Besov regularity for solutions of p-harmonic equations

1 Introduction

In this paper we study the extra fractional differentiability of weak solutions of the following nonlinear elliptic equations in divergence form:

where Ω ⊂ ℝ n , n ≥ 2 , is a domain, u : Ω → ℝ , F : Ω → ℝ n , and 𝒜 : Ω × ℝ n → ℝ n is a Carathéodory function with p - 1 growth. This means that there exist an exponent p ≥ 2 and constants ℓ , L , ν > 0 and 0 ≤ μ ≤ 1 such that

1. 〈 𝒜 ⁢ ( x , ξ ) - 𝒜 ⁢ ( x , η ) , ξ - η 〉 ≥ ν ⁢ ( μ 2 + | ξ | 2 + | η | 2 ) p - 2 2 ⁢ | ξ - η | 2 ,

2. | 𝒜 ⁢ ( x , ξ ) - 𝒜 ⁢ ( x , η ) | ≤ L ⁢ ( μ 2 + | ξ | 2 + | η | 2 ) p - 2 2 ⁢ | ξ - η | ,

3. | 𝒜 ⁢ ( x , ξ ) | ≤ ℓ ⁢ ( μ 2 + | ξ | 2 ) p - 1 2

for every ξ , η ∈ ℝ n and for a.e. x ∈ Ω .

When dealing with p-harmonic equations, regularity results usually refer to the auxiliary function

which takes into account the p-growth of the operator. Obviously, V p ⁢ ( D ⁢ u ) reduces to the gradient of the solution for p = 2 . Heuristically, thinking of the classical p-Laplace equation

and setting w = | V p ⁢ ( D ⁢ u ) | 2 , we have that Dw is a subsolution of a linear elliptic equation (for more details, we refer to [10, p. 272]). Therefore, the function V p ⁢ ( D ⁢ u ) , that takes into account the nonlinearity of the equation, is the natural substitute of the gradient of the solution when passing from the linear to the p-harmonic setting.

It is well known that the Lipschitz continuity of the partial map x → 𝒜 ⁢ ( x , ⋅ ) is a sufficient condition for the higher differentiability of the solutions when the right-hand side of the equation is sufficiently regular (we refer again to [10] for an exhaustive treatment). Also, it is clear that no extra differentiability can be expected for solutions, even if F is smooth, unless some differentiability is assumed on the x-dependence of 𝒜 .

Recent developments show that the Lipschitz regularity of the partial map x → 𝒜 ⁢ ( x , ⋅ ) can be weakened in a W 1 , n assumption on the coefficients, both in the linear and in the nonlinear setting, in order to get higher differentiability of the solution of integer order. In this direction, in [9, 19, 20], the higher differentiability of the function V p ⁢ ( D ⁢ u ) is obtained from a pointwise condition on 𝒜 that is equivalent to the W 1 , n regularity of the map x → 𝒜 ⁢ ( x , ⋅ ) . More precisely, it is assumed that there exists a non negative function g ∈ L loc n ⁢ ( Ω ) such that

for almost every x , y ∈ Ω and every ξ ∈ ℝ n . Related results concerning the planar Beltrami equation [3] and minimizers of non uniformly convex functionals [6, 7, 8] can also be found.

It turns out that the higher differentiability of the solutions can also be analyzed in the case of fractional Sobolev regularity of the coefficients. We mention previous contributions made in [4, 2, 5] for the case of planar Beltrami systems, and [16, 17] for higher dimensional results with not necessarily linear growth. Closer to the subject of the present paper, and assuming that 𝒜 ⁢ ( x , ξ ) has linear growth with respect to the gradient variable and enjoys either a Triebel–Lizorkin or a Besov–Lipschitz smoothness (roughly speaking enjoys a fractional differentiability property) with respect to the x-variable, it is proven in [1] that the fractional differentiability of 𝒜 ⁢ ( x , ⋅ ) transfers to the gradient of the solution with no losses in the order of differentiation.

The aim of this paper is to extend the results of [1] to the case of p-harmonic type operators with p ≥ 2 . More precisely, we will show that a fractional differentiability assumption for the operator 𝒜 with respect to the x-variable yields a fractional differentiability for the solutions. In this case, the fractional differentiability of 𝒜 ⁢ ( x , ⋅ ) transfers to V p ⁢ ( D ⁢ u ) .

Our first result concerns the case of Triebel–Lizorkin coefficients, i.e., we assume that there exists a function g ∈ L loc n α ⁢ ( Ω ) such that

for almost every x , y ∈ Ω , and every ξ ∈ ℝ n .

See Section 2 for the definition of B p , q α and the meaning of locally. It is worth mentioning that there is a jump between (1.2) and (1.3) when stated in terms of the Triebel–Lizorkin scale F p , q α . The jump appears in the q index, when the order of differentiation becomes integer. Indeed, condition (1.2) fully describes equations with coefficients in the Sobolev space W 1 , n , that is, the Triebel–Lizorkin space F n , 2 1 . In contrast, condition (1.3), for 0 < α < 1 , says that 𝒜 ⁢ ( x , ⋅ ) belongs to F n / α , ∞ α . More explanations about this can be found in [15, Remark 3.3] and the references therein.

