous results to Corollaries 2.2 and 2.4 below.

### Lemma 2.1

### Proof

- 1.First let us establish the following identityfor \(u,\phi \in C(\overline{\Omega })\). For any \(x_0\) such that \(\left| u(x_0)\right| =\left\| u\right\| _\infty \),$$\begin{aligned}&\lim _{\epsilon \rightarrow 0^+}\frac{\left\| u+\epsilon \phi \right\| ^p_\infty -\left\| u\right\| ^p_\infty }{\epsilon }\nonumber \\&\quad =p\max \left\{ \left| u(x)\right| ^{p-2}u(x)\phi (x): x\in \overline{\Omega }, \left| u(x)\right| =\left\| u\right\| _\infty \right\} , \end{aligned}$$(2.2)Therefore,$$\begin{aligned} \frac{1}{p}\left\| u+\epsilon \phi \right\| _\infty ^p&\ge \frac{1}{p}\left| u(x_0)+\epsilon \phi (x_0)\right| ^p\\&\ge \frac{1}{p}\left| u(x_0)\right| ^p+\epsilon \left| u(x_0)\right| ^{p-2}u(x_0)\phi (x_0)\\&= \frac{1}{p}\left\| u\right\| _\infty ^p+\epsilon \left| u(x_0)\right| ^{p-2}u(x_0)\phi (x_0). \end{aligned}$$and so “\(\ge \)” holds in (2.2). Now choose a sequence of positive numbers \((\epsilon _j)_{j\in \mathbb {N}}\) tending to 0 such that$$\begin{aligned} \liminf _{\epsilon \rightarrow 0^+}\frac{\left\| u+\epsilon \phi \right\| ^p_\infty -\left\| u\right\| ^p_\infty }{\epsilon }\ge p\left| u(x_0)\right| ^{p-2}u(x_0)\phi (x_0), \end{aligned}$$and select a sequence \((x_{j})_{j\in \mathbb {N}}\) maximizing \(|u+\epsilon _j \phi |\) that converges to a maximizer \(x_0\) of \(\left| u\right| \). Such sequences exist by the continuity of$$\begin{aligned} \limsup _{\epsilon \rightarrow 0^+}\frac{\left\| u+\epsilon \phi \right\| _\infty ^p-\left\| u\right\| _\infty ^p}{\epsilon }=\limsup _{j\rightarrow \infty }\frac{\left\| u+\epsilon _j\phi \right\| _\infty ^p-\left\| u\right\| _\infty ^p}{\epsilon _j}, \end{aligned}$$
*u*and \(\phi \), the compactness of \(\overline{\Omega }\), and the inequalities \(|u(x)+\epsilon _j \phi (x)|\le \Vert u+\epsilon _j \phi \Vert _\infty =|u(x_j)+\epsilon _j\phi (x_j)|\). As \(\mathbb {R}\ni z\mapsto \frac{1}{p}|z|^p\) is continuously differentiable,We conclude “\(\le \)” in (2.2).$$\begin{aligned} \limsup _{j\rightarrow \infty }\frac{\left\| u+\epsilon _j\phi \right\| _\infty ^p-\left\| u\right\| _\infty ^p}{\epsilon _j}&\le \limsup _{j\rightarrow \infty }\frac{\left| u(x_{j})+\epsilon _j\phi (x_{j})\right| ^p-\left| u(x_{j})\right| ^p}{\epsilon _j}\\&=p\left| u(x_0)\right| ^{p-2}u(x_0)\phi (x_0)\\&\le p\max \{\left| u(x)\right| ^{p-2}u(x)\phi (x) : \left| u(x)\right| =\left\| u\right\| _\infty \}. \end{aligned}$$ - 2.Any extremal \(u\in W^{1,p}_0(\Omega ){\setminus }\{0\}\) satisfiesfor each \(\phi \in W^{1,p}_0(\Omega )\) and \(\epsilon >0\) sufficiently small. Exploiting (2.2)$$\begin{aligned} \lambda _p=\frac{\displaystyle \int _\Omega \left| Du\right| ^pdx}{\left\| u\right\| _\infty ^p}\le \frac{\displaystyle \int _\Omega \left| Du+\epsilon D\phi \right| ^pdx}{\left\| u+\epsilon \phi \right\| _\infty ^p}, \end{aligned}$$Canceling the factor \(1/\left\| u \right\| _\infty ^p\) and replacing \(\phi \) with \(-\phi \) gives (2.1).$$\begin{aligned} 0&\le \lim _{\epsilon \rightarrow 0^+}\frac{1}{p\epsilon }\left( \frac{\displaystyle \int _\Omega \left| Du+\epsilon D\phi \right| ^pdx}{\left\| u+\epsilon \phi \right\| _\infty ^p}-\frac{\displaystyle \int _\Omega \left| Du\right| ^pdx}{\left\| u\right\| _\infty ^p}\right) \\&= \frac{\displaystyle \int _\Omega \left| Du\right| ^{p-2}Du\cdot D\phi dx}{\left\| u \right\| _\infty ^p}\\&\quad -\frac{\displaystyle \int _\Omega \left| Du\right| ^pdx}{\left\| u \right\| _\infty ^{2p}}\max \{|u(x)|^{p-2}u(x)\phi (x): x\in \overline{\Omega }, |u(x)|=\left\| u \right\| _\infty \}\\&=\frac{1}{\left\| u \right\| _\infty ^p}\left( \int _\Omega \left| Du\right| ^{p-2}Du\cdot D\phi dx\right. \\&\quad \left. -\lambda _p\max \{|u(x)|^{p-2}u(x)\phi (x): x\in \overline{\Omega }, |u(x)|=\left\| u \right\| _\infty \}\right) . \end{aligned}$$
- 3.
Of course if (2.1) holds, we can choose \(\phi =u\) to verify that

