Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 317–323

A REGULARITY PROPERTY OF p-HARMONIC FUNCTIONS

Luigi Greco and Anna Verde

Università degli Studi di Napoli “Federico II” Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Via Cintia, I-80126 Napoli, Italy greco@matna2.dma.unina.it; averde@matna3.dma.unina.it

Abstract. The paper is concerned with the 𝒜 -harmonic equation

div[⟨G(x)∇u, ∇u⟩^{(p−2)/2}G(x)∇u] = 0

where 1 < p < ∞ and G is a positive definite matrix whose entries are in L^{∞} ∩ VMO. We show that for every r > 1, very weak solutions of class W_{loc}^{1,r} actually belong to W_{loc}^{1,p} and are solutions in the distributional sense, provided p is sufficiently close to 2.

0. Introduction

We consider the familiar 𝒜 -harmonic equation

(0.1)  div 𝒜 (x, ∇u) = 0

in an open set Ω ⊂ R^{n}, with 𝒜 = 𝒜 (x, ξ) verifying usual conditions, in particular the growth condition |𝒜 (x, ξ)| ≈ |ξ|^{p−1} , with p > 1. A distributional solution to (0.1) is a function u ∈ W_{loc}^{1,p}(Ω) such that

(0.2)  ∫_{Ω}⟨𝒜 (x, ∇u), ∇φ⟩ dx = 0

for all φ ∈ C_{0}^{∞}(Ω). Hereafter, ⟨ , ⟩ denotes the scalar product of vectors in R^{n}. Of course, (0.2) extends to arbitrary φ ∈ W^{1,p}(Ω) with compact support. The p-integrability of ∇u is not required for (0.2) to be meaningful, but it is a natural assumption because it is used in studying regularity of solutions. Actually, properties of solutions are often deduced by a suitable choice of test functions in (0.2), typically φ = λu, with λ a cut-off function. Here, we mention only the

1991 Mathematics Subject Classification: Primary 35J60, 35B65.

This paper has been written under the research program “Metodi di rilassamento e di omogeneizzazione nello studio dei materiali compositi” which is part of the project ’95 “Matematica per la tecnologia e la società”.

well-known higher integrability of ∇u proved first in [ME], see [G] and references therein. This result is achieved by the technique of reverse Hölder inequalities which are obtained by testing (0.2) with appropriate φ.

In the paper [IS], the notion of very weak solution is considered, relaxing the natural integrability assumption. A function u ∈ W_{loc}^{1,r}(Ω), with r ≥ max{1, p−1}, is called a very weak solution to (0.1) if it satisfies (0.2), for all φ ∈ C_{0}^{∞}(Ω).

It is interesting to know how far below the natural exponent can r be and still allow for a comprehensive theory of very weak solutions. The first result in this context was obtained in [IS]. They found exponents r_{1} , r_{2} verifying 1 < r_{1} < p < r_{2} and such that any solution u ∈ W_{loc}^{1,r_{1}} actually belongs to W_{loc}^{1,r_{2}} . Thus u is a solution in the distributional sense. Obviously, for very weak solutions (0.2) holds for all ϕ ∈ W^{1,r/(r−p+1)}(Ω) and we no longer can use λu as a test function. In [IS] a test function for (0.2) is produced using the Hodge decomposition. We mention that a different method for constructing test functions is proposed in [L].

In [KZ], the authors considered a particular case of (0.1), namely the p-Laplacian

div(|∇u|^{p−2}∇u) = 0.

Essentially, they observed that the exponent r_{1} in the result of [IS] may be taken arbitrarily close to 1, provided p is sufficiently close to 2. Here, we obtain this kind of result for the more general equation

(0.3)  div[⟨G(x)∇u, ∇u⟩^{(p−2)/2}G(x)∇u] = 0,

where G(x) is a definite positive symmetric matrix whose entries are bounded VMO functions. We use the Hodge decomposition which reduces our problem to linear (p = 2) elliptic equations with VMO coefficients, see [D], [IS2].

Remark. Equations of type (0.3) arise in quasiconformal geometry and nonlinear elasticity, see for instance [I]. These are variational equations for the energy functionals

ℰ [u] = ∫_{Ω}⟨G(x)∇u, ∇u⟩^{p/2} dx.

