The quasisuperminimizing constant for the minimum of two quasisuperminimizers in R^n

TL;DR

This paper investigates the quasisuperminimizing constant for the minimum of two such functions in higher dimensions, extending previous one-dimensional results and exploring cases with different constants.

Contribution

It provides higher-dimensional examples and analyzes the case where the two functions have different quasisuperminimizing constants.

Findings

Higher-dimensional examples of quasisuperminimizers are constructed.

The quasisuperminimizing constant for the minimum is shown to be close to optimal.

The case with different constants for the two functions is analyzed.

Abstract

It was shown in Bj\"orn--Bj\"orn--Korte ("Minima of quasisuperminimizers", Nonlinear Anal. 155 (2017), 264-284) that $u:=\min\{u_1,u_2\}$ is a $\overline{Q}$-quasisuperminimizer if $u_1$ and $u_2$ are $Q$-quasisuperminimizers and $\overline{Q}=2Q^2/(Q+1)$. Moreover, one-dimensional examples therein show that $\overline{Q}$ is close to optimal. In this paper we give similar examples in higher dimensions. The case when $u_1$ and $u_2$ have different quasisuperminimizing constants is considered as well.

Tables

Table 1. Table 1: Optimal quasiminimizing and quasisub/superminimizing constants of log α ⁡ | x | superscript 𝛼 𝑥 \log^{\alpha}|x| in A 1 , γ subscript 𝐴 1 𝛾 A_{1,\gamma} and ( − log ⁡ | x | ) α superscript 𝑥 𝛼 (-\log|x|)^{\alpha} in A γ , 1 subscript 𝐴 𝛾 1 A_{\gamma,1} , as provided by Theorems  2.3 and  2.5 .

	quasi-	quasi-	quasi-
	minimizer	subminimizer	superminimizer
	$Q=Q_{\alpha ,n}$	$Q=Q_{\alpha ,n}$	$Q=1$
	$Q=1$	$Q=1$	$Q=1$
	$Q=Q_{\alpha ,n}$	$Q=1$	$Q=Q_{\alpha ,n}$

Table 2. Table 2: Q ^ ^ 𝑄 \widehat{Q} for certain values of p = n 𝑝 𝑛 p=n with Q 1 = Q 2 = Q subscript 𝑄 1 subscript 𝑄 2 𝑄 Q_{1}=Q_{2}=Q , as well as ¯ ​ Q ¯ absent 𝑄 {\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}$\hss}{Q}}} from Theorem  1.1 .

$Q$	$p=n=2$	$p=n=3$	$p=n=10$	$p=n=100$
$1.001$	1.001480660	1.001480663	1.001480664	1.001480665	1.001500250
1.01	1.014825154	1.014825447	1.014825583	1.014825593	1.015024876
1.125	1.188100103	1.188143910	1.188164386	1.188165836	1.191176471
2	2.619135721	2.621145314	2.622093879	2.622161265	2.666666667
10	17.67321156	17.70495731	17.72058231	17.72170691	18.18181818
100	196.3948537	196.5222958	196.5905036	196.5955633	198.0198020

Table 3. Table 3: Q ~ ~ 𝑄 \widetilde{Q} (given by ( 3.6 )) for certain values of p 𝑝 p , Q 1 subscript 𝑄 1 Q_{1} and Q 2 subscript 𝑄 2 Q_{2} , as well as ¯ ​ Q ¯ absent 𝑄 {\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}$\hss}{Q}}} from Theorem  1.1 . When p 𝑝 p is an integer these are also the values of Q ^ ^ 𝑄 \widehat{Q} by ( 3.7 ).

$Q_1$	$Q_2$	$p=1.2$	$p=2$	$p=10$		$\overline{}Q$
2 10	10 2	10.450759 10.222890	10.474426 10.293133	10.477869 10.309651	10.477946 10.310050	10.526316
9 10	10 9	16.513457 16.473657	16.719374 16.689656	16.762792 16.736154	16.763819 16.737258	17.191011
2 100	100 2	100.427051 100.055345	100.450836 100.111528	100.454265 100.134063	100.454342 100.132692	100.502513
10 100	100 10	107.287586 106.251592	107.542028 106.758915	107.596390 106.910025	107.597679 106.913964	108.108108
90 100	100 90	185.787954 185.723660	186.446301 186.399453	186.634352 186.594523	186.639194 186.599568	188.020891

Equations

$${\int }_{\phi \ne 0}|\nabla u{|}^{p}dx\le Q{\int }_{\phi \ne 0}|\nabla (u+\phi ){|}^{p}dx$$

$${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}$\hss}{Q}}}=\begin{cases}1,&\text{if }Q_{1}=Q_{2}=1,\\ \displaystyle(Q_{1}+Q_{2}-2)\frac{Q_{1}Q_{2}}{Q_{1}Q_{2}-1},&\text{otherwise.}\end{cases}$$

$$-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)=0,$$

$${\Omega }_{\tau }:=\{x:\tau x\in \Omega \}.$$

$${\int }_{\phi \ne 0}|\nabla v{|}^{p}dx$$

$$\le Q{\tau }^{p-n}{\int }_{\tilde{\phi }\ne 0}|\nabla (u+\tilde{\phi }){|}^{p}dx=Q{\int }_{\phi }|\nabla (v+\phi ){|}^{p}dx.$$