*u*is extremal.\(\square \)

### Corollary 2.2

Each extremal function is everywhere positive or everywhere negative in \(\Omega \).

### Proof

*w*is

*p*-superharmonic, \(w\ge 0\) and \(w|_{\partial \Omega }=0\). Since

*w*doesn’t vanish identically, \(w=\left| u\right| >0\) (Theorem 11.1 in [18]). Hence,

*u*doesn’t vanish in \(\Omega \) and so

*u*has a definite sign in \(\Omega \). \(\square \)

Observe that the left hand side of (2.1) is linear in \(\phi \), while the right hand side appears to be nonlinear in \(\phi \). We will argue that this forces the set \(\{x\in \overline{\Omega }: |u(x)|=\left\| u \right\| _\infty \}\) to be a singleton for any extremal function.

### Proposition 2.3

Assume *u* is an extremal function. Then \(\{x\in \Omega : |u(x)|=\left\| u \right\| _\infty \}\) is a singleton.

### Proof

### Corollary 2.4

*u*is an extremal function. Then \(\left| u\right| \) attains its maximum value uniquely at some \(x_0\in \Omega \). Moreover,

*u*is a weak solution of (1.3).

We note that any solution *u* of (1.3) is differentiable with a locally Hölder continuous gradient in \(\Omega {\setminus }\{x_0\}\), see [10, 15, 23]. However, we show below that *u* is not differentiable at \(x_0\).

### Example 2.5

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 \(\frac{p-n}{p-1}\in (0,1]\), which is a slight improvement of the exponent \(\frac{p-n}{p}\) one has from the Sobolev embedding \(W^{1,p}_0(\Omega )\subset C^{1-\frac{n}{p}}(\overline{\Omega })\). We will first argue that solutions of (1.3), and in particular extremals, have exactly this type of continuity at their maximizing or minimizing points.

