Three-spheres theorem for $p$-harmonic mappings

Abstract

Let $u=(u^1,\ldots , u^n)$ be a $p$-harmonic mapping in a domain $\Omega \subset \mathbb {R}^n$ for $n\ge 2$. We investigate level sets for compositions of coordinate functions $u^i$ with convex functions satisfying growth conditions and derive the de Giorgi-type estimates. Our main result is the arithmetic three-spheres theorem for coordinate functions of mapping $u$. The discussion is illustrated by radial $p$-harmonics.

1 Introduction

Consider a subharmonic function $u$ in a planar domain and let $M(r)$ denote the maximum of $u$ over a circle $x^2+y^2=r^2$ concentric with two other circles with radii satisfying $r_1<r<r_2$. Then, the classical Hadamard three-circles theorem asserts that $M(r)$ is a convex function of $r$, see e.g. Chapter 12 in Protter–Weinberger [19] (Hadamard formulated this result for analytic functions without providing a proof, see Hadamard [13]). Namely it holds that

$$\begin{aligned} M(r)\le \frac{\log (r_2/r)}{\log (r_2/r_1)}M(r_1)+\frac{\log (r/r_1)}{\log (r_2/r_1)}M(r_2). \end{aligned}$$

The result can be further generalized in the following directions: by considering higher dimensional analogs, by studying equations more general than the Laplace equation and by investigating different types of inequalities involving $M(r)$ or the norms of functions in subject. Indeed, one studies the setting of concentric spheres in $\mathbb {R}^n, n\ge 3$ see e.g. Theorem 30 in [19] or spheres which need not be concentric, see Arakelian–Matevosyan [5]. The three-circles (or the three-spheres) theorem can be extended to the setting of more general elliptic equations, see discussion in [19, Chapter 12], Brummelhuis [7] for a discussion in the setting of second-order linear elliptic equations, Miklyukov–Rasila–Vuorinen [18] for $p$-harmonic equations, Výborný [22] for quasilinear equations with Lipschitz coefficients; see also Granlund–Marola [12] for studies in the setting of $(A,B)$-equations of Riccati type and for further references. Finally, instead of the above inequality, one studies estimates involving $L^2$ or $L^{\infty }$ norms of solutions to elliptic equations, see e.g. Lin–Nagayasu–Wang [17] and Alessandrini–Rondi–Rosset–Vessella [4]. In the latter publication, among other topics the authors discuss the role of the three-spheres theorems in the studies of the unique continuation problems and ill-posed problems. For related topics and estimates we refer to Colding–De Lellis–Minicozzi [8] and Garofalo–Lin [9].

The main goal of this paper is to prove a variant of three-spheres theorem in the context of coupled elliptic systems of equations represented by $p$-harmonic systems of equations. According to our best knowledge three-spheres theorems have not yet been studied for systems of equations.

A mapping $u=(u^1,\ldots ,u^n)\,\in \,W^{1,p}_{loc}(\Omega ,\mathbb {R}^n)\,$ is called a p-harmonic mapping if it is a solution to the following system of equations:

$$\begin{aligned} \mathrm{div}(|Du|^{p-2}Du)=0,\,\,\,\,u=(u^1,\ldots ,u^n):\Omega \subset \mathbb {R}^N\rightarrow \mathbb {R}^n,\quad 1<p<\infty , \end{aligned}$$

(1)

where $Du$ denotes the Jacobi matrix of $u$, i.e. $Du=(\nabla u^1,\ldots , \nabla u^n)^T$. Equivalently, this system can be written as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{div}(|Du|^{p-2}\nabla u^1)= 0, \\ \qquad \vdots \\ \mathrm{div}(|Du|^{p-2}\nabla u^n)= 0. \end{array} \right. \end{aligned}$$

(2)

The $p$-harmonic system of equations is the Euler-Lagrange system of the associated $p$-Dirichlet energy functional

$$\begin{aligned} \int _{\Omega }|Du|^p. \end{aligned}$$

In the weak form (2) reads

$$\begin{aligned} \int _{\Omega }|Du|^{p-2}\langle \nabla u^i, \nabla \phi ^i \rangle \,=\,0 \quad \hbox { for }\ i=1,\ldots ,n, \end{aligned}$$

(3)

where $\phi ^i \in C_{0}^{\infty }(\Omega )$ are test functions. In what follows we will consider the case of $N=n$. For $p=2$ the system reduces to the well-known harmonic system of equations (such a system is uncoupled). Therefore, one may view a $p$-harmonic system as a natural generalization of the harmonic system to the nonlinear setting. If we let the dimension of the target space be $n=1$, then we retrieve the scalar $p$-harmonic equation. The above system is strongly coupled by the appearance of the full differential $Du$. As a consequence many methods of PDEs known in the linear (harmonic) setting fail for $p\not =2$. This in turn stimulates the development of new methods and new approaches to handle the nonlinear problems.

The $p$-harmonic systems and their generalizations appear naturally in differential geometry, see e.g. Hardt–Lin [14] or in relation to differential forms and quasiregular maps, see e.g. Bonk–Heinonen [6]. As for the applied sciences, the second order coupled elliptic systems are studied in nonlinear elasticity theory, e.g. Iwaniec–Onninen [15], nonlinear fluid dynamics, as well as in astrophysics or climate sciences and several other areas (see Adamowicz [3] for the list of further references).

We will now introduce notation, describe our approach to the three-spheres estimates and formulate auxiliary results and the main result of the paper. The proofs are presented in the remaining sections.

Let $\phi :\mathbb {R}\rightarrow \mathbb {R}$ be a $C^2$ convex function such that there exists a subinterval $I\subset \mathbb {R}$ with properties:

$$\begin{aligned}&\qquad (\phi '(x))^2\le \phi ''(x)\quad \hbox { for all }x\in I \end{aligned}$$

(4)

$$\begin{aligned}&\qquad \hbox { either (i) } \phi '(x)>0 \quad \hbox { in }I \hbox { or (ii) } \phi '(x)<0 \hbox { in }I \end{aligned}$$

(5)

In what follows we will deal several times with the expression $|Du|/|\nabla u^1|$, its reciprocal and integrals involving it. In particular, we will need some conditions to ensure that

$$\begin{aligned} \int _{B_r} \left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\,dx<\infty , \end{aligned}$$

(6)

where $B_r$ is a ball of radius $r$. The case of $1<p<2$ is easy, since then $(\frac{|\nabla u^1|}{|Du|})^{2-p}<1$ and by employing also continuity of $p$-harmonic mappings (see e.g. Tolksdorf [20]) we get that the above integral exists and is finite. The case of $p=2$ is trivial. For $p>2$ we need, however, to be more careful at critical points of $u^1$.

First, recall that results due to Tolksdorf [20, Theorem, Formula (1.14)] and Uhlenbeck [21, Theorem 3.2] applied to $p$-harmonic systems give us, respectively, that

$$\begin{aligned} \sup _{B_r}|Du|&\le \frac{C(p,n)}{r^\frac{n}{p}} \left( |B_{2r}|^{\frac{1}{p}}+\Vert Du\Vert _{L^p(B_{2r})}\right) \quad \hbox { for }\quad 1<p<2 \end{aligned}$$

(7)

$$\begin{aligned} \sup _{B_r}|Du|&\le \frac{C(p,n)}{r^\frac{n}{p}} \Vert Du\Vert _{L^p(B_{2r})}\quad \hbox { for }\quad p\ge 2. \end{aligned}$$

(8)

The discrepancy between formulas (7) and (8) is a consequence of different nature of $p$-harmonic mappings for $1<p<2$ (singular case) and $p>2$ (degenerate elliptic case). In fact (7) holds for $1<p<\infty$, cf. statement of Theorem and Section 3 in [20]. Nevertheless, the fact that inequality (8) is scale invariant with respect to $u$ and is independent of the size of $B_{2r}$ makes (8) still of interest to us and, therefore, in what follows we will appeal to both estimates. The above discussion allows us to continue estimation at (6) for $p>2$ in a following way:

$$\begin{aligned} \int _{B_r} \left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\,dx\le \frac{C(p, n)\Vert Du\Vert ^{p-2}_{L^p(B_{2r})}}{r^{n(1-\frac{2}{p})}}\int _{B_r}\frac{1}{|\nabla u^1|^{p-2}}. \end{aligned}$$

(9)

Suppose that a ball $B_r$ is centered at $x_0$ and that $B_{2r}\subset \Omega$. Furthermore, assume that for $p>2$

$$\begin{aligned} |\nabla u^1(x)|\ge |x-x_0|^{\alpha }\quad \hbox { for }x\in B_r\,\hbox { and some } \,\alpha <\frac{n}{p-2}. \end{aligned}$$

(10)

Then, by (9) we have:

$$\begin{aligned} \int _{B_r} \left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\,dx\le \frac{C(p, n) \Vert Du\Vert ^{p-2}_{L^p(B_{2r})}}{r^{n(1-\frac{2}{p})}}\int _{0}^{r} t^{-\alpha (p-2)+n-1}\,dt<\infty . \end{aligned}$$

(11)

For the sake of simplicity and clarity of discussion all the results in the paper are stated for $u^1$, the first coordinate function of a $p$-harmonic mapping $u$. However, the reader should keep in mind that all the presented results hold as well for all coordinate functions $u^i$, for $i=1,\ldots , n$ upon the necessary reformulations of results. Denote by

$$\begin{aligned} M(r)=\sup _{|x-x_{0}|= r}u^1(x) \end{aligned}$$

and

$$\begin{aligned} m(r)=\inf _{|x-x_{0}|= r}u^1(x). \end{aligned}$$

In Sect. 2 we show the Caccioppoli-type estimates for a composition of a convex function with a coordinate function of a $p$-harmonic mapping (Lemma 1). According to our best knowledge, such estimates and such an approach to $p$-harmonic mappings has not been studied in the literature so far.