Here G: Ω → R^{n×n} is viewed as a metric tensor on Ω. In this sense our equation div 𝒜 (x, ∇u) = 0 is no other than a p-harmonic equation with respect to this metric. In a recent paper [IKM] the authors study mappings with BMO-distortion. It becomes clear that mappings whose distortion tensor G(x) is in VMO will play a central role in further developments.

During the publication of the paper, we learned that S. Zhou in a very recent preprint [Z] has given independently a result similar to our Theorem 1. We thank the referee for pointing this out.

A regularity property of p-harmonic functions 319

1. Hodge decomposition with VMO coefficients

For the definitions and basic properties of the classical spaces BMO and VMO we refer to [JN] and [Sa], where those spaces were introduced. Here we want only to recall that a BMO or VMO function a, defined, for example, on a cube Q, has an extension ã which is BMO or VMO, respectively, on the entire space R^{n}. Also, the extension can be made so that, if a is bounded, ã is bounded and ∥ã∥_{∞} ≤ C∥a∥_{∞,Q}. Therefore, it is not a restriction to assume that the functions in question are defined on R^{n}. For these questions, see e.g. [A].

Now, let Ω be a smooth open subset of R^{n}. In the familiar Hodge decomposition, one expresses a vector field ω ∈ L^{p}(Ω, R^{n}), 1 < p < ∞, as

(1.1)  ω = ∇ϕ + H,

where ϕ ∈ W_{0}^{1,p}(Ω) and H ∈ L^{p}(Ω, R^{n}) is a divergence free vector field. Moreover, the following estimate holds

(1.2)  ∥∇ϕ∥_{p} + ∥H∥_{p} ≤ C_{p}(Ω)∥ω∥_{p}.

This is done by solving the Poisson equation

Δϕ = div ω.

Next, in [I], [IS], a stability property for the Hodge decomposition is proved, which amounts to saying that if ω is a small perturbation of a gradient field, then the term H in (1.1) is small. More precisely, if ω = |∇u|^{ε}∇u, with u ∈ W_{0}^{1,r}(Ω), 1 < r < ∞ and ε is sufficiently small, say −1 < 2ε < r − 1, we have

(1.3)  ∥H∥_{r/(1+ε)} ≤ C_{r}(Ω)|ε|∥∇u∥_{r}^{1+ε}.

In this paper, we shall use a somewhat more general version of the Hodge decomposition, which is more suitable for our purposes. Let A = A(x) and B = B(x) be n × n matrices such that

|ξ|^{2} ≤ ⟨Aξ, ξ⟩ ≤ Λ|ξ|^{2}, |ξ|^{2} ≤ ⟨Bξ, ξ⟩ ≤ Λ|ξ|^{2}.

Also, we assume that the entries of the product matrix G = BA are in VMO. Then we can decompose a vector field ω ∈ L^{p}(Ω, R^{n}) as

(1.4)  ω = A∇ϕ + H,

where ϕ ∈ W_{0}^{1,p}(Ω), and H ∈ L^{p}(Ω, R^{n}) verifies

(1.5)  div BH = 0.

Note that estimate (1.2) still holds. Simply we solve the equation

div(G∇ϕ) = div(Bω)

and use the L^{p} -estimate for variational equations with VMO coefficients, see [D], [IS2]. Moreover, we have the following stability property in our decomposition.

Lemma 1.1. Under the above assumptions, if u ∈ W_{0}^{1,r}(Ω), 1 < r < ∞, −1 < 2ε < r − 1, we have

|A∇u|^{ε}A∇u = A∇ϕ + H

with ϕ ∈ W_{0}^{1,r/(1+ε)}(Ω) and H ∈ L^{r/(1+ε)}(Ω, R^{n}) verifying both (1.5) and (1.3).

Proof. We mimic the proof of stability in the case of the classical Hodge decomposition.

By the uniqueness of the decomposition (1.4) we can define a linear operator 𝒯 : L^{p}(Ω, R^{n}) → L^{p}(Ω, R^{n}), for all 1 < p < ∞, by the rule 𝒯 ω = H. Note that 𝒯 vanishes on vector fields of the form A∇u. Therefore, 𝒯 (|A∇u|^{ε}A∇u) can be written as a commutator

H = 𝒯 (|A∇u|^{ε}A∇u) − |𝒯 A∇u|^{ε}𝒯 A∇u.

Then we conclude the lemma using a result for commutators in [IS].

Remark. If A and B are equal to the identity matrix, we get the classical Hodge decomposition.

Remark. The regularity of Ω is required in order to apply L^{p} theory, compare with [D].

