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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Besov regularity for solutions of p-harmonic equations

Albert Clop / Raffaella GiovaORCID iD: https://orcid.org/0000-0002-7632-4664 / Antonia Passarelli di NapoliORCID iD: https://orcid.org/0000-0002-3371-1090
Published Online: 2017-09-15 | DOI: https://doi.org/10.1515/anona-2017-0030

Abstract

We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., div𝒜(x,Du)=divF, when 𝒜 is a p-harmonic type operator, and under the assumption that x𝒜(x,ξ) belongs to the critical Besov–Lipschitz space Bn/α,qα. We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When divF=0, we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map x𝒜(x,ξ).

Keywords: Nonlinear elliptic equations; higher order fractional differentiability; Besov spaces; Triebel–Lizorkin spaces

MSC 2010: 35B65; 35J60; 42B37; 49N60

1 Introduction

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

div𝒜(x,Du)=divFin Ω,(1.1)

where Ωn, n2, is a domain, u:Ω, F:Ωn, and 𝒜:Ω×nn is a Carathéodory function with p-1 growth. This means that there exist an exponent p2 and constants ,L,ν>0 and 0μ1 such that

  • (A1)

    𝒜(x,ξ)-𝒜(x,η),ξ-ην(μ2+|ξ|2+|η|2)p-22|ξ-η|2,

  • (A2)

    |𝒜(x,ξ)-𝒜(x,η)|L(μ2+|ξ|2+|η|2)p-22|ξ-η|,

  • (A3)

    |𝒜(x,ξ)|(μ2+|ξ|2)p-12

for every ξ,ηn and for a.e. xΩ.

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

Vp(Du)=(μ2+|Du|2)p-24Du,

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

div(|Du|p-2Du)=0

and setting w=|Vp(Du)|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 Vp(Du), 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 x𝒜(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 x𝒜(x,) can be weakened in a W1,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 Vp(Du) is obtained from a pointwise condition on 𝒜 that is equivalent to the W1,n regularity of the map x𝒜(x,). More precisely, it is assumed that there exists a non negative function gLlocn(Ω) such that

|𝒜(x,ξ)-𝒜(y,ξ)||x-y|(g(x)+g(y))(μ2+|ξ|2)p-12(1.2)

for almost every x,yΩ and every ξn. 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 p2. 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 Vp(Du).

Our first result concerns the case of Triebel–Lizorkin coefficients, i.e., we assume that there exists a function gLlocnα(Ω) such that

|𝒜(x,ξ)-𝒜(y,ξ)||x-y|α(g(x)+g(y))(μ2+|ξ|2)p-12(1.3)

for almost every x,yΩ, and every ξn.

Theorem 1.1.

Let 0<α<1. Assume that A satisfies (A1)(A3), and that (1.3) holds. If uWloc1,p(Ω) is a weak solution of

div𝒜(x,Du)=0,(1.4)

then Vp(Du)B2,α, locally, and as a consequence DuBp,2αp, locally.

See Section 2 for the definition of Bp,qα 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 Fp,qα. 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 W1,n, that is, the Triebel–Lizorkin space Fn,21. In contrast, condition (1.3), for 0<α<1, says that 𝒜(x,) belongs to Fn/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 Fn/α,qα 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 1q, we assume that there exists a sequence of measurable non-negative functions gkLnα(Ω) such that

kgkLnα(Ω)q<,

and at the same time the following holds:

  • (A4)

    |𝒜(x,ξ)-𝒜(y,ξ)||x-y|α(gk(x)+gk(y))(μ2+|ξ|2)p-12

for each ξn, and almost every x,yΩ such that 2-k|x-y|<2-k+1. We will shortly write then that (gk)kq(Lnα). If 𝒜(x,ξ)=a(x)|ξ|p-2ξ and Ω=n, then (A4) says that a belongs to Bn/α,qα, 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.

Theorem 1.2.

Let uWloc1,p(Ω) be a weak solution of the equation

div𝒜(x,Du)=divF,(1.5)

under assumptions (A1)(A4), with μ>0. Then the implication

FB2,qβVp(Du)B2,qmin{α,β}

holds locally, provided that 0<β<1 and 1q2nn-2β .

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

𝒜(x,ξ)=a(x)(μ2+|ξ|2)p-22ξ,

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

𝒜(x,ξ)=a(x)|ξ|p-2ξ

for some coefficient νa(x). 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 Vp(Du) 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 Vp(Du), but with a loss in the order of differentiation, even assuming the datum in a Besov space slightly smaller than B2,qβ. More precisely we have the following theorem.

Theorem 1.3.