$${\int }_G|\nabla u{|}^{p}dx=Q{\int }_G|\nabla v{|}^{p}dx<\infty ,$$

$$A_{r_1,r_2}=\{x\in {\mathbf{R}}^n:r_1<|x|<r_2\},\text{where }0\le r_1<r_2\le \infty .$$

$$Q_{\alpha ,n}=\frac{{\alpha }^n}{n\alpha -n+1}$$

$$I_u(A_{r_1,r_2}):={\int }_{\Omega }|\nabla u{|}^{n}dx=c_{n-1}{\int }_{r_1}^{r_2}|{\phi }^{\prime }(r){|}^n{r}^{n-1}dr=:c_{n-1}{\hat{I}}_{\phi }(G),$$

$${\hat{I}}_{\phi }(G)$$

$$=Q_{\alpha ,n}(s_2^{n\alpha -n+1}-s_1^{n\alpha -n+1})=Q_{\alpha ,n}{s}_1^{n\alpha -n+1}(S^{n\alpha -n+1}-1).$$

$${\hat{I}}_{\psi }(G)=s_1(S-1).$$

$$a=\frac{s_2^{\alpha }-s_1^{\alpha }}{s_2-s_1}=s_1^{\alpha -1}\frac{S^{\alpha }-1}{S-1},$$

$$k(S):=\frac{\hat{I}_{\varphi}(G)}{\hat{I}_{\eta}(G)}=\frac{\hat{I}_{\varphi}(G)}{|a|^{n}\hat{I}_{\psi}(G)}=Q_{\alpha,n}\frac{S^{n\alpha-n+1}-1}{S-1}\biggl{(}\frac{S-1}{S^{\alpha}-1}\biggr{)}^{n},$$

$${\hat{I}}_{\phi }(G)$$

$$\displaystyle=k(S)\biggl{(}\frac{\hat{I}_{\varphi}(G_{1})}{k(\sqrt{S})}+\frac{\hat{I}_{\varphi}(G_{2})}{k(\sqrt{S})}\biggr{)}=\frac{k(S)}{k(\sqrt{S})}\hat{I}_{\varphi}(G).$$

$$\underset{S>1}{\sup }k(S)=\underset{S\to \infty }{\lim }k(S)=Q_{\alpha ,n}.$$

$$V=\{x\in (1,\gamma ):\omega (x)\ne \phi (x)\}$$

$${\hat{I}}_{\phi }(V)=\sum _j{\hat{I}}_{\phi }(I_j)\le \sum _j{Q}_{\alpha ,n}{\hat{I}}_{\omega }(I_j)=Q_{\alpha ,n}{\hat{I}}_{\omega }(V).$$

$$\Omega =\{x\in A_{1,\gamma }:v(x)\ne u(x)\}.$$

$$V_{\theta }=\{r:(r,\theta )\in \Omega \}\text{and}{v}_{\theta }(r)=v(r,\theta ).$$

$$I_u(\Omega )$$

$$\displaystyle=Q_{\alpha,n}\int_{\Omega}\biggl{|}\frac{\partial v}{\partial r}\biggr{|}^{n}\,dx\leq Q_{\alpha,n}\int_{\Omega}|\nabla v|^{n}\,dx=Q_{\alpha,n}I_{v}(\Omega),$$

$${\hat{I}}_{\phi }(A_{1,\gamma })=Q_{\alpha ,n}(\log \gamma {)}^{n\alpha -n+1}$$

$$-\mathrm{div}(|\nabla u(x){|}^{n-2}\nabla u(x))=\alpha (1-\alpha )|\alpha {|}^{n-2}(n-1)(\log |x|{)}^{(n-1)\alpha -n}|x{|}^{-n}$$

$$Q_{\alpha ,n}=\frac{{\alpha }^n}{n\alpha -n+1}$$

$$A:=A_{1/e,1}=\{x\in {\mathbf{R}}^n:1/e<|x|<1\}.$$

$$Q_{\alpha ,n}=\frac{{\alpha }^n}{n\alpha -n+1}.$$

$$u_Q(x)={\log }^{\alpha }|ex|\text{and}{\bar{u}}_Q(x)=1-(-\log |x|{)}^{{\alpha }^{\prime }}.$$

$$I_u:={\int }_A|\nabla u{|}^{n}dx>Q_2{I}_v,$$

$$I_v={\int }_A|\nabla v{|}^{n}dx=c_{n-1}{\int }_{1/e}^1{r}^{-n}{r}^{n-1}dr=c_{n-1}.$$

Full text

The quasisuperminimizing constant for the minimum

of two quasisuperminimizers in $\mathbf{R}^{n}$

Anders Björn, Jana Björn and Ismail Mirumbe

  

Anders Björn

*Department of Mathematics, Linköping University,

SE-581 83 Linköping, Sweden; anders.bjorn@liu.se

Jana Björn

Department of Mathematics, Linköping University,

SE-581 83 Linköping, Sweden; jana.bjorn@liu.se

Ismail Mirumbe

Department of Mathematics, Makerere University,

P.O. Box 7062, Kampala, Uganda; mirumbe@cns.mak.ac.ug

(Preliminary version, )

**Abstract. It was shown in Björn–Björn–Korte (“Minima of quasisuperminimizers”, Nonlinear Anal. 155 (2017), 264–284) that $u:=\min\{u_{1},u_{2}\}$ is a {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}}-quasisuperminimizer if $u_{1}$ and $u_{2}$ are $Q$-quasisuperminimizers and {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}}=2Q^{2}/(Q+1). Moreover, one-dimensional examples therein show that {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}} is close to optimal. In this paper we give similar examples in higher dimensions. The case when $u_{1}$ and $u_{2}$ have different quasisuperminimizing constants is considered as well. **

Key words and phrases: nonlinear potential theory, quasiminimizer, quasisuperminimizer.