Remark. The following Hodge decomposition is also worth noting. Let a ∈ L^{∞} and u ∈ W^{1,r}(R^{n}); then

a|∇u|^{ε}∇u = ∇ϕ + H

with H ∈ L^{r/(1+ε)}(R, R^{n}) verifying div H = 0 and

∥H∥_{r/(1+ε)} ≤ C_{r}(∥a∥_{BMO} + ∥a∥_{∞} |ε|)∥∇u∥_{r}^{1+ε}.

In fact the operator 𝒯 defined above is a singular integral operator and we have

\begin{aligned}H = 𝒯 (a|∇u|^{ε}∇u) & = 𝒯 (a |∇u|^{ε}∇u) − a𝒯 (|∇u|^{ε}∇u) \\ & + a𝒯 (|∇u|^{ε}∇u) − a|𝒯 ∇u|^{ε}𝒯 ∇u. \end{aligned}

Therefore, we are in a position to use the result of [CRW] on the commutators of 𝒯 with multiplication by a BMO function.

2. The main results

Here we need to consider the nonhomogeneous counterpart of equation (0.3), that is,

(2.1)  div[⟨G(∇u + g), ∇u + g⟩^{(p−2)/2}G(∇u + g)] = div h,

on Ω, with g ∈ L^{r}(Ω, R^{n}) and h ∈ L^{r/(p−1)}(Ω, R^{n}) for some r > max{1, p − 1}. The matrix G = G(x) is assumed to be symmetric and verifying

⟨Gξ, ξ⟩ ≥ |ξ|^{2} for all ξ ∈ R^{n}.

The entries of G are in L^{∞} ∩ VMO.

Theorem 1. Under these assumptions, for each 1 < r < 2, there exist δ > 0 and C > 0 such that, if |p − 2| < δ then for all solutions u ∈ W_{0}^{1,r}(Ω) to (2.1) the following estimate holds:

(2.2)  ∫_{Ω} |∇u|^{r} dx ≤ C ∫_{Ω}(|g|^{r} + |h|^{r/(p−1)}) dx.

Here, δ and C depend on n, r, Λ, the VMO-modulus of A and Ω. In the case that Ω is a ball or a cube, δ and C can be found independent of Ω.

Proof. Let A be the positive square root of G, i.e. G = A^{2} . Then ⟨Aξ, ξ⟩ ≥ |ξ|^{2} , for all ξ ∈ R^{n} and equation (2.1) can be rewritten as

(2.3)  div[|A(∇u + g)|^{p−2}G(∇u + g)] = div h.

We have

(2.4)  ∫_{Ω} |∇u|^{r} ≤ ∫_{Ω}⟨|∇u|^{r−p}∇u, |A∇u|^{p−2}G∇u⟩.

Using Hodge decomposition, we write

(2.5)  |∇u|^{r−p}∇u = ∇ϕ + H

where ϕ ∈ W_{0}^{1,r/(r−p+1)}(Ω). Here H ∈ L^{r/(r−p+1)}(Ω, R^{n}) with ∥H∥_{r/(r−p+1)} ≤ C∥∇u∥_{r}^{r−p+1} and satisfies

div(GH) = 0.

Inserting (2.5) in (2.4) and using equation (2.3) yields

∫_{Ω} |∇u|^{r} ≤ ∫_{Ω}⟨∇ϕ + H, |A∇u|^{p−2}G∇u⟩
= ∫_{Ω}⟨H, |A∇u|^{p−2}G∇u⟩ + ∫_{Ω}⟨∇ϕ, h⟩
+ ∫_{Ω}⟨∇ϕ, |A∇u|^{p−2}G∇u − |A(∇u + g)|^{p−2}G(∇u + g)⟩
= I_{1} + I_{2} + I_{3}.

