Three-spheres theorem for p -harmonic mappings

Let u = ( u 1 , . . . , u n ) be a p -harmonic mapping in a domain (cid:2) ⊂ R n for n ≥ 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


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 M(r ) ≤ log(r 2 /r ) log(r 2 /r 1 ) M(r 1 ) + log(r/r 1 ) log(r 2 /r 1 ) M(r 2 ).
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 R n , n ≥ 3 see e.g. Theorem 30 Communicated by L. Ambrosio.
T. Adamowicz  in [19] or spheres which need not be concentric, see Arakelian-Matevosyan [5]. The threecircles (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 pharmonic 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 ∞ 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 , . . . , u n ) ∈ W 1, p loc ( , R n ) is called a p-harmonic mapping if it is a solution to the following system of equations: div(|Du| p−2 Du) = 0, u = (u 1 , . . . , u n ) : where Du denotes the Jacobi matrix of u, i.e. Du = (∇u 1 , . . . , ∇u n ) T . Equivalently, this system can be written as follows: The p-harmonic system of equations is the Euler-Lagrange system of the associated p-Dirichlet energy functional In the weak form (2) reads 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 = 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 φ : R → R be a C 2 convex function such that there exists a subinterval I ⊂ R with properties: In what follows we will deal several times with the expression |Du|/|∇u 1 |, its reciprocal and integrals involving it. In particular, we will need some conditions to ensure that where B r is a ball of radius r . The case of 1 < p < 2 is easy, since then ( |∇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 The discrepancy between formulas (7) and (8) is a consequence of different nature of pharmonic mappings for 1 < p < 2 (singular case) and p > 2 (degenerate elliptic case). In fact (7) holds for 1 < p < ∞, 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: Suppose that a ball B r is centered at x 0 and that B 2r ⊂ . Furthermore, assume that for Then, by (9) we have: 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, . . . , n upon the necessary reformulations of results. Denote by 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
for p > 1. Assume, furthermore, that a convex function φ : I → R satisfies conditions (4) and (5). Then the following estimates hold for every ball B r ⊂ and 0 < δ < 1: If p ≥ 2, then Furthermore, for p > 2 the growth condition (10) ensures finitness of the last integral.
In the next result we study the behavior of φ(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 ≥ 0. Upon the above notation we define

Lemma 2 Let u 1 be the first coordinate function of a p-harmonic mapping u in a domain
Assume, furthermore, that conditions (4) and (5) hold for a convex function φ : I → R. Then the following estimates hold for every ball B r ⊂ and 0 < δ < 1: Furthermore, the following inequality holds for 1 < p < 2: where the constant . (17) If, additionally, there exists a positive constant c, such that then the following inequality holds for p > 2 and n > 2: sup with constant 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 ≤ n and u = (u 1 , . . . , u n ) : ⊂ R n → R n be a p-harmonic mapping. Consider three concentric balls centered at x 0 ∈ with radii 0 < r 1 < r < r 2 such that B 2r 2 ⊂ and for some fixed c. If 1 < p < 2, then let us assume that for a given α > 0, the coordinate function u 1 satisfies the following growth condition: If 2 < p < n, then let us assume instead that there is a positive constant c 1 such that Then there exist a constant C and a radius r 3 such that if B r 3 ⊂ , then the following inequalities hold: For 1 < p < 2 constant C depends on n, p, c S the constant in the Sobolev embedding theorem, r 1 , r 2 , c, α and Du L p (B 2r 2 ) , while r 3 > r 2 depends on p, n, r 2 , α, and Du L p (B 2r 2 ) . For 2 < p ≤ n constant C depends on n, p, c S , r 1 , r 2 , c, c 1 and Du 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 R n of the p-harmonic system (2) instead in the domain . In such a case one may neglect the assumption that B r 3 ⊂ .
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, . . . , 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 < ∞ 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 = 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. φ (u 1 (x)) > 0 for all x ∈ B r . Take a nonnegative function ξ ∈ C ∞ 0 (B r ) and define a test function We use ∇η in the first equation of p-harmonic system (3) and by using (4) together with (5) we obtain For some 0 < < 1, whose value will be determined later, we rewrite the right-hand side of (24) and apply the Young inequality: In the last inequality we also estimated p p−1 ≤ . Now, we use (25) in (24) and by taking e.g. = 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: Denote c( p) := p2 p+1 p−1 . Case 1: 1 < p < 2. The left-hand side of the above inequality can be written as follows.
If (5) (ii) holds, i.e. φ (u 1 (x)) < 0 for all x ∈ B r , then as a test function we take η(x) = (−φ (u 1 (x))) p−1 ξ p (x) for x ∈ B r and claims of the lemma follow the same way as previously.
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).
In such a case (28) reads: Similarly, for p > 2 the Uhlenbeck inequality (8) results in the following estimate.
(B) In fact estimate (26) gives rise to the following inequality for p > 1: We now turn to considering a special class of p-harmonic mappings, namely radial transformations in the form where H ∈ C 2 ( ) and |x| stands for the magnitude of x. For such mappings p-harmonic system (2) becomes a nonlinear second order ODE: 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] 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 |∇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 pharmonic mapping u in ⊂ R n , n > 1 for p > 1. Assume that a convex function φ : I → R satisfies conditions (4) and (5). Furthermore, assume the following: Then the following estimate holds for every ball B r ⊂ and 0 < δ < 1: Conditions (31) Hence . Let x 0 ∈ B r and consider the following cases.
Therefore we conclude that g is bounded by a constant c(diam( ), n, c , c ) for all x ∈ B r . Then assertion (33) follows immediately from (12) and (13).