Mathematics Subject Classification (2010): Primary: 31C45; Secondary: 35J60.

1 Introduction

Let $\Omega\subset\mathbf{R}^{n}$ be a nonempty open set and $1<p<\infty$. A function $u\in W^{1,p}_{\rm loc}(\Omega)$ is a $Q$-quasi(super)minimizer in $\Omega$, with $Q\geq 1$, if

[TABLE]

for all (nonnegative) $\varphi\in W^{1,p}_{0}(\Omega)$. Quasiminimizers were introduced by Giaquinta–Giusti [6] as a tool for a unified treatment of variational integrals, elliptic equations and systems, and quasiregular mappings on $\mathbf{R}^{n}$.

Quasi(super)minimizers have an interesting theory already in the one-dimensional case, see e.g. [6] and Martio–Sbordone [10]. Kinnunen–Martio [9] showed that one can build a rich potential theory based on quasiminimizers. In particular, they introduced quasisuperharmonic functions, which are related to quasisuperminimizers in a similar way as superharmonic functions are related to supersolutions. See Björn–Björn–Korte [5] for further references.

Kinnunen–Martio [9, Lemmas 3.6 and 3.7] also showed that if $u_{j}$ is a $Q_{j}$-quasisuperminimizer in $\Omega\subset\mathbf{R}^{n}$ (or in a metric space), $j=1,2$, then $u:=\min\{u_{1},u_{2}\}$ is a $\min\{Q_{1}+Q_{2},Q_{1}Q_{2}\}$-quasisuperminimizer. Björn–Björn–Korte [5] improved upon this result in the following way.

Theorem 1.1**.**

(Theorem 1.2 in [5])* Let $u_{j}$ be a $Q_{j}$-quasisuperminimizer in $\Omega\subset\mathbf{R}^{n}$ (or in a metric space), $j=1,2$. Then $u:=\min\{u_{1},u_{2}\}$ is a {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}}-quasisuperminimizer in $\Omega$, where*

[TABLE]

In particular, if $Q_{1}=Q_{2}$, then {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}}=2Q_{1}^{2}/(Q_{1}+1).

It is not known whether {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}} is optimal, but it is the best upper bound known. On the other hand, that $u$ is (in general) not better than a $\max\{Q_{1},Q_{2}\}$-quasisuperminimizer is rather obvious.

The first examples (and so far the only ones) showing that $u$ is (in general) not a $\max\{Q_{1},Q_{2}\}$-quasisuperminimizer were given in [5]. More precisely, if $1<Q_{1}\leq Q_{2}$ then there are $Q_{j}$-quasisuperminimizers $u_{j}$ on $(0,1)\subset\mathbf{R}$ such that $u:=\min\{u_{1},u_{2}\}$ is not a $Q_{2}$-quasisuperminimizer. Estimates and concrete examples, showing that the constant {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}} above is not too far from being optimal, were also given in [5]. All examples therein were on $(0,1)\subset\mathbf{R}$ and our aim in this paper is to obtain similar examples in higher dimensions, i.e. for subsets of $\mathbf{R}^{n}$, $n\geq 2$.

The examples in [5] (giving the best lower bounds) were based on power functions $x\mapsto x^{\alpha}$ and reflections of such functions. For such functions, also in the higher-dimensional case on $\mathbf{R}^{n}$, $n\geq 2$, optimal quasi(super)minimizing constants $Q(\alpha,p,n)$ were obtained in Björn–Björn [2], and these formulas for $Q(\alpha,p,n)$ (with $n=1$) were used in the calculations in [5].

As power-type functions $x\mapsto|x|^{\alpha}$ only have point singularities, it seems difficult to use them for higher-dimensional analogues of the examples in [5]. Instead we base our examples on log-power functions $\log^{\alpha}|x|$ and $(-\log|x|)^{\alpha}$. Since $\log|ex|=1-(-\log|x|)$, we are able to scale and translate them and create higher-dimensional examples on annuli, in the spirit of [5]. For this to be possible we need the log-powers to be quasisuperminimizers which requires $p$ to be equal to the conformal dimension $n$. In particular we obtain the following result.

Theorem 1.2**.**

Let $p=n\geq 2$ and $1<Q_{1}\leq Q_{2}$ be given. Then there are functions $u_{1}$ and $u_{2}$ on $A:=\{x\in\mathbf{R}^{n}:1/e<|x|<1\}$ such that $u_{j}$ is a $Q_{j}$-quasisuperminimizer in $A$, $j=1,2$, but $\min\{u_{1},u_{2}\}$ is not a $Q_{2}$-quasiminimizer in $A$.