Lemma 1

Let $u^1$ be the first coordinate function of a $p$-harmonic mapping $u$ in a domain $\Omega$ for $p>1$. Assume, furthermore, that a convex function $\phi : I\rightarrow \mathbb {R}$ satisfies conditions (4) and (5). Then the following estimates hold for every ball $\overline{B_r}\subset \Omega$ and $0<\delta <1$:

$$\begin{aligned}&\mathrm{If }1<p<2, \hbox { then }\int _{B_{(1-\delta )r}} \left( \frac{|\nabla u^1|}{|Du|}\right) ^{2-p}|\nabla \phi (u^1)|^p \le \frac{c(p,n)}{\delta ^p}r^{n-p}. \end{aligned}$$

(12)

$$\begin{aligned}&\mathrm{If }p\ge 2, \mathrm{then }\int _{B_{(1-\delta )r}} |\nabla \phi (u^1)|^p \le \frac{c(p)}{(\delta r)^p}\int _{B_r}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}. \end{aligned}$$

(13)

Furthermore, for $p>2$ the growth condition (10) ensures finitness of the last integral.

In the next result we study the behavior of $\phi (u^1)$ over the level sets and investigate the de Giorgi type estimates. Such estimates are well-known for solutions of elliptic equations, see e.g. Giusti [10]. In the setting of vector functions and systems of equations such estimates require extra attention and effort. The results of Lemma 2 can be used in further analysis of level sets for coordinate functions of $p$-harmonic mappings (see Sect. 3 for the proof of Lemma 2).

Let $k\ge 0$. Upon the above notation we define

$$\begin{aligned} A_{k, r}:=\{x\in B_r: \phi (u^1(x))>k\}. \end{aligned}$$

Lemma 2

Let $u^1$ be the first coordinate function of a $p$-harmonic mapping $u$ in a domain $\Omega \subset \mathbb {R}^n$ for $1<p\le n$. Assume, furthermore, that conditions (4) and (5) hold for a convex function $\phi : I\rightarrow \mathbb {R}$. Then the following estimates hold for every ball $\overline{B_r}\subset \Omega$ and $0<\delta <1$:

$$\begin{aligned}&\int _{A_{k, (1-\delta )r}} \left( \frac{|\nabla u^1|}{|Du|}\right) ^{2-p}|\nabla \phi (u^1)|^p \le \frac{c(p)}{(\delta r)^p}\int _{A_{k, r}}(\phi (u^1)-k)^p\quad \mathrm{for }\quad 1<p\le 2, \end{aligned}$$

(14)

$$\begin{aligned}&\int _{A_{k, (1-\delta )r}} |\nabla \phi (u^1)|^p \le \frac{c(p)}{(\delta r)^p}\int _{A_{k, r}}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}(\phi (u^1)-k)^p\quad \mathrm{for }\quad p\ge 2 \ \mathrm{and} \ n>2. \end{aligned}$$

(15)

Furthermore, the following inequality holds for $1<p<2$:

$$\begin{aligned} \big (\sup _{B_{(1-\delta )r}} \phi (u^1)\big )^p \le \frac{C^{\frac{n}{p}}}{(\delta r)^{n}}\int _{B_r}(\phi (u^1))^p+r^{p-n}, \end{aligned}$$

(16)

where the constant

$$\begin{aligned} C=C(p, n, c_S)\left[ 1+\Vert \phi '\Vert ^p_{L^{\infty }(B_{r})}\left( r^n+ \Vert Du\Vert ^p_{L^p(B_{2r})}\right) \right] . \end{aligned}$$

(17)

If, additionally, there exists a positive constant $c$, such that

$$\begin{aligned} |\nabla u^1|>c\quad \mathrm{in} \ B_r, \end{aligned}$$

(18)

then the following inequality holds for $p>2$ and $n>2$:

$$\begin{aligned} \big (\sup _{B_{(1-\delta )r}} \phi (u^1)\big )^p \le \frac{C^{\frac{n}{p}}}{(\delta r)^{n}}\int _{B_r}(\phi (u^1))^p, \end{aligned}$$

(19)

with constant

$$\begin{aligned} C=c(p,n,c_S)\left( \frac{\Vert Du\Vert _{L^p(B_{2r})}}{cr^\frac{n}{p}}\right) ^{\frac{n}{p}(p-2)}. \end{aligned}$$

(20)

Here, $c_S$ stands for the constant in the Sobolev embedding theorem. In the harmonic case $p=2$, assumption (18) can be neglected and (19) holds with constant $C=c(p,n,c_S)$ in (20).

The main result of this paper is the following version of the arithmetic three-spheres theorem for coordinate functions of $p$-harmonic mappings. We prove Theorem in Sect. 4 as well as comment on the existence of $p$-harmonic mappings satisfying assumptions of Theorem.

Theorem

(The arithmetic three-spheres theorem) Let $1<p\le n$ and $u=(u^1,\ldots , u^n):\Omega \subset \mathbb {R}^n\rightarrow \mathbb {R}^n$ be a $p$-harmonic mapping. Consider three concentric balls centered at $x_0\in \Omega$ with radii $0<r_1<r<r_2$ such that $\overline{B_{2r_2}}\subset \Omega$ and

$$\begin{aligned} 0<c\le \frac{r_1}{r},\quad \frac{r_1}{r_2}<c<1, \end{aligned}$$

(21)

for some fixed $c$.

If $1<p<2$, then let us assume that for a given $\alpha >0$, the coordinate function $u^1$ satisfies the following growth condition:

$$\begin{aligned} | u^1(x)-u^1(x_0)|\ge C|x-x_0|^{\alpha }\quad \mathrm{for} \ x\in \Omega \setminus B_{r_2}. \end{aligned}$$

(22)

If $2<p<n$, then let us assume instead that there is a positive constant $c_1$ such that

$$\begin{aligned} |\nabla u^1|>c_1\quad \mathrm{in} \ B_{r_2}. \end{aligned}$$

(23)

Then there exist a constant $C$ and a radius $r_3$ such that if $\overline{B_{r_3}}\subset \Omega$, then the following inequalities hold:

$$\begin{aligned} M(r)&\le C M(r_1)+(1-C)M(r_3),\nonumber \\ m(r)&\ge C m(r_1)+(1-C)m(r_3). \end{aligned}$$

For $1<p<2$ constant $C$ depends on $n, p$, $c_S$ the constant in the Sobolev embedding theorem, $r_1, r_2$, $c, \alpha$ and $\Vert Du\Vert _{L^p(B_{2r_2})}$, while $r_3>r_2$ depends on $p, n, r_2, \alpha$, and $\Vert Du\Vert _{L^p(B_{2r_2})}$. For $2<p\le n$ constant $C$ depends on $n, p$, $c_S$, $r_1, r_2$, $c, c_1$ and $\Vert Du\Vert _{L^p(B_{2r_2})}$, while $r_3=r_2$. For $p=2<n$ condition (23) is obsolete, $r_3=r_2$ and constant $C$ depends only on $n, p, c_S$ and $c, r_1, r_2$.

In the statement of Theorem one may consider local solutions in $\mathbb {R}^n$ of the $p$-harmonic system (2) instead in the domain $\Omega$. In such a case one may neglect the assumption that $\overline{B_{r_3}}\subset \Omega$.

If $p=2$, then the $p$-harmonic system (2) reduces to the uncoupled system of harmonic equations satisfied by coordinate functions $u^i$ for $i=1,\ldots ,n$. In such a case the arithmetic three-spheres theorem holds for each $u^i$, see Theorem 30 in Protter–Weinberger [19, Chapter 12].