Let uWloc1,p(Ω) be a weak solution of the equation

div𝒜(x,Du)=divF,

under assumptions (A1)(A4), with μ=0. Let 0<α,β<1 and p=pp-1. Then the implication

FB2,qp/2βVp(Du)B2,qmin{α,βp2}

holds locally, provided that 1qp22nn-2β.

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 B2,qα or F2,qα. 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 Bs,qα and Fs,qα for some s2 sufficiently close to 2.

All our theorems rely on the basic fact that the Besov spaces Bn/α,qα and the Triebel–Lizorkin space Fn/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 Lp theory (see [12] for n=2 or [13] for n2). A first nonlinear growth counterpart was found in [14] for 𝒜(x,ξ)=A(x)ξ,ξp-22A(x)ξ, 2pn, 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 n will be the standard euclidean one and it will be denoted by ||. In particular, for the vectors ξ, ηn, we write ξ,η for the usual inner product and |ξ|:=ξ,ξ12 for the corresponding euclidean norm.

In what follows, B(x,r)=Br(x)={yn:|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 Vp, defined for all ξn as

Vp(ξ):=(μ2+|ξ|2)p-24ξ,

where μ0 and p1 are parameters, we record the following estimate (see the proof of [10, Lemma 8.3]).

Lemma 2.1.

Let 1<p< and 0μ1. There exists a constant c>0, depending only on n,p but not on μ>0, such that

c-1(μ2+|ξ|2+|η|2)p-22|Vp(ξ)-Vp(η)|2|ξ-η|2c(μ2+|ξ|2+|η|2)p-22

for any ξ, ηRn such that |ξ-η|0.

Noticing now that for p2, one has

|ξ-η|p=|ξ-η|2|ξ-η|p-2|ξ-η|2(|ξ|+|η|)p-2c|ξ-η|2(μ2+|ξ|2+|η|2)p-22,

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

|ξ-η|pc|Vp(ξ)-Vp(η)|2(2.1)

for every ξ,ηn.

2.1 Besov–Lipschitz spaces

Given hn and v:n, let τhv(x)=v(x+h) and Δhv(x)=v(x+h)-v(x). As in [21, Section 2.5.12], given 0<α<1 and 1p,q<, we say that v belongs to the Besov space Bp,qα(n) if vLp(n) and

vBp,qα(n)=vLp(n)+[v]B˙p,qα(n)<,

where

[v]B˙p,qα(n)=(n(n|v(x+h)-v(x)|p|h|αpdx)qpdh|h|n)1q<.

Equivalently, we could simply say that vLp(n) and Δhv|h|αLq(dh|h|n;Lp(n)). As usually, if one simply integrates for hB(0,δ) for a fixed δ>0, then an equivalent norm is obtained because

({|h|δ}(n|v(x+h)-v(x)|p|h|αpdx)qpdh|h|n)1qc(n,α,p,q,δ)vLp(n).

Similarly, we say that vBp,α(n) if vLp(n) and

[v]B˙p,α(n)=suphn(n|v(x+h)-v(x)|p|h|αpdx)1p<.

Again, one can simply take the supremum over |h|δ and obtain an equivalent norm. By construction, Bp,qα(n)Lp(n). One also has the following version of the Sobolev embeddings (a proof can be found in [11, Proposition 7.12], taking into account that Lr=Fr,20, with 1<r<+).

Lemma 2.2.

Suppose that 0<α<1.

  • (a)

    If 1<p<nα and 1qpα=:npn-αp , then there exists a continuous embedding Bp,qα(n)Lpα(n).

  • (b)

    If p=nα and 1q , then there exists a continuous embedding Bp,qα(n)BMO(n).

Given a domain Ωn, we say that v belongs to the local Besov space Bp,q,locα if φv belongs to the global Besov space Bp,qα(n) whenever φ belongs to the class 𝒞c(Ω) of smooth functions with compact support contained in Ω. The following lemma is an easy exercise.

Lemma 2.3.

A function vLlocp(Ω) belongs to the local Besov space Bp,q,locα if and only if

Δhv|h|αLq(dh|h|n;Lp(B))<

for any ball B2BΩ with radius rB. Here the measure dh|h|n is restricted to the ball B(0,rB) on the h-space.

Proof.

Let us fix a smooth and compactly supported test function φ. We have the pointwise identity

Δh(φv)(x)|h|α=v(x+h)Δhφ(x)|h|α+Δhv(x)|h|αφ(x).

It is clear that

|v(x+h)Δhφ(x)|h|α||v(x+h)|φ|h|1-α,

and therefore one always has Δhφ|h|αLq(dh|h|n;Lp(n)). As a consequence, we have the equivalence

φvBp,qα(n)Δhv|h|αφLq(dh|h|n;Lp(n)).