As in [5] we also give lower bounds for the increase in the quasisuperminimizing constant and show that these lower bounds are the same as in the 1-dimensional case considered in [5], see Section 3. In Section 4 we show that one can add dummy variables to these examples, as well as to those in [5]; this is nontrivial and partly relies on results from Björn–Björn [4].

Acknowledgement. A.B. and J.B. were supported by the Swedish Research Council grants 2016-03424 and 621-2014-3974, respectively, while I.M. was supported by the SIDA (Swedish International Development Cooperation Agency) project 316-2014 “Capacity building in Mathematics and its applications” under the SIDA bilateral program with the Makerere University 2015–2020.

2 Quasi(sub/super)minimizers

Above we defined what quasiminimizers and quasisuperminimizers are. A function $u$ is a $Q$-quasisubminimizer if $-u$ is a $Q$-quasisuperminimizer. Our definition of quasiminimizers (and quasisub- and quasisuperminimizers) is one of several equivalent possibilities, see Proposition 3.2 in A. Björn [1]. In particular, we will use that it is enough to require (1.1) to hold for all (nonnegative) $\varphi\in{\operatorname{Lip}_{c}}(\Omega)$, where ${\operatorname{Lip}_{c}}(\Omega)$ denotes the space of all Lipschitz functions with compact support in $\Omega$.

When $Q=1$ we usually drop “quasi” and say (sub/super)minimizer. Being a (sub/super)minimizer is the same as being a (weak) (sub/super)solution of the $p\mspace{1.0mu}$-Laplace equation

[TABLE]

see Chapter 5 in Heinonen–Kilpeläinen–Martio [8]. The function $u$ is a supersolution of this equation if the left-hand side is nonnegative in a weak sense.

By Giaquinta–Giusti [7, Theorem 4.2], a $Q$-quasiminimizer can be modified on a set of measure zero so that it becomes locally Hölder continuous in $\Omega$. This continuous $Q$-quasiminimizer is called a $Q$-quasiharmonic function, and a $p\mspace{1.0mu}$-harmonic function is a continuous minimizer.

If $u$ is a quasisuperminimizer, then $au+b$ is also a quasisuperminimizer whenever $a\geq 0$ and $b\in\mathbf{R}$. Also, $u$ is a $Q$-quasiminimizer if and only if it is both a $Q$-quasisubminimizer and a $Q$-quasisuperminimizer. Quasisuperminimizers are invariant under scaling in the following way.

Lemma 2.1**.**

Let $\Omega\subset\mathbf{R}^{n}$ be open and $\tau>0$. If $u:\Omega\rightarrow\mathbf{R}^{n}$ is a $Q$-quasisuperminimizer in $\Omega$, then $v(x):=u(\tau x)$ is a $Q$-quasisuperminimizer in

[TABLE]

• Proof.

Let $\varphi\in W_{0}^{1,p}(\Omega_{\tau})$ be nonnegative. Then ${\widetilde{\varphi}}(x):=\varphi(x/\tau)\in W_{0}^{1,p}(\Omega)$. Hence, as $u:\Omega\rightarrow\mathbf{R}^{n}$ is a $Q$-quasisuperminimizer in $\Omega$,

[TABLE]

Thus $v$ is a $Q$-quasisuperminimizer in $\Omega_{\tau}$. ∎

The following definition will play a role in Section 3. The finiteness requirement is important for this to be useful, and is always fulfilled when {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}\hss}{G}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}\hss}{G}}} is a compact subset of $\Omega$, by the definition of quasiminimizers.

Definition 2.2**.**

If $u$ is a $Q$-quasiminimizer in $\Omega\subset\mathbf{R}^{n}$ we say that $u$ has the maximal $p\mspace{1.0mu}$-energy allowed by $Q$ on the open set $G\subset\Omega$ if

[TABLE]

where $v$ is the minimizer in $G$ with boundary values $v=u$ on $\partial G$.

For further discussion on quasi(super)minimizers, as well as references to the literature, we refer to Björn–Björn [2] and Björn–Björn–Korte [5]. We will mainly be interested in radially symmetric functions on $\mathbf{R}^{n}$, $n\geq 2$.

For the rest of this section, as well as in most of Section 3, we will only consider the case when $p=n$, the conformal dimension.

Define the annulus

[TABLE]

In the conformal case ($p=n$) the logarithm $\log|x|$ is an $n$-harmonic function, and log-powers are quasiminimizers as we shall see. These log-powers and their optimal quasisuperminimizing constants will be the crucial ingredients in Section 3, when investigating the increase in the quasisuperminimizing constant for the minimum of two quasisuperminimizers. The optimal quasiminimizing and quasisub/superminimizing constants for power functions and log-powers $(-\log|x|)^{\alpha}$ on punctured unit balls were obtained in Björn–Björn [2]. These are rather easily shown to apply also to the annuli $A_{\gamma,1}$ with $0<\gamma<1$, see Theorem 2.5 below. We also need to consider log-powers $\log^{\alpha}|x|:=(\log|x|)^{\alpha}$ on the annuli $A_{1,\gamma}$, $\gamma>1$, and their optimal quasiminimizing and quasisub/superminimizing constants provided by the following result.

Theorem 2.3**.**

Let $1<\gamma\leq\infty$, $\alpha>1-1/n$ and $u(x)=\log^{\alpha}|x|$. Then $u$ is a quasiminimizer in $A_{1,\gamma}$ with

[TABLE]

being the best quasiminimizing constant.