### Proposition 2.6

*u*is a solution of (1.3) and that \(B_r(x_0)\subset \Omega \subset B_R(x_0)\). Then

### Proof

*u*and

*v*are

*p*-harmonic in \(B_r(x_0){\setminus }\{x_0\}\), \(u(x_0)=v(x_0)\) and \(u\ge v\) on \(\partial B_r(x_0)\). By weak comparison, \(u\ge v\) in \(B_r(x_0)\). That is

*u*and

*w*are

*p*-harmonic in \(\Omega {\setminus }\{x_0\}\), \(u(x_0)=w(x_0)\) and \(u\le w\) on \(\partial \Omega \) as \(\Omega \subset B_R(x_0)\). By weak comparison, \(u\le w\) in \(\Omega \). That is

### Corollary 2.7

Suppose that *u* is a non-zero solution of (1.3). Then *u* is not differentiable at \(x_0\).

### Proof

*R*so large that \(\Omega \subset B_R(x_0)\), we have by the previous proposition that

*u*is then of the form (2.4). \(\square \)

We will now refine the above estimates to deduce the exact behavior of a solution *u* of (1.3) near \(x_0\). The following proposition relies on the results of Kichenassamy and Veron in [12].

### Proposition 2.8

*u*is a solution of (1.3). Then

### Proof

*u*is positive in \(\Omega \) and that \(u(x_0)=1\). Recall that

*u*is

*p*-harmonic in \(\Omega {\setminus } \{x_0\}\); and in view of Proposition 2.6,

*u*satisfies \(0\le 1-u(x)\le C|x-x_0|^\frac{p-n}{p-1}\) in \(\Omega \) for some constant

*C*. This permits us to use Theorem 1.1 and Remark 1.6 in [12] to conclude that there is \(\gamma > 0\) such that

### Remark 2.9

*potential*function. Observe that every extremal is a multiple of a potential function but not vice versa. For instance, if \(\Omega =B_1(0)\), then

*w*is an extremal if and only if \(x_0=0\).

*p*-harmonic functions implies that \(0<w<1\) in \(\Omega {\setminus }\{x_0\}\). 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

*w*with \(\lambda \) replacing \(\lambda _p\).

## 3 Support function of an extremal

*u*are smooth.

*support function*of

*u*as

*x*, \(D_\xi h(\xi ,t)\) is the inverse image of the Gauss map at

*x*. Moreover, as \(D^2_\xi h(\xi ,t)\xi =0\), the restriction of the linear transformation \(D^2_\xi h(\xi ,t):\mathbb {R}^n\rightarrow \mathbb {R}^n\) to \(\xi ^\perp :=\{z\in \mathbb {R}^n: z\cdot \xi =0\}\) 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, \(D^2_\xi h(\xi ,t)|_{\xi ^\perp }:\xi ^\perp \rightarrow \xi ^\perp \) is positive definite and its eigenvalues are the reciprocals of the principle curvatures of \(\{u=t\}\) at

*x*.

*h*satisfies

## 4 Convex domains

*Minkowski combination*of \(u_0\) and \(u_1\)

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 [11]. We also were particularly inspired to utilize the Minkowski combination after we became aware of the work of Colesanti and Salani in [7], 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 (4.1), we will need some other useful properties of \(u_\rho \).

### Proposition 4.1

- (
*i*) -
\(u_\rho (x_\rho )=\left\| u_\rho \right\| _\infty =1\).

- (
*ii*) -
\(u_\rho |_{\partial \Omega }=0\).

- (
*iii*) -
\(u_\rho \in C^\infty (\Omega {\setminus }\left\{ x_\rho \right\} )\cap C(\overline{\Omega })\).

- (
*iv*) - For each \(z\in \Omega {\setminus }\left\{ x_\rho \right\} \), there are \(x\in \Omega {\setminus }\left\{ x_0\right\} \) and \(y\in \Omega {\setminus }\left\{ x_1\right\} \) such that$$\begin{aligned} z&=(1-\rho )x+\rho y,&\\ u_\rho (z)&=u_0(x)=u_1(y),&\\ \frac{Du_\rho (z)}{\left| Du_\rho (z)\right| }&=\frac{Du_0(x)}{\left| Du_0(x)\right| }=\frac{Du_1(y)}{\left| Du_1(y)\right| },&\\ \frac{1}{\left| Du_\rho (z)\right| }&=(1-\rho )\frac{1}{\left| Du_0(x)\right| }+\rho \frac{1}{\left| Du_1(y)\right| },&\\ \frac{D^2u_\rho (z)}{\left| Du_\rho (z)\right| ^3}&\ge (1-\rho )\frac{D^2u_0(x)}{\left| Du_0(x)\right| ^3}+\rho \frac{D^2u_1(y)}{|Du_1(y)|^3}.&\end{aligned}$$