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 ≥ 0 and define ψ(x) = max{φ(u 1 (x))− k, 0} for x ∈ B r with for B r ⊂ . Take ξ ∈ C ∞ 0 ( ) and define a test function Using ∇η in the first equation of p-harmonic system (3) we get the following inequality.
We invoke property (4) of function φ and use it in the second integral on the left-hand side of the inequality. Since ψ ≥ 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)): Upon choosing small enough value of 0 < < 1, e.g. = 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): We choose function ξ so that it satisfies: supp ξ ⊂ B r , 0 ≤ ξ ≤ 1, ξ ≡ 1 in B (1−δ)r and |∇ξ | ≤ c δr in B r . Note also, that by definition ψ ≡ 0 in B r \ A k,r . This and the choice of ξ in (35) lead us to the following inequality (cf. estimate (26) in Lemma 1): 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 := |Du| (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| = |∇u 1 |, and so g ≡ 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.
for function ψ as in the first part of the proof. By the Hölder and the Sobolev inequalities we get Using the definition of ξ we compute that ∇ξ = ψ∇η + η∇ψ and hence Let α = p 4 (2 − p) and β = p(1 − p 4 ). Then, the Young inequality applied with exponents 2− p α = 4 p and its conjugate 2− p α = 2− p 2− p−α = 4 4− p gives us the following: In order to estimate further (38) for p in the given range we appeal to Tolksdorf's estimate [20] and note that |∇u 1 | |Du| 2 p 4− p < 1. Then We use this inequality in the second integral on the right-hand side of (38). Moreover, we observe that under assumptions on σ and τ it holds that σ +τ 2 ≤ τ and thus A k, σ +τ 2 r ⊂ A k,τr . We apply estimate (14) with r := τr and δ such that (1 − δ)r τ := σ +τ 2 r in the first integral on the right-hand side of (38). In a consequence, estimate (37) takes the following form: If h < k, then Using these in (39) we obtain the following estimate: 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: Hence Finally, let With this notation inequality (40) reads: 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, By taking B = 2 p(1+ p/n) , α = p n and c =
We proceed similarly to the previous case. Upon using estimate (15) in (37) together with the fact that |Du| |∇u 1 | is bounded by (8) and our assumptions, we obtain that Constant C sup = c( p, n, c S )( 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).
If p = 2 and n > 2, then we follow the proof for 2 < p < n and since |Du| 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 This leads to the estimate similar to (41) and (19), while the resulting constant C depends on p, n and c S .
As in the previous case, we use [16,Corollary 1.57] and obtain the following We follow the proof for the case of 2 < p < n and get C sup = c(n) Hence, the proof of Lemma 2 is completed.

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 ⊂ R n it holds that L n (B 2R ) ≤ CL n (B R ), where C = 2 n . Below, we also appeal to the (1, p) 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 ∈ W 1, p ( , R n ) be a p-harmonic mapping in the domain ⊂ R n . If for some u i , i = 1, . . . , n there exists k ∈ R such that u i ≤ k on ∂ , then u i ≤ k in .
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 Du L p and φ L ∞ requires additional efforts and caution.
Define the following function (keeping in mind that the exact value of r 3 will be determined later): for any but fixed > 0. It is easy to check that φ is C 2 convex and satisfies conditions (4) and (5). Indeed, by computations we see that Furthermore, since φ(u 1 ) < 0 on B r 1 , it holds that function ψ := max{φ(u 1 ), 0} satisfies ψ ≡ 0 on B r 1 , thus also the mean value of ψ vanishes, ψ 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 We consider two cases. Case 1: 1 < p < 2. Recall the supremum estimate (16) from Lemma 2: where C sup is constant in (17), cf. statement of Lemma 2. We apply this estimate with R = (r + r 2 )/2 and 0 < δ = (r 2 − r )/(r 2 + r ) <1, then use the definition of function ψ and Poincaré-type estimate (44).
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 1 α = 1 + 1 and β = 1 + , for some ∈ (0, 1): The second integral on the right-hand side of (46) can be easily estimated as follows: We use (12) from Lemma 1 with with r 2 and δ = r 2 −r 2r 2 together with properties of radii (21). Then estimate (46) for the p-energy of φ(u 1 ) takes the following form: 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 φ, see (43) 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): 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 φ is strictly increasing and so we have that Note that in both cases: 1 < p < 2 and p ≥ 2, constant A is independent of . Upon simplifying this inequality we arrive at the following: Letting → 0 + we reach the first assertion of theorem. In order to prove the second assertion, we define a function Similarly to the proof of the first assertion, we verify that φ is C 2 convex and satisfies conditions (4) and (5). Indeed, by computations we see that As in the case of maxima, we introduce a function ψ := max{φ(u 1 ), 0} and show that ψ ≡ 0 on B r 1 . Then, following the steps of the proof for M(r ) we reach conclusion that The second assertion of the theorem now follows from taking → 0 + and the proof of Theorem is completed.