We point out that the case of scalar $p$-harmonic functions for $1<p<\infty$ can be handled by the approach very similar to the one for harmonic functions, as one has on the disposal comparison principles and the Harnack-type estimates. Such tools are not known in the setting of coordinate functions of $p$-harmonic mappings.

For the case $p=n$, Granlund–Marola [12] proved a variant of the three-spheres theorem in the setting of $(A, B)$-quasilinear equations, in particular for a $p$-harmonic equation, cf. Theorem 5.4 [12]. However, their approach is based on the existence of the strong maximum (minimum) principle and the Harnack inequality for solutions of the considered $(A,B)$-equation. Similar results are not known in the setting of coupled $p$-harmonic mappings ($p\not =2$).

2 The proof of Lemma 1

In this section we prove Lemma 1 and then illustrate the discussion by the class of radial $p$-harmonic mappings.

Proof

We begin the proof as in Granlund–Marola [12]. For the sake of simplicity, let us assume that (5) (i) holds, i.e. $\phi '(u^1(x))>0$ for all $x\in B_r$. Take a nonnegative function $\xi \in C_{0}^{\infty }(B_r)$ and define a test function $\eta (x)=\phi '(u^1(x))^{p-1}\xi ^p(x)$ for $x\in B_r$. Then

$$\begin{aligned} \nabla \eta = (p-1)(\phi '(u^1))^{p-2}\phi ''(u^1)\xi ^p\nabla u^1+p\xi ^{p-1}(\phi '(u^1))^{p-1}\nabla \xi . \end{aligned}$$

We use $\nabla \eta$ in the first equation of $p$-harmonic system (3) and by using (4) together with (5) we obtain

$$\begin{aligned} (p-1)\int _{B_r} |Du|^{p-2}|\nabla u^1|^2\phi '(u^1)^p\xi ^p\le p\int _{B_r}|Du|^{p-2}|\nabla u^1|\phi '(u^1)^{p-1}\xi ^{p-1}|\nabla \xi |. \end{aligned}$$

(24)

For some $0<\epsilon <1$, whose value will be determined later, we rewrite the right-hand side of (24) and apply the Young inequality:

$$\begin{aligned}&p\int _{B_r}\!\!|Du|^{p-2}|\nabla u^1|\phi '(u^1)^{p-1}\xi ^{p-1}|\nabla \xi |\nonumber \\&\quad =p\int _{B_r}\!\!\left( \epsilon |Du|^{(p-2)\frac{p-1}{p}}|\nabla u^1|^{2\frac{p-1}{p}}\phi '(u^1)^{p-1}\xi ^{p-1} \right) \left( \frac{1}{\epsilon }|Du|^{\frac{p-2}{p}}|\nabla u^1|^{\frac{2-p}{p}}|\nabla \xi |\right) \nonumber \\&\quad \le (p-1)\epsilon \int _{B_r}|Du|^{p-2}|\nabla u^1|^2\phi '(u^1)^{p}\xi ^{p} +p\epsilon ^{-p}\int _{B_r}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}|\nabla \xi |^p. \end{aligned}$$

(25)

In the last inequality we also estimated $\epsilon ^{\frac{p}{p-1}}\le \epsilon$. Now, we use (25) in (24) and by taking e.g. $\epsilon =\frac{1}{2}$ we may include the first integral on the right-hand side of (25) into the left-hand side of (24). In a consequence we arrive at the following estimate:

$$\begin{aligned} \int _{B_r} |Du|^{p-2}|\nabla u^1|^2\phi '(u^1)^p\xi ^p\le \frac{p2^{p+1}}{p-1} \int _{B_r}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}|\nabla \xi |^p. \end{aligned}$$

(26)

Denote $c(p):=\frac{p2^{p+1}}{p-1}$.

Case 1: $1<p<2.$ The left-hand side of the above inequality can be written as follows.

$$\begin{aligned} \int _{B_r} |Du|^{p-2}|\nabla u^1|^2\phi '(u^1)^p\xi ^p= \int _{B_r} \left( \frac{|\nabla u^1|}{|Du|}\right) ^{2-p}|\nabla \phi (u^1)|^p\xi ^p. \end{aligned}$$

(27)

Using (27) and the fact that $|\nabla u^1|\le |Du|$ in $\Omega$ we observe that for $1<p<2$ inequality (26) becomes:

$$\begin{aligned} \int _{B_r} \left( \frac{|\nabla u^1|}{|Du|}\right) ^{2-p}|\nabla \phi (u^1)|^p\xi ^p \le c(p)\int _{B_r}|\nabla \xi |^p. \end{aligned}$$

(28)

Case 2: $p\ge 2.$ We have that $|\nabla u^1|^{p-2}\le |Du|^{p-2}$ in $\Omega$ and hence (26) takes the following form.

$$\begin{aligned} \int _{B_r} |\nabla \phi (u^1)|^p\xi ^p \le c(p)\int _{B_r}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}|\nabla \xi |^p. \end{aligned}$$

(29)

If we additionally assume (10), then (11) holds and the last integral is finite. In the definition of $\eta$ we take $\xi$ such that $0\le \xi \le 1$, $\mathrm{supp}\,\xi \subset B_r$, $\xi \equiv 1$ in $B_{(1-\delta )r}$ and $|\nabla \xi |\le \frac{c}{\delta r}$ in $B_r$. Using such $\xi$ in (28) and (29) we arrive at claims (12) and, respectively, (13) of Lemma 1.

If (5) (ii) holds, i.e. $\phi '(u^1(x))<0$ for all $x\in B_r$, then as a test function we take $\eta (x)=(-\phi '(u^1(x)))^{p-1}\xi ^p(x)$ for $x\in B_r$ and claims of the lemma follow the same way as previously. $\square$

We remark that assertions (28) and (29) of Lemma 1 can be further refined. Namely, the following remark holds by Lemma 1 and the discussion at formulas (7) and (8).

Remark 1

(A) Let $1<p<2$. Then (7) implies that for $x\in B_r$

$$\begin{aligned} \frac{1}{|Du(x)|}\ge \frac{1}{\sup _{B_r}|Du|}\ge C(p,n, \mathrm{diam}(\Omega ), \Vert Du\Vert _{L^p(B_{2r})})r^\frac{n}{p}. \end{aligned}$$

In such a case (28) reads:

$$\begin{aligned} \int _{B_r} \left( |\nabla u^1| r^\frac{n}{p}\right) ^{2-p}|\nabla \phi (u^1)|^p\xi ^p \le c(p)\int _{B_r}|\nabla \xi |^p. \end{aligned}$$

Similarly, for $p>2$ the Uhlenbeck inequality (8) results in the following estimate.

$$\begin{aligned} \int _{B_r} |\nabla \phi (u^1)|^p\xi ^p \le C(p, n, \Vert Du\Vert _{L^p(B_{2r})})\int _{B_r}\frac{1}{r^{n(1-\frac{2}{p})}|\nabla u^1|^{p-2}}|\nabla \xi |^p. \end{aligned}$$

(B) In fact estimate (26) gives rise to the following inequality for $p>1$:

$$\begin{aligned} \int _{B_r} \left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2} |\nabla \phi (u^1)|^p\xi ^p\le \frac{p2^{p+1}}{p-1} \int _{B_r}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}|\nabla \xi |^p. \end{aligned}$$

We now turn to considering a special class of $p$-harmonic mappings, namely radial transformations in the form

$$\begin{aligned} u(x)=H(|x|)(x_1,\ldots ,x_n),\quad \hbox { for } x=(x_1,\ldots ,x_n)\in \Omega \subset \mathbb {R}^n, \end{aligned}$$

where $H\in C^2(\Omega )$ and $|x|$ stands for the magnitude of $x$. For such mappings $p$-harmonic system (2) becomes a nonlinear second order ODE:

$$\begin{aligned}&(p-1)H''(H')^2r^3+(2p+n-3)(H')^3r^2+2(p-1)HH'H''r^2 \nonumber \\&\quad +(np+3p-4)H(H')^2r + (p+n-2)H^2H''r + (n+1)(p+n-2)H^2H'=0. \end{aligned}$$

(30)

The problem of finding suitable examples is the general feature of the $p$-harmonic world, as we know only few classes of $p$-harmonic maps and few explicit solutions of the $p$-harmonic system of equations, namely affine, radial and quasiradial, see e.g. Adamowicz [2, 3], Iwaniec–Onninen [15, Part 1] for various applications of radial $p$-harmonics, and Adamowicz [1, Chapter 2] for the definition of quasiradial $p$-harmonic mappings.

For the class of radial $p$-harmonic mappings, Lemma 1 can be refined. Indeed, for radial mappings we can formulate simple conditions for integrability of ratio $|\nabla u^1|/|Du|$ and in turn estimates (12) and (13) reduce to the following result.