We omit the proof of the above proposition. However, we remark that (*i*) and (*ii*) are elementary; Theorem 4 of [7] and Theorem 1 of [14] together imply (*iii*); and (*iv*) follows from Sect. 2, [3] or Sect. 7 of [16]. Using these properties we will verify that \(u_\rho \) itself is an extremal for each \(\rho \in (0,1)\).

### Lemma 4.2

\(u_\rho \) is extremal.

### Proof

*p*-subharmonic and integrate by parts to derive an upper bound on the integral \(\int _\Omega |Du_\rho |^pdz\). Then we show that \(u_\rho \) satisfies the limits in Proposition 2.8 (that are also satisfied by every extremal function). Finally, we combine the upper bound and limits to arrive at the desired conclusion.

- 1.Let \(z\in \Omega {\setminus }\{x_\rho \}\), and select \(x\in \Omega {\setminus }\{x_0\}\) and \(y\in \Omega {\setminus }\{x_1\}\) such thatand$$\begin{aligned} e:=\frac{Du_\rho (z)}{\left| Du_\rho (z)\right| }=\frac{Du_0(x)}{\left| Du_0(x)\right| }=\frac{Du_1(y)}{\left| Du_1(y)\right| } \end{aligned}$$Recall that such$$\begin{aligned} \frac{D^2u_\rho (z)}{\left| Du_\rho (z)\right| ^3}\ge (1-\rho )\frac{D^2u_0(x)}{\left| Du_0(x)\right| ^3}+\rho \frac{D^2u_1(y)}{\left| Du_1(y)\right| ^3}. \end{aligned}$$
*x*,*y*exist by Proposition 4.1. We haveNote that \(\min \{1,p-1\}>0\) is a lower bound on the eigenvalues of the matrix \(I_n+(p-2)e\otimes e\). Therefore,$$\begin{aligned} \left| Du_\rho (z)\right| ^{-(p+1)}\Delta _pu_\rho (z)&=\frac{\Delta u_\rho (z)}{\left| Du_\rho (z)\right| ^3}+(p-2)\frac{D^2u_\rho (z)e\cdot e}{\left| Du_\rho (z)\right| ^3}\\&=\left( I_n+(p-2)e\otimes e\right) \cdot \frac{D^2u_\rho (z)}{|Du_\rho (z)|^3}. \end{aligned}$$Consequently, \(-\Delta _p u_\rho \le 0\) in \(\Omega {\setminus }\{x_\rho \}\).$$\begin{aligned}&\left| Du_\rho (z)\right| ^{-(p+1)}\Delta _pu_\rho (z)\\&\quad \ge \left( I_n+(p-2)e\otimes e\right) \cdot \left( (1-\rho )\frac{D^2u_0(x)}{\left| Du_0(x)\right| ^3}+\rho \frac{D^2u_1(y)}{\left| Du_1(y)\right| ^3}\right) \\&\quad = (1-\rho )\left| Du_0(x)\right| ^{-(p+1)}\Delta _pu_0(x)+ \rho \left| Du_1(y)\right| ^{-(p+1)}\Delta _pu_1(y)\\&\quad = (1-\rho )\cdot 0+ \rho \cdot 0\\&\quad =0. \end{aligned}$$ - 2.The divergence theorem givesOn the other hand, since \(u_\rho \) is a positive$$\begin{aligned}&\int _{\Omega {\setminus } B_r(x_\rho )}\text {div}(u_\rho \left| Du_\rho \right| ^{p-2}Du_\rho )dz\\&\quad =\int _{\partial B_r(x_\rho )}u_\rho \left| Du_\rho \right| ^{p-2}Du_\rho \cdot \left( -\frac{z-x_\rho }{|z-x_\rho |}\right) d\sigma . \end{aligned}$$