Moreover, $u$ is a $Q$-quasi(sub/super)minimizer in $A_{1,\gamma}$ as given in Table 1, where $Q$ in Table 1 is the best quasi(sub/super)minimizing constant. Furthermore, $u$ has the maximal $n$-energy allowed by $Q_{\alpha,n}$ on $A_{1,\gamma}$ if $\gamma<\infty$.

The proof is a modification of the proofs of Theorems 7.3 and 7.4 in [2]. For the reader’s convenience we provide the details.

• Proof.

Let $\varphi(r)=\log^{\alpha}r$, $1<r_{1}<r_{2}<\gamma$, $G=(r_{1},r_{2})$, $s_{1}=\log r_{1}$, $s_{2}=\log r_{2}$ and $S=s_{2}/s_{1}>1$.

The $n$-energy of $u$ in $A_{r_{1},r_{2}}$ is given by

[TABLE]

where $c_{n-1}$ is the $(n-1)$-dimensional surface area on the sphere $\mathbf{S}^{n-1}$. Moreover

[TABLE]

A minimizer is given by $\psi(r)=\log r$, and we have letting $\alpha=1$ above,

[TABLE]

We want to compare the energy $\hat{I}_{\varphi}(G)$ with the energy $\hat{I}_{\eta}(G)$ of the minimizer $\eta=a\psi+b$ having the same boundary values on $\partial G$ as $\varphi$. As

[TABLE]

their quotient is

[TABLE]

which only depends on $S$.

Let $s=\sqrt{s_{1}s_{2}}$ and let $\eta_{1}$ and $\eta_{2}$ be the minimizers of the $\hat{I}$-energy on $G_{1}=(e^{s_{1}},e^{s})$ resp.  $G_{2}=(e^{s},e^{s_{2}})$ having the same boundary values as $\varphi$ on $\partial G_{1}$ resp. $\partial G_{2}$. Also let ${\tilde{\eta}}=\eta_{1}\chi_{G_{1}}+\eta_{2}\chi_{G\setminus G_{1}}$. Then, as $s/s_{1}=s_{2}/s=\sqrt{S}$, we see that

[TABLE]

As $0<\hat{I}_{\varphi}(G)<\infty$, we find that $k(S)\geq k(\sqrt{S})$, and thus

[TABLE]

Comparing $u$ with $x\mapsto\eta(|x|)$ shows that the quasiminimizing constant for $u$ cannot be less than $Q_{\alpha,n}$.

To show that $Q_{\alpha,n}$ will do, let $\omega$ be a function such that $\omega-\varphi\in{\operatorname{Lip}_{c}}((1,\gamma))$. The open set

[TABLE]

can be written as a countable (or finite) union of pairwise disjoint intervals $\{I_{j}\}_{j}$. We find from (2.2) and (2.3) that

[TABLE]

Hence $\varphi$ is indeed a $Q_{\alpha,n}$-quasiminimizer for the energy $\hat{I}$ on $(1,\gamma)$.

Next, we turn to $u$. Let $v$ be such that $v-u\in{\operatorname{Lip}_{c}}(A_{1,\gamma})$. Also let

[TABLE]

Using polar coordinates $x=(r,\theta)$, where $r>1$ and $\theta\in\mathbf{S}^{n-1}$, let

[TABLE]

We then find, applying (2.4) to $G=V_{\theta}$, that

[TABLE]

showing that $u$ is indeed a $Q_{\alpha,n}$-quasiminimizer in $A_{1,\gamma}$.

It follows directly that the constants in the quasiminimizer column in Table 1 are correct. By Lemma 2.4 below, $u$ is a subminimizer if $\alpha\geq 1$ and a superminimizer if $1-1/n<\alpha\leq 1$. As $u$ is a $Q$-quasiminimizer if and only if it is both a $Q$-quasisubminimizer and a $Q$-quasisuperminimizer, it follows that $Q_{\alpha,n}$ is the optimal quasisubminimizing constant when $1-1/n<\alpha\leq 1$, and the optimal quasisuperminimizing constant when $\alpha\geq 1$.

Finally, if $\gamma<\infty$, then it follows from (2.1) that

[TABLE]

and that the minimizer with the same boundary values has $n$-energy $(\log\gamma)^{n\alpha-n+1}$, i.e. $u$ has the maximal $n$-energy allowed by $Q_{\alpha,n}$ on $A_{1,\gamma}$. ∎

Lemma 2.4**.**

Let $u(x)=\log^{\alpha}|x|$. Then $u$ is a superminimizer in $A_{1,\infty}$ if and only if $0\leq\alpha\leq 1$. Similarly, $u$ is a subminimizer in $A_{1,\infty}$ if and only if $\alpha\leq 0$ or $\alpha\geq 1$.

• Proof.

A straightforward calculation shows that

[TABLE]

for $x\in A_{1,\infty}$. The sign of this expression is the same as of $\alpha(1-\alpha)$. The function $u$ is, by definition, a superminimizer if this expression is nonnegative, and a subminimizer if it is nonpositive throughout $A_{1,\infty}$, which thus happens exactly as stated. ∎

We also need the following result, which is essentially from Björn–Björn [2].