Proposition 1

(Radial Lemma 1) Let $u^1$ be the first coordinate function of a radial $p$-harmonic mapping $u$ in $\Omega \subset \mathbb {R}^n, n>1$ for $p>1$. Assume that a convex function $\phi : I\rightarrow \mathbb {R}$ satisfies conditions (4) and (5). Furthermore, assume the following:

$$\begin{aligned}&(1)~\mathrm{if}~x~\mathrm{is~such~that}~H(x)=0, \mathrm{then} \ H'(x)\not =0,\nonumber \\&(2)~\mathrm{there~exist~constant}~c_{\Omega }, c'_{\Omega }>0 \ \mathrm{such \ that} \ x_1^2<c_{\Omega }(|x|^2-x_1^2) \ \hbox { and } \ x_1>c'_{\Omega } \nonumber \\&\quad \hbox { for all } \ x\in \Omega , \end{aligned}$$

(31)

$$\begin{aligned}&(3)~\mathrm{there~exists}~C>0 \hbox { such that } \eta (x):=\frac{H'(|x|)}{H(|x|)}|x|<C~\hbox { for all }~x\in \Omega . \end{aligned}$$

(32)

Then the following estimate holds for every ball $\overline{B_r}\subset \Omega$ and $0<\delta <1$:

$$\begin{aligned} \int _{B_{(1-\delta )r}} |\nabla \phi (u^1)|^p \le \frac{c(\mathrm{diam} (\Omega ),p, n, c_{\Omega }, c'_{\Omega })}{\delta ^p}r^{n-p}. \end{aligned}$$

(33)

Conditions (31) and (32) geometrically mean that we require domain $\Omega$ to lie inside a cone symmetric about $x_1$-axis and to consist of points with a positive distance to the hyperplane $\{x_1=0\}$.

Proof

Let $u(x)=H(|x|)(x_1,\ldots , x_n)$ be a radial $p$-harmonic mapping in $\Omega$. Denote $r:=|x|$ the magnitude of vector $x$. Then $u^i(x)=H(r)x_i$ for $i=1,\ldots , n$ and

$$\begin{aligned} u^i_{x_j}(x)= \left\{ \begin{array}{ll} H'(r)\frac{x_i x_j}{r},&{}\quad \hbox { for } j\not =i,\\ H'(r)\frac{x_i^2}{r}+H(r),&{}\quad \hbox { for } j=i. \end{array}\right. \end{aligned}$$

Hence $|\nabla u^i|^2=\sum _{j\not =i}(H'\frac{x_i x_j}{r})^2+(H'\frac{x_i^2}{r}+H)^2=(H')^2x_i^2+2HH'\frac{x_i^2}{r}+H^2$. Upon computing $|\nabla u^i|^2/|\nabla u^1|^2$ and summing over $i=1,\ldots , n$ we get that

$$\begin{aligned} g:=\frac{|Du|^2}{|\nabla u^1|^2}= \frac{|\nabla u^1|^2+\ldots +|\nabla u^n|^2}{|\nabla u^1|^2}=\frac{(H')^2r^2+2HH'r+nH^2}{(H')^2x_1^2+2HH'\frac{x_1^2}{r}+H^2}. \end{aligned}$$

Let $x_0\in B_r$ and consider the following cases.

Case 1: $H(x_0)=0$. Since, by assumptions we have that $H'(|x_0|)\not =0$, then $g(x_0)=\frac{r^2}{(x_0)_1^2}$. Under the second part of assumption (31) we obtain that $|g(x_0)|\le c(\mathrm{diam}(\Omega ), c_{\Omega }).$

Case 2: $H(x_0)\not =0$ and $H'(x_0)=0$. Then $g(x_0)=n$.

Case 3: $H(x_0)\not =0$ and $H'(x_0)\not =0$. Then for $\eta :=\eta (x_0)=\frac{H'(|x_0|)}{H(|x_0|)}|x_0|$ we have

$$\begin{aligned} g(x_0)=\frac{\eta \left( \eta +2\right) +n}{\frac{|(x_0)_1|^2}{|x_0|^2}\eta \left( \eta +2\right) +1}. \end{aligned}$$

Depending on the sign of $\eta (\eta +2)$ we distinguish two cases:

• (a) If $\eta \ge 0$ or $\eta \le -2$, then $g(x_0)\le \eta (\eta +2)+n=(\eta +1)^2+n-1$ and assumption (32) results in $g(x_0)\le (C+1)^2+n-1$.

• (b) If $-2\le \eta \le 0$, then $|g(x_0)|\le n\frac{|x_0|^2}{||x_0|^2-|(x_0)_1|^2|}\le n c'_{\Omega }$. In the last inequality we also used the first part of assumption (31).

Therefore we conclude that $g$ is bounded by a constant $c(\mathrm{diam} (\Omega ), n, c_{\Omega }, c'_{\Omega })$ for all $x\in \overline{B_r}$. Then assertion (33) follows immediately from (12) and (13). $\square$

3 The proof of Lemma 2

The first part of the proof is similar to the one of Lemma 1 (see also Granlund [11, Section 2] and Granlund–Marola [12, Lemma 2.6]). Let $k\ge 0$ and define $\psi (x)\!=\!\max \{\phi (u^1(x))\!-k, 0\}$ for $x\in B_r$ with for $\overline{B_{r}}\subset \Omega$. Take $\xi \in C_{0}^{\infty }(\Omega )$ and define a test function $\eta (x)=\psi (x)(\phi '(u^1(x)))^{p-1}\xi ^p(x)$. Then

$$\begin{aligned} \nabla \eta&= \phi '(u^1)(\phi '(u^1))^{p-1}\xi ^p\nabla u^1+(p-1)\psi \phi ''(u^1)(\phi '(u^1))^{p-2}\xi ^p\nabla u^1\nonumber \\&+p\psi (\phi '(u^1))^{p-1}\xi ^{p-1}\nabla \xi . \end{aligned}$$

Using $\nabla \eta$ in the first equation of $p$-harmonic system (3) we get the following inequality.

$$\begin{aligned}&\int _{B_r}|Du|^{p-2}|\nabla u^1|^2(\phi '(u^1))^{p}\xi ^{p}+(p-1)\int _{B_r}|Du|^{p-2}|\nabla u^1|^2\psi \phi ''(u^1) (\phi '(u^1))^{p-2}\xi ^{p} \nonumber \\&\quad \le p\int _{B_r}|Du|^{p-2}|\nabla u^1||\nabla \xi |\psi (\phi '(u^1))^{p-1}\xi ^{p-1}. \end{aligned}$$

(34)

We invoke property (4) of function $\phi$ and use it in the second integral on the left-hand side of the inequality. Since $\psi \ge 0$ in $B_r$ we can drop the aforementioned integral. The Young inequality applied to the integral on the right-hand side gives us the estimate (cf. inequality (25)):

$$\begin{aligned}&p\int _{B_r}|Du|^{p-2}|\nabla u^1||\nabla \xi |\psi (\phi '(u^1))^{p-1}\xi ^{p-1} \nonumber \\&\quad = p\int _{B_r}\left( |Du|^{(p-2)\frac{p-1}{p}}|\nabla u^1|^{2\frac{p-1}{p}}\phi '(u^1)^{p-1}\xi ^{p-1}\right) \left( |Du|^{\frac{p-2}{p}}|\nabla u^1|^{1-2\frac{p-1}{p}}\psi |\nabla \xi |\right) \nonumber \\&\quad \le (p-1)\epsilon \int _{B_r}|Du|^{p-2}|\nabla u^1|^2\phi '(u^1)^{p}\xi ^{p} +\epsilon ^{-p}\int _{B_r}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\psi ^p|\nabla \xi |^p. \end{aligned}$$

Upon choosing small enough value of $0<\epsilon <1$, e.g. $\epsilon =\frac{1}{2(p-1)}$ we include the first integral on the right-hand side of the inequality into the integral on the left-hand side of (34) and arrive at the following estimate (cf. (26) in Lemma 1):

$$\begin{aligned} \int _{B_r} |Du|^{p-2}|\nabla u^1|^2\phi '(u^1)^p\xi ^p\le c(p)\int _{B_r}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\psi ^p|\nabla \xi |^p. \end{aligned}$$

(35)

We choose function $\xi$ so that it satisfies: $\mathrm{supp}\,\xi \subset B_r$, $0\le \xi \le 1$, $\xi \equiv 1$ in $B_{(1-\delta )r}$ and $|\nabla \xi |\le \frac{c}{\delta r}$ in $B_r$. Note also, that by definition $\psi \equiv 0$ in $B_r\setminus A_{k, r}$. This and the choice of $\xi$ in (35) lead us to the following inequality (cf. estimate (26) in Lemma 1):

$$\begin{aligned} \int _{A_{k, (1-\delta )r}} |Du|^{p-2}|\nabla u^1|^2 \phi '(u^1)^{p} \le \frac{c(p)}{(\delta r)^p}\int _{A_{k, r}}\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\psi ^p. \end{aligned}$$

The discussion similar to the one for Cases 1 and 2 in Lemma 1 (cf. inequalities (28) and (29)) gives us assertions (14) and (15).