*p*-subharmonic function in \(\Omega {\setminus }\{x_\rho \}\)As \(u_\rho \le 1\),$$\begin{aligned} \int _{\Omega {\setminus } B_r(x_\rho )}\text {div}(u_\rho \left| Du_\rho \right| ^{p-2}Du_\rho )dz&=\int _{\Omega {\setminus } B_r(x_\rho )}\left( u_\rho \Delta _p u_\rho +\left| Du_\rho \right| ^{p}\right) dz \\&\ge \int _{\Omega {\setminus } B_r(x_\rho )}\left| Du_\rho \right| ^{p}dz. \end{aligned}$$$$\begin{aligned} \int _{\Omega }\left| Du_\rho \right| ^pdz\le \liminf _{r\rightarrow 0^+}\int _{\partial B_r(x_\rho )}\left| Du_\rho \right| ^{p-1}d\sigma . \end{aligned}$$(4.3) - 3.Let \(w_\rho \) be a solution of the PDE (2.9) with \(x_\rho \) replacing \(x_0\). As \(u_\rho \) is
*p*-subharmonic, \(u_\rho (x_\rho )=1\) and \(u_\rho |_{\partial \Omega }=0\), weak comparison implies \(u_\rho \le w_\rho \). This is a version of Borell’s inequality; see [1, 3]. In particular,whenever \(z=(1-\rho )x+\rho y\). Now let \(z^k\rightarrow x_\rho \) with \(z^k\ne x_\rho \) for all \(k\in \mathbb {N}\) sufficiently large. Set$$\begin{aligned} w_\rho (z)\ge u_\rho (z)\ge \min \left\{ u_0(x),u_1(y)\right\} , \end{aligned}$$(4.4)for each \(k\in \mathbb {N}\). Observe \(z^k=(1-\rho )x^k+\rho y^k\) and$$\begin{aligned} {\left\{ \begin{array}{ll} x^k:=z^k+\left( x_0-x_\rho \right) \\ y^k:=z^k+\left( x_1-x_\rho \right) \end{array}\right. } \end{aligned}$$Setting \(\lambda := \int _\Omega |Dw_\rho |^pdz\), we have from Proposition 2.8, Remark 2.9 and (4.4) that$$\begin{aligned} \left| z^k-x_\rho \right| =\left| x^k-x_0\right| =\left| y^k-x_1\right| . \end{aligned}$$It follows that \(\lambda = \lambda _p\). In view of (4.4), and since the sequence \(z^k\) was arbitrary,$$\begin{aligned} \left( \frac{p-1}{p-n}\right) \left( \frac{\lambda }{n\omega _n}\right) ^\frac{1}{p-1}&=\lim _{k\rightarrow \infty } \frac{1-w_\rho (z^k)}{|z^k-x_\rho |^{\frac{p-n}{p-1}}}\\&\le \lim _{k\rightarrow \infty } \max \left\{ \frac{1-u_0(x^k)}{|x^k-x_0|^{\frac{p-n}{p-1}}}, \frac{1-u_1(y^k)}{|y^k-x_1|^{\frac{p-n}{p-1}}}\right\} \\&=\left( \frac{p-1}{p-n}\right) \left( \frac{\lambda _p}{n\omega _n}\right) ^\frac{1}{p-1}. \end{aligned}$$$$\begin{aligned} \lim _{z\rightarrow x_\rho }\frac{1-u_\rho (z)}{\left| z-x_\rho \right| ^{\frac{p-n}{p-1}}}=\left( \frac{p-1}{p-n}\right) \left( \frac{\lambda _p}{n\omega _n}\right) ^\frac{1}{p-1}. \end{aligned}$$(4.5) - 4.Again let \(z^k\rightarrow x_\rho \) with \(z^k\ne x_\rho \) for all \(k\in \mathbb {N}\) sufficiently large. By Proposition 4.1, there are \(x^k\in \Omega {\setminus }\{x_0\}\) and \(y^k\in \Omega {\setminus }\{x_1\}\) such that \(z^k=(1-\rho )x^k+\rho y^k\),and$$\begin{aligned} u_\rho \left( z^k\right) =u_0\left( x^k\right) =u_1\left( y^k\right) , \end{aligned}$$(4.6)Since \(u_\rho (z^k)\rightarrow 1\), (4.6) implies that \(x^k\rightarrow x_0\) and \(y^k\rightarrow x_1\) as \(u_0\) and \(u_1\) are uniquely maximized as these points, respectively. Combining this fact with Proposition 2.8, (4.5) and again with (4.6) also gives$$\begin{aligned} \frac{1}{\left| Du_\rho (z^k)\right| }=(1-\rho )\frac{1}{\left| Du_0(x^k)\right| }+\rho \frac{1}{\left| Du_1(y^k)\right| }. \end{aligned}$$(4.7)By (4.7),$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{\left| y^k-x_1\right| }{\left| z^k-x_\rho \right| }=\lim _{k\rightarrow \infty } \frac{\left| x^k-x_0\right| }{\left| z^k-x_\rho \right| }=1. \end{aligned}$$(4.8)We can now employ the second limit in Proposition 2.8 and (4.8) to obtain$$\begin{aligned} \frac{\left| z^k-x_\rho \right| ^{\frac{p-n}{p-1}-1}}{\left| Du_\rho (z^k)\right| }&=(1-\rho )\frac{ \left| z^k-x_\rho \right| ^{\frac{p-n}{p-1}-1}}{\left| Du_0(x^k)\right| }+\rho \frac{\left| z^k-x_\rho \right| ^{\frac{p-n}{p-1}-1}}{\left| Du_1(y^k)\right| }\\&=\left( \frac{\left| z^k-x_\rho \right| }{\left| x^k-x_0\right| }\right) ^{\frac{p-n}{p-1}-1}(1-\rho )\frac{ \left| x^k-x_0\right| ^{\frac{p-n}{p-1}-1}}{\left| Du_0(x^k)\right| }\\&\quad + \left( \frac{\left| z^k-x_\rho \right| }{\left| y^k-x_1\right| }\right) ^{\frac{p-n}{p-1}-1}\rho \frac{\left| y^k-x_1\right| ^{\frac{p-n}{p-1}-1}}{\left| Du_1(y^k)\right| }. \end{aligned}$$And since \(z^k\) was arbitrary,$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{\left| Du_\rho (z^k)\right| }{\left| z^k-x_\rho \right| ^{\frac{p-n}{p-1}-1}}=\left( \frac{\lambda _p}{n\omega _n}\right) ^\frac{1}{p-1}. \end{aligned}$$$$\begin{aligned} \lim _{z\rightarrow x_\rho } \frac{\left| Du_\rho (z)\right| }{\left| z-x_\rho \right| ^{\frac{p-n}{p-1}-1}}=\left( \frac{\lambda _p}{n\omega _n}\right) ^\frac{1}{p-1}. \end{aligned}$$(4.9)
- 5.Using the upper bound (4.3) and the limits (4.5) and (4.9), we can proceed with the same arguments as in the proof of Proposition 2.8 to conclude\(\square \)$$\begin{aligned} \int _{\Omega }\left| Du_\rho \right| ^pdz&\le \liminf _{r\rightarrow 0^+}\int _{\partial B_r(x_\rho )}\left| Du_\rho \right| ^{p-1}d\sigma = \lambda _p. \end{aligned}$$