The proof of Theorem 1.1 does not seem to work when the coefficients are assumed to belong to F n / α , q α for finite values of q. As in the linear case (see [1]), the Besov setting fits better in this context. To be precise, given 0 < α < 1 and 1 ≤ q ≤ ∞ , we assume that there exists a sequence of measurable non-negative functions g k ∈ L n α ⁢ ( Ω ) such that

and at the same time the following holds:

1. | 𝒜 ⁢ ( x , ξ ) - 𝒜 ⁢ ( y , ξ ) | ≤ | x - y | α ⁢ ( g k ⁢ ( x ) + g k ⁢ ( y ) ) ⁢ ( μ 2 + | ξ | 2 ) p - 1 2

for each ξ ∈ ℝ n , and almost every x , y ∈ Ω such that 2 - k ≤ | x - y | < 2 - k + 1 . We will shortly write then that ( g k ) k ∈ ℓ q ⁢ ( L n α ) . If 𝒜 ⁢ ( x , ξ ) = a ⁢ ( x ) ⁢ | ξ | p - 2 ⁢ ξ and Ω = ℝ n , then (A4) says that a belongs to B n / α , q α , see [15, Theorem 1.2].

Under (A4), we are able to deal with non-homogeneous equations and we prove that the extra differentiability of the solutions is related to the regularity of the datum and of the coefficients, both measured in the Besov scale. More precisely, we have the following result.

The parameter μ in assumption (A1) plays a very important role. When μ > 0 , the equation is non-degenerate elliptic while the case μ = 0 corresponds to degenerate cases. For instance, a model case for μ > 0 is given by

while a typical degenerate problem is the weighted p-Laplace equation

for some coefficient ν ≤ a ⁢ ( x ) ≤ ℓ . In the degenerate case, the ellipticity assumption (A1) is lost when | ξ | approaches zero, and the estimates worsen even in the classical theory (see [22]). Actually, in this case we are not able to prove an extra fractional differentiability of the function V p ⁢ ( D ⁢ u ) completely analogous to our previous theorem. Instead, due to the degeneracy μ = 0 , we have the following weaker result in the sense that the differentiability of the datum F still transfers to the function V p ⁢ ( D ⁢ u ) , but with a loss in the order of differentiation, even assuming the datum in a Besov space slightly smaller than B 2 , q β . More precisely we have the following theorem.

Note that, for p = 2 , Theorems 1.2 and 1.3 both recover [1, Theorem 3] at the energy space and in the case α = β . Actually, when dealing with equations with linear growth, the natural degree of integrability of the gradient of the solutions as well as of their extra α fractional Hajlasz gradients is 2. Therefore, the higher fractional differentiability results at the energy space are those proving that Du belongs to B 2 , q α or F 2 , q α . In [1], extra fractional differentiability results for equations with linear growth have been established also in spaces different from the natural ones, i.e., it has been proven that Du belongs to B s , q α and F s , q α for some s ≠ 2 sufficiently close to 2.

All our theorems rely on the basic fact that the Besov spaces B n / α , q α and the Triebel–Lizorkin space F n / α , ∞ α continuously embed into the VMO space of Sarason (e.g., [11, Proposition 7.12]). Linear equations with VMO coefficients are known to have a nice L p theory (see [12] for n = 2 or [13] for n ≥ 2 ). A first nonlinear growth counterpart was found in [14] for 𝒜 ⁢ ( x , ξ ) = 〈 A ⁢ ( x ) ⁢ ξ , ξ 〉 p - 2 2 ⁢ A ⁢ ( x ) ⁢ ξ , 2 ≤ p ≤ n , see also [17, 18]. For proving Theorems 1.1, 1.2 and 1.3, we shall use a result proved in [1] (see Theorem 2.5 in Section 2.2 below), and combine it with the Sobolev type embedding for Besov Lipschitz spaces to obtain the higher integrability of the gradient of the solutions of equation (1.1). Such higher integrability allows us to estimate the difference quotient of order α of the gradient of the solutions that yields their Besov type regularity.

The paper is structured as follows. In Section 2 we give some preliminaries on Harmonic Analysis. In Section 3 we prove Theorem 1.1, and in Section 4 we prove Theorems 1.2 and 1.3.

2 Notations and preliminary results

In this paper we follow the usual convention and denote by c a general positive constant that may vary on different occasions, even within the same line of estimates. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts. The norm we use on ℝ n will be the standard euclidean one and it will be denoted by | ⋅ | . In particular, for the vectors ξ, η ∈ ℝ n , we write 〈 ξ , η 〉 for the usual inner product and | ξ | := 〈 ξ , ξ 〉 1 2 for the corresponding euclidean norm.

In what follows, B ⁢ ( x , r ) = B r ⁢ ( x ) = { y ∈ ℝ n : | y - x | < r } will denote the ball centered at x of radius r. We shall omit the dependence on the center and on the radius when no confusion arises.

For the auxiliary function V p , defined for all ξ ∈ ℝ n as

where μ ≥ 0 and p ≥ 1 are parameters, we record the following estimate (see the proof of [10, Lemma 8.3]).

Noticing now that for p ≥ 2 , one has

and combining this with Lemma 2.1, we find that there exists a constant c > 0 such that

for every ξ , η ∈ ℝ n .

3 Proof of Theorem 1.1

We first prove that if (1.3) is satisfied, then 𝒜 has the locally uniform VMO property (2.2). The proof goes exactly as that of [1, Lemma 17], concerning the case of an operator with linear growth. We report it here for the sake of completeness.