The proofs of supremum estimates (16) (in both cases) require extra attention due to the appearance of expression $g:=\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}$ under integrals (14) and (15) exploited in derivation of the supremum estimate. Note, that when the $p$-harmonic system (1) reduces to a single $p$-harmonic equation, then $|Du|=|\nabla u^1|$, and so $g\equiv 1$. In such a case we retrieve estimates from Lemma 2.6 in Granlund–Marola [12]. Here, we instead follow the method from a book by Giusti [10, Theorem 7.2] and adapt it to the vectorial case.

Using the notation analogous to [10], let $(1-\delta )r\le \sigma r\le \tau r\le r$. At this point the discussion splits into four cases: (1) $1 < p < 2$, (2) $p >$ 2 according to estimates (14) and (15), respectively, (3) $p$ $=$ 2 and (4) $p = n$.

Case 1: $1<p<2.$

Let $\eta \in C_{0}^{\infty }(B_{\frac{\sigma +\tau }{2}r})$ such that $\eta \equiv 1$ on $B_{\sigma r}$ and $|\nabla \eta |\le \frac{c}{(\tau -\sigma )r}$. Define $\xi (x)= \eta (x)\psi (x)$ for function $\psi$ as in the first part of the proof. By the Hölder and the Sobolev inequalities we get

$$\begin{aligned} \int _{A_{k,\sigma r}}\psi ^p \le \int _{A_{k,\sigma r}}\xi ^p \le \left( \int _{A_{k,\sigma r}} \xi ^{\frac{np}{n-p}}\right) ^{1-\frac{p}{n}} |A_{k,\sigma r}|^{\frac{p}{n}}\le c_S|A_{k,\tau r}|^{\frac{p}{n}}\int _{A_{k,\sigma r}} |\nabla \xi |^p. \end{aligned}$$

(36)

Using the definition of $\xi$ we compute that $\nabla \xi =\psi \nabla \eta +\eta \nabla \psi$ and hence

$$\begin{aligned} \int _{A_{k,\sigma r}}\psi ^p \le c_S|A_{k,\tau r}|^{\frac{p}{n}}\bigg (\int _{A_{k, \frac{\sigma +\tau }{2} r}} |\nabla \phi (u^1)|^p+ \frac{1}{(\tau -\sigma )^p r^p}\int _{A_{k, \frac{\sigma +\tau }{2}r}}\psi ^p\bigg ). \end{aligned}$$

(37)

Let $\alpha =\frac{p}{4}(2-p)$ and $\beta =p(1-\frac{p}{4})$. Then, the Young inequality applied with exponents $\frac{2-p}{\alpha }=\frac{4}{p}$ and its conjugate $\left( \frac{2-p}{\alpha }\right) '=\frac{2-p}{2-p-\alpha }=\frac{4}{4-p}$ gives us the following:

$$\begin{aligned} \int \limits _{A_{k, \frac{\sigma +\tau }{2}r}}\!\!\!\!|\nabla \phi (u^1)|^p&= \int \limits _{A_{k, \frac{\sigma +\tau }{2}r}} \left( \,|\nabla \phi (u^1)|^{p-\beta }\left( \frac{|\nabla u^1|}{|Du|}\right) ^{\alpha }\,\right) \left( \,|\phi '(u^1)|^{\beta }|Du|^\beta \left( \frac{|\nabla u^1|}{|Du|}\right) ^{\beta -\alpha }\,\right) \nonumber \\&\le \frac{p}{4} \int \limits _{A_{k, \frac{\sigma +\tau }{2}r}}\left( \frac{|\nabla u^1|}{|Du|}\right) ^{2-p}|\nabla \phi (u^1)|^p\nonumber \\&\quad +(1-\frac{p}{4})\int \limits _{A_{k, \frac{\sigma +\tau }{2}r}} \left( \frac{|\nabla u^1|}{|Du|}\right) ^{\frac{2p}{4-p}}|\phi '(u^1)|^p |Du|^p. \end{aligned}$$

(38)

In order to estimate further (38) for $p$ in the given range we appeal to Tolksdorf’s estimate [20] and note that $\left( \frac{|\nabla u^1|}{|Du|}\right) ^{\frac{2p}{4-p}}<1$. Then

$$\begin{aligned}&\int _{\!A_{k,\!\frac{\sigma +\tau }{2}r}}\!\!\! \left( \!\frac{|\nabla u^1|}{|Du|}\!\right) ^{\frac{2p}{4-p}}\!\!|\phi '(u^1)|^p |Du|^p\nonumber \\&\quad \le C(p,n)\,\Vert \phi '\Vert ^p_{L^{\infty }(\!B_{\frac{\sigma +\tau }{2}r}\!)}\left( \!\frac{2}{(\sigma +\tau )r}\!\right) ^n\! \left( |B_{(\sigma +\tau )r}|+\Vert Du\Vert ^p_{L^p(B_{(\sigma +\tau )r})}\right) |A_{k, \frac{\sigma +\tau }{2}r}|. \end{aligned}$$

We use this inequality in the second integral on the right-hand side of (38). Moreover, we observe that under assumptions on $\sigma$ and $\tau$ it holds that $\frac{\sigma +\tau }{2}\le \tau$ and thus $A_{k, \frac{\sigma +\tau }{2}r}\subset A_{k, \tau r}$. We apply estimate (14) with $r:=\tau r$ and $\delta$ such that $(1-\delta )r\tau :=\frac{\sigma +\tau }{2}r$ in the first integral on the right-hand side of (38). In a consequence, estimate (37) takes the following form:

$$\begin{aligned}&\int _{A_{k,\sigma r}}\!\!\!\!\psi ^p \le C(p, n, c_S)|A_{k,\tau r}|^{\frac{p}{n}} \left( \frac{2^p}{(\tau -\sigma )^pr^p} \int _{A_{k, \tau r}}\!\!\!\!\!\psi ^p + \Vert \phi '\Vert ^p_{L^{\infty }(B_{\frac{\sigma +\tau }{2}r})}\right. \nonumber \\&\quad \left. \times \frac{2^n}{(\sigma +\tau )^nr^n} (r^n+\Vert Du\Vert ^p_{L^p(B_{2r})})|A_{k, \frac{\sigma +\tau }{2}r}| +\frac{1}{(\tau -\sigma )^pr^p}\int _{A_{k, \frac{\sigma +\tau }{2}r}}\psi ^p\right) .\qquad \end{aligned}$$

(39)

If $h<k$, then

$$\begin{aligned} (k-h)^p|A_{k,\tau r}|\le \int _{A_{h,\tau r}} (\phi (u^1)-h)^p\quad \hbox { and }\quad \int _{A_{k,\tau r}} (\phi (u^1)-k)^p \le \int _{A_{h,\tau r}} (\phi (u^1)-h)^p. \end{aligned}$$

Denote

$$\begin{aligned} C=C(p, n, c_S)\left[ 1+\Vert \phi '\Vert ^p_{L^{\infty }(B_{r})}(r^n+ \Vert Du\Vert ^p_{L^p(B_{2r})})\right] . \end{aligned}$$

Using these in (39) we obtain the following estimate:

$$\begin{aligned}&\int _{A_{k,\sigma r}}\!(\phi (u^1)-k)^p\le \frac{C|A_{k,\tau r}|^{\frac{p}{n}}}{(\sigma -\tau )^pr^p} \left( \int _{A_{h,\tau r}} (\phi (u^1)-h)^p+ \frac{(\sigma -\tau )^pr^p}{(\sigma +\tau )^nr^n} |A_{k, \tau r}|\right) \nonumber \\&\le \frac{C}{(k-h)^{\frac{p^2}{n}}(\sigma -\tau )^pr^p}\left( \int _{A_{h,\tau r}} (\phi (u^1)-h)^p\right) ^{1+\frac{p}{n}} \left[ 1+\frac{1}{(k-h)^p}\frac{(\sigma -\tau )^pr^p}{(\sigma +\tau )^nr^n}\right] . \end{aligned}$$

(40)

We are in a position to use the iteration scheme as in Lemma 7.1 in Giusti book [10]. Indeed, for some $d>0$, to be determined later, let us consider the following quantities:

$$\begin{aligned}&k_i:=2d(1-2^{-i}),\quad \hbox {for }i=0,1,\ldots , \qquad k=k_{i+1},\quad h=k_{i},\quad k-h=\frac{d}{2^i}\nonumber \\&\sigma _i:= \delta +(1-\delta )2^{-i},\quad \hbox {for }i=0,1,\ldots , \qquad \sigma =\sigma _{i+1},\qquad \tau =\sigma _i. \end{aligned}$$