Theorem 2.5**.**

([2, Theorems 7.3 and 7.4])* Let $0\leq\gamma<1$, $\alpha>1-1/n$ and $u(x)=(-\log|x|)^{\alpha}$. Then $u$ is a quasiminimizer in $A_{1,\gamma}$ with*

[TABLE]

being the best quasiminimizing constant.

Furthermore, $u$ is a $Q$-quasi(sub/super)minimizer in $A_{\gamma,1}$ as given in Table 1, where $Q$ in Table 1 is the best quasi(sub/super)minimizing constant. Also, $u$ has the maximal $n$-energy allowed by $Q_{\alpha,n}$ on $A_{\gamma,1}$ if $0<\gamma<1$.

• Proof.

When $\gamma=0$, the first part follows from Theorem 7.3 in [2]. The proof therein holds equally well when $\gamma>0$ as $S$ therein still ranges over $1<S<\infty$, cf. the proof of Theorem 2.3 above.

Similarly, the second part follows from Theorem 7.4 in [2], where again the arguments hold also when $\gamma>0$. Finally, the last part follows from the formula at the bottom of p. 314 in [2], cf. the end of the proof of Theorem 2.3 above. ∎

3 The increase in the

quasisuperminimizing constant

In this section we are going to use the log-powers considered in Section 2 to construct higher-dimensional analogues of the examples in Björn–Björn–Korte [5, Section 3], concerning the optimality of (1.2) in Theorem 1.1.

Fix $n\geq 2$. We study quasisuperminimizers on the annulus

[TABLE]

As before, we let $p=n$ be the conformal dimension.

Example 3.1*.*

Given $Q>1$, there are exactly two exponents $1-1/n<\alpha^{\prime}<1<\alpha$ such that $Q=Q_{\alpha,n}=Q_{\alpha^{\prime},n}$, where

[TABLE]

This is easily shown by differentiating (3.1) with respect to $\alpha$ and noting that the derivative is negative for $\alpha<1$ and positive for $\alpha>1$, and that $Q_{\alpha}\to\infty$ as $\alpha\to 1-1/n$ and as $\alpha\to\infty$.

We let

[TABLE]

Then $u_{Q}(y)=\bar{u}_{Q}(y)=0$ if $|y|=1/e$, and $u_{Q}(y)=\bar{u}_{Q}(y)=1$ if $|y|=1$. By Theorem 2.3 and Lemma 2.1, $u_{Q}$ is a subminimizer and a $Q$-quasisuperminimizer in $A$. The same is true for $\bar{u}_{Q}$ by Theorem 2.5.

It follows from Theorems 2.3 and 2.5 that $u_{Q}$ has the maximal $n$-energy allowed by $Q$ on each annulus $A_{1/e,\gamma}$, $1/e<\gamma\leq 1$, while $\bar{u}_{Q}$ has the maximal $n$-energy allowed by $Q$ on each annulus $A_{\gamma,1}$, $1/e\leq\gamma<1$. This will be of crucial importance when proving Theorem 1.2, which we will do now.

• Proof of Theorem 1.2.

Let $1-1/n<\alpha_{2}<1<\alpha_{1}$ be such that $Q_{1}=Q_{\alpha_{1},n}$ and $Q_{2}=Q_{\alpha_{2},n}$ as in (3.1), and let $u_{1}:=u_{Q_{1}}$ and $u_{2}:=\bar{u}_{Q_{2}}$ be the corresponding quasiminimizers as in (3.2). By Example 3.1, $u_{j}$ is a subminimizer and a $Q_{j}$-quasisuperminimizer in $A$, $j=1,2$.

By Theorem 1.1, the function $u:=\min\{u_{1},u_{2}\}$ is a {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}}-quasisuperminimizer in $A$ with the quasisuperminimizing constant {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}\hss}{Q}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern 0.0pt}\hss}{Q}}} given by (1.2). Let $v=\log|ex|$, which is the minimizer on $A$ with the same boundary values as $u$, $u_{1}$ and $u_{2}$. As $u_{1}$ and $u_{2}$ are subminimizers they are less than $v$, which can also be seen directly. Thus $u<v$ in $A$.

We are going to show that $u$ is not a $Q_{2}$-quasisuperminimizer on $A$. As $u<v$, to do this it suffices to show that the $n$-energy

[TABLE]

where

[TABLE]

For convenience, write $u_{j}(r)=u_{j}(x)$ when $r=|x|$. There is a unique number $r_{0}\in(1/e,1)$ such that $u_{1}(r_{0})=u_{2}(r_{0})$ (see below), i.e. such that

[TABLE]

To see that there is a unique solution, we consider $w=u_{2}-u_{1}$ and note that $w(1/e)=w(1)=0$. Since $w^{\prime}(1/e)>0$ and $w^{\prime}(1)=\infty$, there is at least one $r\in(1/e,1)$ such that $w(r)=0$. Moreover, $w^{\prime}(r)=0$ if and only if

[TABLE]

Note that $f^{\prime}(r)>0$ if and only if $0<r<e^{-\beta/(\beta+1)}$ and that $f(r)$ attains its maximum at (and only at) $r=e^{-\beta/(\beta+1)}$. Since $f(1/e)=f(1)=0$, there are at most two solutions to $w^{\prime}(r)=0$, and thus there can be at most one solution to (3.4) which must lie in between the two local extrema of $w$.