### Proof of Corollary 1.2

## 5 Nonuniqueness

We will now explain that uniqueness does not hold for general domains by providing a few explicit examples. 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.

### Example 5.1

*u*that achieves is maximum at a single point \(x_0\in \Omega _{r_1,r_2}\). Notice that for any \(n\times n\) orthogonal matrix

*O*, \(v:=u\circ O\) is a positive extremal and \(\Vert v\Vert _\infty =\left\| u \right\| _\infty \). Consequently, for each \(y_0\in \Omega _{r_1,r_2}\) with \(|y_0|=|x_0|\), there is a distinct positive extremal with supremum norm equal to \(\left\| u \right\| _\infty \). Thus, uniqueness of extremals does not hold for annuli as showed in Fig. 2.

### Example 5.2

*p*-Laplacian are invariant with respect to reflection about the \(x_1\) and \(x_2\) axes.

As a result, we conclude that there is some \(\epsilon >0\) such that \(u_\epsilon \) does not achieve its maximum value at (0, 0). For this value of \(\epsilon \), \(\Omega _\epsilon \) will have a least two positive extremals with supremum norm equal to 1.

### Example 5.3

## 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 \(\Omega \subset \mathbb {R}^n\) is a convex domain that is invariant with respect to reflection across the hyperplanes \(\left\{ x\in \mathbb {R}^n: x_j=0\right\} \) for \(j=1,\dots , n\). Then any positive (negative) extremal achieves its maximum (minimum) value at \(0\in \mathbb {R}^n\).

### Proof

*z*. As \(\Omega \) is invariant with respect to \(\{x\in \mathbb {R}^n: x_1=0\}\), the function

*z*about the plane \(\{x\in \mathbb {R}^n: x_1=0\}\). By Theorem 1.1, \(u_1=u\) which forces \(z\in \{x\in \mathbb {R}^n: x_1=0\}\). Repeating this argument for \(j=2,\dots , n\), we find \(z\in \{x\in \mathbb {R}^n: x_j=0\}\) for \(j=1,\dots , n\). As a result, \(z=0\). \(\square \)

We now seek to extend this observation. We will show below that certain symmetric two dimensional domains have unique 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 [6] 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 \(\mathbb {R}^2\). For a given \(A\subset \mathbb {R}^2\) and \(a\in \mathbb {R}\), we will denote \(A\cap \{x_1=a\}\) as the intersection of *A* with the vertical line \(x_1=a\). We also will write \(\mathcal{L}^m\) for the outer Lebesgue measure defined on all subsets of \(\mathbb {R}^m\)\((m=1,2)\).

### Definition 6.2

*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 \(A^*_1=A\).

*u*

*Steiner rearrangement*of

*u*with respect to the \(x_1\) axis. Observe that

*Pólya-Szegö inequality*

Our main assertion regarding the uniqueness of extremals on Steiner symmetric domains is as follows.

### Proposition 6.3

Assume \(\Omega \subset \mathbb {R}^2\) is a bounded domain that is equal to its Steiner symmetrization with respect to the \(x_1\) and \(x_2\) axes. Then any positive (negative) extremal achieves its maximum (minimum) value at \(0\in \mathbb {R}^2\).

### Proof

*u*satisfies (6.3). Once we verify this assertion, we would have \(u=u^*_1\) which implies \(u(x_1,x_2)=u(x_1,-x_2)\) for all \((x_1,x_2)\in \Omega \). As a result

*z*belongs to the \(x_1\) axis, and very similarly we would have that

*z*also belongs to the \(x_2\) axis. Therefore, \(z=0\in \mathbb {R}^2\).

*u*is

*p*-harmonic in \(\Omega {\setminus }\{z\}\) and therefore, \(u\in C^1_{\text {loc}}(\Omega {\setminus }\{z\})\). By the results of Manfredi in [17], we know the zeros of

*Du*are isolated in \(\Omega {\setminus }\{z\}\). Consequently,

*u*is locally real analytic in \(S:=\Omega {\setminus }\left( \{z\}\cup \{\left| Du\right| =0\}\right) \), which is an open set of full measure. In particular, \(u_{x_2}\) is also locally real analytic in

*S*. Therefore, if

*S*; see section 3.1 of [13]. Since \(u_{x_2}\) is continuous in \(\Omega {\setminus }\{z\}\), it would then follow that \(u_{x_2}\equiv 0\) in \(\Omega {\setminus }\{z\}\), as well. However, this is clearly not possible as the function

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