Hence

$$\begin{aligned} \tau -\sigma =(1-\delta )2^{-i-1},\qquad \tau +\sigma =2\delta +(1-\delta )3\cdot 2^{-i-1}. \end{aligned}$$

Finally, let

$$\begin{aligned} \phi _{i}:=\int _{A_{k_i,\sigma _{i}}}(\phi (u^1)-k_i)^p. \end{aligned}$$

With this notation inequality (40) reads:

$$\begin{aligned} \phi _{i+1}\le \frac{C}{(1-\delta )^pr^p}d^{-\frac{p^2}{n}}(1+d^{-p}r^{p-n})2^{pi(1+p/n)}\, \phi _{i}^{1+\frac{p}{n}},\quad \hbox {for }i=0,1,\ldots . \end{aligned}$$

The claim of the second part of Lemma 2 for $1<p<2$ follows from [10, Lemma 7.1], cf. the proof of Theorem 7.2 in [10]. Indeed, we set that for some $a>0$,

$$\begin{aligned} 1+d^{-p}r^{p-n}\le a. \end{aligned}$$

By taking $B=2^{p(1+p/n)}, \alpha =\frac{p}{n}$ and $c=\frac{Ca}{(1-\delta )^pr^p}d^{-\frac{p^2}{n}}$ we verify that the assumption of [10, Lemma 7.1] that $\phi _{0}\le c^{-1/\alpha }B^{-1/(\alpha ^2)}$ leads to the following conditions:

$$\begin{aligned} d^p\ge \frac{(Ca)^{\frac{n}{p}}}{((1-\delta )r)^{n}}\int _{B_r}(\phi (u^1))^p\quad \hbox { and }\quad (a-1)d^p\ge r^{p-n}, \end{aligned}$$

as $A_{0, r}=B_r$. Thus, by taking e.g. $a=2$ we get that the above inequalities for

$$\begin{aligned} d^p=\frac{C^{\frac{n}{p}}}{((1-\delta )r)^{n}}\int _{B_r}(\phi (u^1))^p+r^{p-n} \end{aligned}$$

and the claim follows, since by [10, Lemma 7.1] $\lim _{i\rightarrow \infty }\phi _i=0$ and so $\sup _{B_{\delta r}} \phi (u^1)\le 2d$.

Case 2: $2<p<n.$

We proceed similarly to the previous case. Upon using estimate (15) in (37) together with the fact that $\frac{|Du|}{|\nabla u^1|}$ is bounded by (8) and our assumptions, we obtain that

$$\begin{aligned} \int _{A_{k,\sigma r}}(\phi (u^1)-k)^p \le \frac{C_{\sup }}{((\sigma -\tau )r)^p}\left( \frac{1}{(k-h)^p}\int _{A_{h,\tau r}} (\phi (u^1)-h)^p\right) ^{1+\frac{p}{n}}. \end{aligned}$$

(41)

Constant $C_{\sup }=c(p, n, c_S)(\frac{\Vert Du\Vert _{L^p(B_{2r})}}{c}r^{-\frac{n}{p}})^{\frac{n}{p}(p-2)}$. The reasoning similar to the previous case gives us the claim for $2<p<n$. However, in this case we get the homogeneous estimate (19).

Case 3: $p=2.$

If $p=2$ and $n>2$, then we follow the proof for $2<p<n$ and since $\left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\equiv 1$, assumption (18) and discussion at (38) are not needed and we obtain (19) with constant $C=c(p, n, c_S)$ in (20).

If $p=2$ and $n=2$, then in (36) we use a variant of the Sobolev inequality (see e.g. Corollary 1.57 in Malý–Ziemer [16]) and get

$$\begin{aligned} \int _{A_{k,\sigma r}}\psi ^2 \le \int _{A_{k,\sigma r}}\xi ^2 \le |A_{k,\tau r}| \int _{A_{k,\sigma r}} |\nabla \xi |^2. \end{aligned}$$

This leads to the estimate similar to (41) and (19), while the resulting constant $C$ depends on $p, n$ and $c_S$.

Case 4: $p=n.$

As in the previous case, we use [16, Corollary 1.57] and obtain the following

$$\begin{aligned} \int _{A_{k,\sigma r}}\psi ^n \le \int _{A_{k,\sigma r}}\xi ^n \le |A_{k,\tau r}| \int _{A_{k,\sigma r}} |\nabla \xi |^n. \end{aligned}$$

We follow the proof for the case of $2<p<n$ and get $C_{\sup }=c(n)\left( \frac{\Vert Du\Vert _{L^n(B_{2r})}}{cr}\right) ^{n-2}$.

Hence, the proof of Lemma 2 is completed. $\square$

4 The proof of Theorem

In the proof of Theorem we use the doubling property of the Lebesgue measure. This means, that for any ball $B_R\subset \mathbb {R}^n$ it holds that $\mathcal {L}^n(B_{2R})\le C \mathcal {L}^n(B_{R})$, where $C=2^n$. Below, we also appeal to the $(1,p)$-Poincaré inequality: if $v\in W^{1,p}_{loc}(\Omega )$, then

$$\begin{aligned} \int _{B_r}|v-v_{B_r}|^p\le Cr^p \int _{B_r}|\nabla v|^p, \end{aligned}$$

where $v_{B_r}$ denotes the mean value of $v$ over the ball $B_r$ and $C$ depends on $n$ and $p$.

Finally, the following auxiliary result is used in the proof of Theorem as well, see Theorem 4.20 in Adamowicz [1] and Appendix A.2 in Adamowicz [3].

Lemma 3

[3, Observation 2] Let $u\in W^{1,p}(\Omega ,{\mathbb R}^n)$ be a $p$-harmonic mapping in the domain $\Omega \subset \mathbb {R}^n$. If for some $u^i$, $i=1,\ldots , n$ there exists $k\in {\mathbb R}$ such that $u^i\,\le \,k$ on $\partial \Omega$, then $u^i\,\le \,k$ in $\Omega$.

Proof of Theorem

Our approach is based on Theorem 1.3 in Granlund–Marola [12]. However, the fact that now we are in the setting of mappings instead of scalar functions requires new type of the de Giorgi estimates (cf. Lemmas 1 and 2). Moreover, the dependence of those estimates on $\Vert Du\Vert _{L^p}$ and $\Vert \phi '\Vert _{L^{\infty }}$ requires additional efforts and caution.

Define the following function (keeping in mind that the exact value of $r_3$ will be determined later):

$$\begin{aligned} \phi (t):=-\log \left( \frac{M(r_3)-t+\epsilon }{M(r_3)-M(r_1)+\epsilon }\right) \quad \hbox {for }t\in (-\infty , M(r_3)), \end{aligned}$$

(42)

for any but fixed $\epsilon >0$. It is easy to check that $\phi$ is $C^2$ convex and satisfies conditions (4) and (5). Indeed, by computations we see that

$$\begin{aligned} \phi '(t)=\frac{1}{M(r_3)-t+\epsilon }>0\quad \hbox { and }\quad \phi ''(t)=\frac{1}{(M(r_3)-t+\epsilon )^2} \quad \hbox { in }(-\infty , M(r_3)). \end{aligned}$$

(43)

Furthermore, since $\phi (u^1)<0$ on $B_{r_1}$, it holds that function $\psi :=\max \{\phi (u^1),0\}$ satisfies $\psi \equiv 0$ on $B_{r_1}$, thus also the mean value of $\psi$ vanishes, $\psi _{B_{r_1}}=0$. This together with the $(1,p)$-Poincaré inequality and the doubling property of the Lebesgue measure (with doubling constant $2^n$) implies

$$\begin{aligned} \mathop {\int \!\!\!\!\!~\!\!\!-}\nolimits _{B_{\frac{r+r_2}{2}}}\psi ^p&\le \mathop {\int \!\!\!\!\!~\!\!\!-}\nolimits _{B_{\frac{r+r_2}{2}}}\big |\psi -\psi _{B_{(r+r_2)/2}}+\big (\psi _{B_{(r+r_2)/2}}-\psi _{B_{r_1}}\big )\big |^p \nonumber \\&\le 2^p\mathop {\int \!\!\!\!\!~\!\!\!-}\nolimits _{B_{\frac{r+r_2}{2}}}\big |\psi -\psi _{B_{(r+r_2)/2}}\big |^p +2^{p+n}\bigg (\mathop {\int \!\!\!\!\!~\!\!\!-}\nolimits _{B_{\frac{r+r_2}{2}}}\big |\psi -\psi _{B_{(r+r_2)/2}}\big |\bigg )^p\nonumber \\&\le C(p, n)\bigg (\frac{r+r_2}{2}\bigg )^p\mathop {\int \!\!\!\!\!~\!\!\!-}\nolimits _{B_{\frac{r+r_2}{2}}} |\nabla \psi |^p. \end{aligned}$$

(44)

We consider two cases.