Since $u_{2}$ is a subminimizer in $A$ we see that

[TABLE]

where the strict inequality follows from the uniqueness of solutions to obstacle problems (see e.g. Björn–Björn [3, Theorem 7.2]) and from the fact that $u_{1}$ and $u_{2}$ differ on a set of positive measure. Hence

[TABLE]

where the last equality follows from the fact that $u_{2}$ has the maximal $n$-energy allowed by $Q_{2}$ on $A$. As $Q_{2}\geq Q_{1}$ this concludes the proof. ∎

Theorem 1.2 shows that in general there is an increase in the quasisuperminimizing constant when taking the minimum of two quasiminimizers but does not give any quantitative estimate of the increase. Next, we are going to analyse the construction in the proof of Theorem 1.2 more carefully to get explicit lower bounds for the increase in the quasisuperminimizing constant.

Given $Q_{1},Q_{2}>1$ and $p=n\geq 2$, let $1-1/n<\alpha_{2}<1<\alpha_{1}$ be such that $Q_{1}=Q_{\alpha_{1},n}$ and $Q_{2}=Q_{\alpha_{2},n}$ as in (3.1), and let $u_{1}:=u_{Q_{1}}$ and $u_{2}:=\bar{u}_{Q_{2}}$ be the corresponding quasiminimizers as in (3.2). Let $r_{0}\in(1/e,1)$ be as in (3.4). Contrary to Theorem 1.2 we here allow for $Q_{1}>Q_{2}$ (the assumption $Q_{1}\leq Q_{2}$ in Theorem 1.2 is only used at the very end of its proof).

It follows from Theorems 2.5 and 2.3 that $u_{1}=u_{Q_{1}}$ has the maximal $n$-energy allowed by $Q_{1}$ on $A_{1/e,r_{0}}$, while $u_{2}=\bar{u}_{Q_{2}}$ has the maximal $n$-energy allowed by $Q_{2}$ on $A_{r_{0},1}$. Using this, we can calculate the $n$-energy of $u=\min\{u_{1},u_{2}\}$ as

[TABLE]

Comparing this value with the energy $I_{v}=c_{n-1}$, given by (3.3), of the minimizer $v$ with the same boundary values as $u$ on $A$ we see that $u$ is not a $Q$-quasisuperminimizer for any $Q<\widehat{Q}$.

For specific values of $Q_{1}$, $Q_{2}$ and $p=n$, one can calculate $\widehat{Q}$ numerically, (after first calculating $\alpha_{1}$, $\alpha_{2}$ and $r_{0}$ numerically), which we have done using Maple 18. These results are presented in Table 2, which shows the values of $\widehat{Q}$ for certain values of $p=n$ and with $Q_{1}=Q_{2}=Q$.

Remark 3.2*.*

The figures for $\widehat{Q}$ in the columns for $p=2$ and $p=100$ in Table 2 above are identical to the corresponding columns for $\widetilde{Q}$ in Table 2 in [5] (there are no columns for $p=3$ and $p=10$ therein). This suggests that the increase in quasisuperminimizing constant in the example above is identical to the increase in the $1$-dimensional example in [5, pp. 271–272]. This is indeed true, as we shall now show, not only when $Q_{1}=Q_{2}$.

Let as before $p=n\geq 2$ be an integer. (In the example in [5, pp. 271–272], the underlying space is $\mathbf{R}$, but $p$ can be an arbitrary real number $>1$. To compare it with our construction above we need $p$ to be an integer.)

Let as above $Q_{1},Q_{2}>1$ be given, and choose $1-1/n<\alpha_{2}<1<\alpha_{1}$ such that $Q_{1}=Q_{\alpha_{1},n}$ and $Q_{2}=Q_{\alpha_{2},n}$ as in (3.1). With $p=n$ this choice of $\alpha_{1}$ and $\alpha_{2}$ also satisfies (3.2) in [5]. Next choose $r_{0}\in(1/e,1)$ to be the unique solution of (3.4). Then $\widehat{Q}$ is given by (3.5).

To relate this to $\widetilde{Q}$ in [5], we let $x_{0}=\log er_{0}\in(0,1)$. It then follows from (3.4) that

[TABLE]

i.e. $x_{0}$ is the unique solution of this equation, which is the same as equation (3.4) in [5]. (That the solution is unique was shown in [5], but also follows from the uniqueness of $r_{0}\in(1/e,1)$.) Using (3.6) in [5] (with $p=n$) we see that

[TABLE]

Remark 3.3*.*

The function $\widetilde{Q}$ depends on $Q_{1}$, $Q_{2}$ and $p$, i.e. $\widetilde{Q}=\widetilde{Q}(Q_{1},Q_{2},p)$. Given $1<Q_{1}<Q_{2}$ (and $p$) it is natural to ask which is larger of $\widetilde{Q}(Q_{1},Q_{2},p)$ and $\widetilde{Q}(Q_{2},Q_{1},p)$. We have calculated some values of $\widetilde{Q}(Q_{1},Q_{2},p)$ using Maple 18, see Table 3. They all indicate that

[TABLE]

Due to the intricate formula (3.6) for $\widetilde{Q}$, involving $x_{0}$, we have not been able to show that this is always the case.