To estimate I_{1} , we write

I_{1} = ∫_{Ω}⟨H, |A∇u|^{p−2}G∇u⟩ = ∫_{Ω}⟨AH, |A∇u|^{p−2}A∇u⟩

and then decompose

|A∇w|^{p−2}A∇w = A∇ψ + K,

where ψ ∈ W_{0}^{1,r/(p−1)}(Ω), and K ∈ L^{r/(p−1)}(Ω, R^{n}) verifies

\begin{aligned}div AK & = 0 \\ ∥K∥_{r/(p−1)} & ≤ C|p − 2| ∥∇u∥_{r}^{p−1}. \end{aligned}

Hence

I_{1} = ∫_{Ω}⟨AH, A∇ψ + K⟩ = ∫_{Ω}⟨AH, A∇ψ⟩ + ∫_{Ω}⟨AH, K⟩
= ∫_{Ω}⟨∇ψ, GH⟩ + ∫_{Ω}⟨AH, K⟩ = ∫_{Ω}⟨H, AK⟩
= ∫_{Ω}⟨|∇u|^{r−p}∇u − ∇ϕ, AK⟩ = ∫_{Ω}⟨|∇u|^{r−p}∇u, AK⟩
≤ ∥A∥_{∞}∥K∥_{r/(p−1)}∥ |∇u|^{r−p+1}∥_{r/(r−p+1)} ≤ C|p − 2| ∥∇u∥_{r}^{r}.

The terms I_{2} and I_{3} can be estimated like in [IS]–[KZ]. In this way, we end up with the following estimate

{1 − C|p − 2| − θ}∥∇u∥_{r}^{r} ≤ C_{θ}(∥g∥_{r}^{r} + ∥h∥_{r/(p−1)}^{r/(p−1)})

where 0 < θ < 1 is arbitrary. This clearly concludes the proof.

Finally, we consider local solutions to (0.3). The technique of reverse Hölder inequalities as used in [IS] can be easily applied to conclude the following

Theorem 2. Let Ω be an open subset of R^{n} and G ∈ VMO(Ω, R^{n×n}) ∩ L^{∞}(Ω, R^{n×n}). For any 1 < r_{1} < 2 there exists δ > 0 such that, if |p−2| < δ and u ∈ W_{loc}^{1,r_{1}}(Ω) is a solution to (0.3), then u ∈ W_{loc}^{1,r_{2}}(Ω) for all exponents r_{2} < ∞.

Remark. That we need G to be in VMO is clear. Examples in [S] show that, even in the linear case, corresponding to p = 2, boundedness of G is not sufficient to obtain r_{1} close to 1 and r_{2} close to ∞.

References

[A] Acquistapace, P.: On BMO regularity for linear elliptic systems. - Ann. Mat. Pura Appl. (4) CLXI, 1992, 231–269.

[CRW] Coifman, R.R., R. Rochberg, and G. Weiss: Factorization theorems for Hardy spaces in several variables. - Ann. of Math. 103, 1976, 611–635.

[D] Di Fazio, G.: L^{p} estimates for divergence form elliptic equations with discontinuous coefficients. - Boll. Un. Mat. Ital. A 7, 1996, 409–420.

[G] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. - Annals of Math. Stud. 105, Princeton University Press, Princeton, NJ, 1983.

[I] Iwaniec, T.: p-harmonic tensors and quasiregular mappings. - Ann. of Math. 136, 1992, 589–624.

[IKM] Iwaniec, T., P. Koskela, and G. Martin: Mappings of BMO-distortion and Beltrami type operators. - To appear.

[IS1] Iwaniec, T., and C. Sbordone: Weak minima of variational integrals. - J. Reine Angew. Math. 454, 1994, 143–161.

[IS2] Iwaniec, T., and C. Sbordone: Riesz transforms and elliptic PDEs with VMO coefficients. - J. Analyse Math. 74, 1998, 183–212.

[JN] John, F., and L. Nirenberg: On functions of bounded mean oscillation. - Comm. Pure Appl. Math. 14, 1961, 415–426.

[KZ] Kinnunen, J., and S. Zhou: A note on a theorem by Iwaniec and Sbordone. - Electron. J. Differential Equations 25, 1997, 1–4.

[L] Lewis, J.: On very weak solutions of certain elliptic systems. - Comm. Partial Differential Equations 18:9–10, 1993, 1515–1537.

[ME] Meyers, N., and A. Elcrat: Some results on regularity for solutions of nonlinear elliptic systems and quasiregular functions. - Duke Math. J. 42:1, 1975, 121–136.

[S] Serrin, J.: Pathological solutions of elliptic equations. - Ann. Scuola Norm. Sup. Pisa 18, 1964, 385–387.

[Sa] Sarason, D.: Functions of vanishing mean oscillation. - Trans. Amer. Math. Soc. 207, 1975, 391–405.

[Z] Zhoun, S.: Very weak solutions of p-Laplacian type equations with VMO coefficients. - Research Report of Peking University, Beijing, China 43, 1998.

Received 23 July 1998