Case 1: $1<p<2.$ Recall the supremum estimate (16) from Lemma 2:

$$\begin{aligned} \big (\sup _{B_{(1-\delta )R}} \phi (u^1)\big )^p \le \frac{C_{\sup }}{(\delta R)^n}\int _{B_{R}}\phi (u^1)^p+R^{p-n}, \end{aligned}$$

where $C_{\sup }$ is constant in (17), cf. statement of Lemma 2. We apply this estimate with $R=(r+r_2)/2$ and $0<\delta =(r_2-r)/(r_2+r)<$1, then use the definition of function $\psi$ and Poincaré-type estimate (44).

$$\begin{aligned} \big (\sup _{B_{r}} \phi (u^1)\big )^p&\le C_{\sup }\left( \frac{2}{r_2-r}\right) ^n\int _{B_{\frac{r+r_2}{2}}}\phi (u^1)^p + \left( \frac{r+r_2}{2}\right) ^{p-n}\nonumber \\&\le C_{\sup }\left( \frac{r+r_2}{r_2-r}\right) ^n\left( \frac{2}{r+r_2}\right) ^n\int _{B_{\frac{r+r_2}{2}}} \psi ^p +\left( \frac{r+r_2}{2}\right) ^{p-n}\nonumber \\&\le C_{\sup }\left( \frac{2cr_2}{cr_2-r_1}\right) ^n \bigg (\frac{r+r_2}{2}\bigg )^{p-n}\int _{B_{\frac{r+r_2}{2}}} |\nabla \phi (u^1)|^p +\left( \frac{r+r_2}{2}\right) ^{p-n}\nonumber \\&\le C_{\sup }\bigg (\frac{2}{c+1}\bigg )^{p-n}\frac{1}{r_2^{n-p}}\int _{B_{\frac{r+r_2}{2}}} |\nabla \phi (u^1)|^p+\bigg (\frac{2}{c+1}\bigg )^{p-n}\frac{1}{r_2^{n-p}}. \end{aligned}$$

(45)

In the last step we included expression with $r_1, r_2, c$ into the constant $C_{\sup }$. In order to use estimate (12) in (45) we need to use first the Young inequality with exponents $\frac{1}{\alpha }=1+\frac{1}{\epsilon }$ and $\beta =1+\epsilon$, for some $\epsilon \in (0,1)$:

$$\begin{aligned} \int _{B_{\frac{r+r_2}{2}}} |\nabla \phi (u^1)|^p&= \int _{B_{\frac{r+r_2}{2}}} \left( \frac{|\nabla u^1|}{|Du|}\right) ^{(2-p)\alpha }|\nabla \phi (u^1)|^{p\alpha } \left( \frac{|\nabla u^1|}{|Du|}\right) ^{(p-2)\alpha }|\nabla \phi (u^1)|^{p(1-\alpha )}\nonumber \\&\le \int _{B_{\frac{r+r_2}{2}}} \left( \frac{|\nabla u^1|}{|Du|}\right) ^{2-p}|\nabla \phi (u^1)|^p +\int _{B_{\frac{r+r_2}{2}}} \left( \frac{|Du|}{|\nabla u^1|}\right) ^{(2-p)\epsilon }|\nabla \phi (u^1)|^p. \end{aligned}$$

(46)

The second integral on the right-hand side of (46) can be easily estimated as follows:

$$\begin{aligned} \int _{B_{\frac{r+r_2}{2}}} \left( \frac{|Du|}{|\nabla u^1|}\right) ^{(2-p)\epsilon }|\nabla \phi (u^1)|^p \le \Vert \phi '\Vert ^p_{L^{\infty }(B_{r_2})} \Vert Du\Vert ^p_{L^p(B_{r_2})}. \end{aligned}$$

We use (12) from Lemma 1 with with $r_2$ and $\delta =\frac{r_2-r}{2r_2}$ together with properties of radii (21). Then estimate (46) for the $p$-energy of $\phi (u^1)$ takes the following form:

$$\begin{aligned} \int _{B_{\frac{r+r_2}{2}}} |\nabla \phi (u^1)|^p&\le \left( \frac{2c}{cr_2-r_1}\right) ^p r_2^{n-p}+\Vert \phi '\Vert ^p_{L^{\infty }(B_{r_2})} \Vert Du\Vert ^p_{L^p(B_{r_2})}. \end{aligned}$$

The expression on the right-hand side is similar to the one in $C_{\sup }$. We use this observation and the last inequality in estimate (45) and obtain the following bound, which we in turn estimate using properties of function $\phi$, see (43):

$$\begin{aligned}&\big (\sup _{B_{r}} \phi (u^1)\big )^p \le C_{\sup } \frac{2^{p-n}}{(c+1)^{p-n}}\frac{1}{r_2^{n-p}} \left( C(p, r_1, r_2, c)r_2^{n-p}+\Vert \phi '\Vert ^p_{L^{\infty }(B_{r_2})} \Vert Du\Vert ^p_{L^p(B_{r_2})}\right) \nonumber \\&\qquad +\frac{2^{p-n}}{(c+1)^{p-n}}\frac{1}{r_2^{n-p}} \nonumber \\&\quad \!\le \! \frac{C}{r_2^{n-p}}\left\{ \left( 1\!+\!\Vert \phi '\Vert ^p_{L^{\infty }(B_{r_2})}\left( r_2^n\!+\! \Vert Du\Vert ^p_{L^p(B_{2r_2})}\right) \right) \left( r_2^{n-p}\!+\!\Vert \phi '\Vert ^p_{L^{\infty }(B_{r_2})} \Vert Du\Vert ^p_{L^p(B_{r_2})}\!+\!1\right) \right\} \nonumber \\&\quad \le C \left( 1+\Vert \phi '\Vert ^p_{L^{\infty }(B_{r_2})}\left( r_2^n+ \Vert Du\Vert ^p_{L^p(B_{2r_2})}\right) \right) ^2\max \{1, r_2^{p-n}\}\nonumber \\&\quad \le C \left( 1+\frac{1}{|M(r_3)-M(r_2)|^{2p}}\left( r_2^{2n}+ \Vert Du\Vert ^{2p}_{L^p(B_{2r_2})}\right) \right) \max \{1, r_2^{p-n}\}. \end{aligned}$$

(47)

Here $C=C(p, n, c, c_S, r_1, r_2)$. By the weak maximum principle in Lemma 3, it holds that $M(r_2)<M(r_3)$ for a non-constant $u^1$ and some $r_3>r_2$. The continuity of $u$ implies that maxima $M(r_2)$ and $M(r_3)$ are attained at some points $x_3\in S_{r_3}$ and $x_2\in S_{r_2}$. By the mean value theorem and the Tolksdorf estimate we get that

$$\begin{aligned} |u^1(x_2)-u^1(x_0)| \le \sup _{B_{r_2}}|Du|\,|x_2-x_0| \le C(p,n) (r_2^\frac{1}{p}+\Vert Du\Vert _{L^{p}(B_{2r_2})})r_2^{1-\frac{n}{p}}. \end{aligned}$$

We now appeal to growth condition (22), to obtain the following estimate:

$$\begin{aligned} |M(r_3)-M(r_2)|=|u^1(x_3)-u^1(x_2)|&\ge |u^1(x_3)-u^1(x_0)|-|u^1(x_2)-u^1(x_0)|\nonumber \\&\ge C r_3^{\alpha }-C(p,n) (r_2^\frac{1}{p}+\Vert Du\Vert _{L^{p}(B_{2r_2})})r_2^{1-\frac{n}{p}}. \end{aligned}$$

We use this inequality on the right-hand side of (47) and notice that by taking sufficiently large $r_3$, for instance such that

$$\begin{aligned} Cr_3^{\alpha }\ge (1+r_2^{2n}+\Vert Du\Vert ^{2p}_{L^{p}(B_{2r_2})})^{1/(2p)}+C(p,n)(r_2^\frac{1}{p}+\Vert Du\Vert _{L^{p}(B_{2r_2})})r_2^{1-\frac{n}{p}} \end{aligned}$$

we get that the right-hand side of (47) can now be estimated by $A:=C(p, n, c, c_S, r_1, r_2) \max \{1, r_2^{p-n}\}$. Observe that $r_3$ depends on $n, p$, $r_2$ and $\Vert Du\Vert _{L^{p}(B_{2r_2})}$, but not on $r$.