The formula for $\widetilde{Q}$ is valid also for nonintegers $p>1$, and $p=1.2$ is included in Table 3. However, if $p$ is an integer then $\widehat{Q}=\widetilde{Q}$, by (3.7), and the same reasoning about the comparison in (3.8) applies to $\widehat{Q}$.

4 Adding dimensions

One way of making higher-dimensional examples from lower-dimensional ones is to add dummy variables. The tensor product $u_{1}\otimes u_{2}(x,y)=u_{1}(x)u_{2}(y)$ and tensor sum $u_{1}\oplus u_{2}=u_{1}(x)+u_{2}(y)$ of two harmonic functions is again harmonic, a fact that is well known and easy to prove. The corresponding fact is false for $p\mspace{1.0mu}$-harmonic functions, but it was observed in Björn–Björn [4] that the tensor product and sum are quasiminimizers. They moreover showed that the tensor product and sum of quasiminimizers are again quasiminimizers, but typically with an increase in the quasiminimizing constant even if both are $1$. However, if one of the quasiminimizers is constant then the increase in the quasiminimizing constant can be avoided, a fact that we shall use. We first recall the following consequence of the results in [4].

Theorem 4.1**.**

Let $u$ be a $Q$-quasisuperminimizer in $\Omega\subset\mathbf{R}^{n}$ and let $I\subset\mathbf{R}$ be an interval. Then $\bar{u}=u\otimes 1$ is a $Q$-quasisuperminimizer in $\Omega\times I$.

This result is true also if the first space $\mathbf{R}^{n}$ is equipped with a so-called $p\mspace{1.0mu}$-admissible weight $w$, see [4]. In particular, by Theorem 3 in [4], $w\otimes 1$ is then a $p\mspace{1.0mu}$-admissible weight on $\mathbf{R}^{n+1}$.

• Proof.

This is a special case of Theorem 7 in [4], with $u_{1}=u$, $u_{2}\equiv 1$, $Q_{1}=Q$ and $Q_{2}=0$. As mentioned in [4, p. 5196], one is allowed to let $Q_{2}=0$ if $u_{2}$ is a constant function. ∎

For the purposes in this paper, this is not enough since, typically, the obtained $Q$ is not the optimal quasisuperminimizing constant for $\bar{u}$ even if it is for $u$. But if $I$ is in addition unbounded then $Q$ is optimal for $\bar{u}$ if it is for $u$, as we shall now show.

Theorem 4.2**.**

Let $u$ be a $Q$-quasisuperminimizer in $\Omega\subset\mathbf{R}^{n}$, where $Q$ is the optimal quasisuperminimizing constant. Let $I\subset\mathbf{R}$ be an unbounded interval. Then $\bar{u}=u\otimes 1$ is a $Q$-quasisuperminimizer in $\Omega\times I$, with $Q$ again being optimal.

• Proof.

By Theorem 4.1, $\bar{u}$ is a $Q$-quasisuperminimizer, so it is only the optimality of $Q$ that needs to be shown. If $Q=1$ there is nothing to prove, so we can assume that $Q>1$.

Let $0<\varepsilon<Q$. Since $Q$ is optimal, there is a nonnegative $\varphi\in{\operatorname{Lip}_{c}}(\Omega)$ such that

[TABLE]

see the beginning of Section 2. As the integral on the left-hand side is positive, also the integral on the right-hand side must be positive, as otherwise $u$ would not be a quasisuperminimizer at all.

Next, let $m>0$ be given. Since $I$ is unbounded we can find $a\in\mathbf{R}$ so that $[a,a+m+2]\subset I$. Assume without loss of generality that $a=0$, and let

[TABLE]

Then

[TABLE]

while

[TABLE]

Letting $m\to\infty$ and then $\varepsilon\to 0$ shows that $Q$ is optimal, since the last integral is nonzero. ∎

It now follows directly from Theorem 4.2 that we can add a dummy variable to the examples constructed in Section 3 and in Björn–Björn–Korte [5, Section 3]. As long as we consider the dummy variable taken over an unbounded interval, we obtain a new example with the same increase in the quasisuperminimizing constant. This can be iterated so that we can add an arbitrary (but finite) number of dummy variables. This way we get higher-dimensional examples on unbounded sets. However, it follows from the following lemma that by taking Cartesian products with large enough bounded intervals, one can obtain similar bounded examples with an increase in the quasisuperminimizing constant, which is arbitrarily close to the increase in Section 3 resp. [5, Section 3].

Lemma 4.3**.**

Let $\Omega_{1}\subset\Omega_{2}\subset\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}\subset\Omega=\bigcup_{j=1}^{\infty}\Omega_{j}$ be an increasing sequence of open subsets of $\mathbf{R}^{n}$. If $u$ is a $Q$-quasisuperminimizer in $\Omega_{j}$ for every $j$, then it is also a $Q$-quasisuperminimizer in $\Omega$.

• Proof.

As mentioned at the beginning of Section 2 it is enough to test (1.1) with nonnegative $\varphi\in{\operatorname{Lip}_{c}}(\Omega)$. By compactness, $\varphi\in{\operatorname{Lip}_{c}}(\Omega_{j})$ for some $j$, and thus (1.1) holds for this particular $\varphi$ as $u$ is a $Q$-quasisuperminimizer in $\Omega_{j}$. ∎