Case 2: $2\le p<n.$ We start from the estimate similar to (45). Namely, the supremum estimate (19) leads to the following inequality:

$$\begin{aligned} \big (\sup _{B_{r}} \phi (u^1)\big )^p \le C_{\sup }\bigg (\frac{2}{c+1}\bigg )^{p-n}\frac{1}{r_2^{n-p}}\int _{B_{\frac{r+r_2}{2}}} |\nabla \phi (u^1)|^p. \end{aligned}$$

We use Lemma 1 with $r_2$ and $\delta =\frac{r_2-r}{2r_2}$ together with constant (20) from Lemma 2 and properties of radii (21) to obtain the following inequalities:

$$\begin{aligned} \big (\sup _{B_{r}} \phi (u^1)\big )^p&\le C_{\sup }\bigg (\frac{2}{c+1}\bigg )^{p-n}\frac{1}{r_2^{n-p}}\int _{B_{\frac{r+r_2}{2}}} |\nabla \phi (u^1)|^p\nonumber \\&\le C_{\sup }\bigg (\frac{2}{c+1}\bigg )^{p-n}\frac{1}{r_2^{n-p}}\left( \frac{2}{r_2-r}\right) ^p\int _{B_{r_2}} \left( \frac{|Du|}{|\nabla u^1|}\right) ^{p-2}\nonumber \\&\le c(p,n,c_S)\left( \frac{\Vert Du\Vert _{L^p(B_{r+r_2})}}{c_1\left( \frac{r+r_2}{2}\right) ^\frac{n}{p}}\right) ^{\frac{n}{p}(p-2)} \left( \frac{2}{r_2-r_1/c}\right) ^p\frac{1}{r_2^{n-p}} \left( \frac{\Vert Du\Vert _{L^p(B_{2r_2})}}{c_1 r_2^\frac{n}{p}}\right) ^{\frac{n}{p}(p-2)}\nonumber \\&\le c(p, n, c_S, c, r_1, r_2)\left( \frac{\Vert Du\Vert _{L^p(B_{2r_2})}}{c_1}\right) ^{2\frac{n}{p}(p-2)}. \end{aligned}$$

(48)

Similarly to the previous case, we denote the constant on the right-hand side of the above inequality by A. Observe that for $p=2$ the constant on the right-hand side of (48) depends only on $p, n, c_S$ and $c, r_1, r_2$ due to (21).

Case 3: $p=n$. We discuss this case separately due to the importance of $n$-harmonic mappings in nonlinear analysis. As in the previous case we obtain the estimate similar to (45):

$$\begin{aligned} \big (\sup _{B_{r}} \phi (u^1)\big )^n \le C_{\sup }\left( \frac{2}{r_2-r}\right) ^n \int _{B_{\frac{r+r_2}{2}}} |\nabla \phi (u^1)|^n. \end{aligned}$$

The reasoning analogous to Case 2 gives us that

$$\begin{aligned} \big (\sup _{B_{r}} \phi (u^1)\big )^n \le c(n)\left( \frac{\Vert Du\Vert _{L^n(B_{2r_2})}}{c_1}\right) ^{2(n-2)}\left( \frac{r_1}{c}+r_2\right) ^2\frac{1}{r_2^{n-2}}. \end{aligned}$$

As in the previous cases, we denote the constant on the right-hand side of the above inequality by A.

We are now in a position to complete the proof of Theorem. By our assumptions $\phi$ is strictly increasing and so we have that

$$\begin{aligned} \log \left( \frac{M(r_3)-M(r)+\epsilon }{M(r_3)-M(r_1)+\epsilon }\right) =- \sup _{B_{r}} \phi (u^1)\ge -A. \end{aligned}$$

Note that in both cases: $1<p<2$ and $p\ge 2$, constant $A$ is independent of $\epsilon$. Upon simplifying this inequality we arrive at the following:

$$\begin{aligned} M(r)\le e^{-A}M(r_1)+(1-e^{-A})M(r_3)+(1-e^{-A})\epsilon . \end{aligned}$$

Letting $\epsilon \rightarrow 0^+$ we reach the first assertion of theorem.

In order to prove the second assertion, we define a function

$$\begin{aligned} \phi (t)=-\log \left( \frac{t-m(r_3)+\epsilon }{m(r_1)-m(r_3)+\epsilon }\right) \quad \hbox { for }t\in [m(r_3),\infty ). \end{aligned}$$

(49)

Similarly to the proof of the first assertion, we verify that $\phi$ is $C^2$ convex and satisfies conditions (4) and (5). Indeed, by computations we see that

$$\begin{aligned} \phi '(t)=-\frac{1}{t-m(r_3)+\epsilon }<0\quad \hbox { and }\quad \phi ''(t)=\frac{1}{(t-m(r_3)+\epsilon )^2} \quad \hbox { in }[m(r_3), \infty ). \end{aligned}$$

As in the case of maxima, we introduce a function $\psi :=\max \{\phi (u^1),0\}$ and show that $\psi \equiv 0$ on $B_{r_1}$. Then, following the steps of the proof for $M(r)$ we reach conclusion that

$$\begin{aligned} \log \left( \frac{u^1(x)-m(r_3)+\epsilon }{m(r_1)-m(r_3)+\epsilon }\right) =- \sup _{x\in B_{r}} \phi (u^1(x))\ge -A\qquad x\in B_r. \end{aligned}$$

Thus,

$$\begin{aligned} u^1(x)\ge e^{-A}m(r_1)+(1-e^{-A})m(r_3)-(1-e^{-A})\epsilon . \end{aligned}$$

The second assertion of the theorem now follows from taking $\epsilon \rightarrow 0^+$ and the proof of Theorem is completed. $\square$

Example 1

Let us comment on the existence of $p$-harmonic mappings satisfying assumptions (22) and (23) of Theorem. In order to do so, we employ radial $p$-harmonic mappings, cf. Lemma 1 and discussion in Sect. 2. Under the notation of Theorem, let us suppose that $1<p<2$ and $u=H(|x|)x$ is a radial $p$-harmonic mapping in $\Omega \subset \mathbb {R}^n$ such that $H(x_0)=0$ for $x_0\in \Omega$. Then (22) reads:

$$\begin{aligned} |H(|x|)||x_1|\ge C|x|^{\alpha } \quad \hbox { for }x\in \mathbb {R}^n\setminus B_{r_2}(x_0). \end{aligned}$$

For instance, let $\Omega$ be such that $\mathrm{dist}(\Omega , \{x\in \mathbb {R}^n: x_1=0\})>c$ and $H(|x|)=|x|^{\frac{2-p-n}{p-1}}+1$, then computations at (30) reveal that $u=H(|x|)x$ is $p$-harmonic in $\Omega$ and the above condition holds for $\alpha =\left| \frac{2-p-n}{p-1}\right|$, see also Adamowicz [1, Chapter 4.1] for further discussion on radial $p$-harmonics.

As for $p>2$ and assumption (23), recall that by the proof of Lemma 1 we have that

$$\begin{aligned} |\nabla u^1|^2=(H')^2x_1^2+2HH'\frac{x_1^2}{r}+H^2=\frac{x_1^2}{r^2}\left( \frac{H'}{H}+1\right) ^2+1-\frac{x_1^2}{r^2} \ge 1-\frac{x_1^2}{r^2}. \end{aligned}$$

From this we infer, that $|\nabla u^1|>c$ follows from $(1-\frac{x_1^2}{r^2})^{1/2}>c$, which in turn is satisfied e.g. if $\Omega$ is contained in cone-type domain $\{x\in \mathbb {R}^n: \frac{x_1^2}{r^2}<1-c\}$ provided that $0<c<1$.

Remark 2

For $1<p<2$, Theorem can be proven in a modified version with radius $r_3=r_2$ and without imposing the growth condition (22). Namely, for the proof of the first assertion we define function [cf. (42)]:

$$\begin{aligned} \phi (t):=-\log \left( \frac{M(r_3)-t+1}{M(r_3)-M(r_1)+1}\right) \quad \hbox {for }t\in (-\infty , M(r_3)), \end{aligned}$$

and the analogous function for the proof of the second assertion, cf. (49). Then $\Vert \phi '\Vert _{L^{\infty }}<1$ and estimate (47) simplifies as follows:

$$\begin{aligned} \big (\sup _{B_{r}} \phi (u^1)\big )^p \le C(p, n, c, C_S) \left( 1+r_2^{n}+ \Vert Du\Vert ^{2p}_{L^p(B_{2r_2})}\right) \max \{1, r_2^{p-n}\}. \end{aligned}$$

In such a case no additional growth restriction on $u^1$ is needed. However, the first assertion of Theorem takes the form:

$$\begin{aligned} M(r)\le C M(r_1)+(1-C)M(r_2)+1-C, \end{aligned}$$

where $C=C(n,p, c_S, r_1, r_2, c, \Vert Du\Vert ^{2p}_{L^p(B_{2r_2})})$.