However, it is clear that Δhv|h|αφLq(dh|h|n;Lp(n)) for each φCc(Ω) if and only if the same happens for every φ=χB and every ball B2BΩ. The claim follows. ∎

As in [21, Section 2.5.10], we say that a function v:n belongs to the Triebel–Lizorkin space Fp,qα(n) if vLp(n) and

vFp,qα(n)=vLp(n)+[v]F˙αp,q(n)<,

where

[v]F˙αp,q(n)=(n(n|v(x+h)-v(x)|q|h|n+αqdh)pqdx)1p.

Equivalently, we could simply say that vLp(n) and Δhv|h|αLp(dx;Lq(dh|h|n)).

It turns out that Besov–Lipschitz and Triebel–Lizorkin spaces of fractional order α(0,1) can be characterized in pointwise terms. Given a measurable function v:n, a fractional α-Hajlasz gradient for v is a sequence (gk)k of measurable non-negative functions gk:n, together with a null set Nn, such that the inequality

|v(x)-v(y)||x-y|α(gk(x)+gk(y))

holds whenever k, and x,ynN are such that 2-k|x-y|<2-k+1. We say that (gk)q(;Lp(n)) if

(gk)kq(Lp)=(kgkLp(n)q)1q<.

Similarly, we write (gk)Lp(n;q()) if

(gk)kLp(q)=(n(gk(x))kq()pdx)1p<.

The following result was proven in [15].

Theorem 2.4.

Let 0<α<1, 1p< and 1q. Let vLp(Rn).

  • (i)

    One has vBp,qα(n) if and only if there exists a fractional α -Hajlasz gradient (gk)kq(;Lp(n)) for v . Moreover,

    vBp,qα(n)inf(gk)kq(Lp),

    where the infimum runs over all possible fractional α -Hajlasz gradients for v.

  • (ii)

    One has vFp,qα(n) if and only if there exists a fractional α -Hajlasz gradient (gk)kLp(n;q()) for v . Moreover,

    vFp,qα(n)inf(gk)kLp(q),

    where the infimum runs over all possible fractional α -Hajlasz gradients for v.

2.2 VMO coefficients in n

In this section, we recall a regularity result, proven in [1], that will be crucial in our proofs. Let n2 and let 𝒜:Ω×nn be a Carathéodory function such that assumptions (A1)(A3) hold. We also require a control on the oscillations, which is described as follows. Given a ball BΩ, let us denote

𝒜B(ξ)=B𝒜(x,ξ)𝑑x.

One can easily check that the operator 𝒜B(ξ) also satisfies assumptions (A1)(A3). Now set

V(x,B)=supξ0|𝒜(x,ξ)-𝒜B(ξ)|(μ2+|ξ|2)p-12

for xΩ and BΩ. If 𝒜 is given by the weighted p-laplacian, that is, 𝒜(x,ξ)=a(x)|ξ|p-2ξ, one obtains

V(x,B)=|a(x)-aB|,where aB=Ba(y)𝑑y,

and so any reasonable VMO condition on a(x) requires that the mean value of V(x,B) on B goes to 0 as |B|0. Our VMO assumption on general Carathéodory functions 𝒜 consists of a uniform version of this fact. Namely, we will say that x𝒜(x,ξ) is locally uniformly in VMO if for each compact set KΩ, we have that

limR0supr(B)<Rsupc(B)KBV(x,B)𝑑x=0.(2.2)

Here c(B) denotes the center of the ball B and r(B) its radius.

The following theorem, proved in [1], is a regularity result for weak solutions of p-harmonic equations with VMO coefficients.

Theorem 2.5.

Let 2pn and q>p. Assume that (A1)(A3) hold, and that xA(x,ξ) is locally uniformly in VMO. If uWloc1,p(Ω) is a weak solution of

div𝒜(x,Du)=divF,

with FLlocqp-1, then DuLlocq. Moreover, there exists a constant λ>1 such that the Caccioppoli inequality

B|Du|qC(n,λ,ν,,L,p,q)(1+1|B|qnλB|u|q+λB|F|qp-1)

holds for any ball B such that λBΩ.

2.3 Difference quotient

We recall some properties of the finite difference operator that will be needed in the sequel. We start with the description of some elementary properties that can be found, for example, in [10].

Proposition 2.6.

Let F and G be two functions such that F,GW1,p(Ω), with p1, and let us consider the set

Ω|h|:={xΩ:dist(x,Ω)>|h|}.

Then the following hold:

  • (1)

    ΔhFW1,p(Ω|h|) and Di(ΔhF)=Δh(DiF).

  • (2)

    If at least one of the functions F and G has support contained in Ω|h| , then

    ΩFΔhG𝑑x=-ΩGΔ-hF𝑑x.

  • (3)

    We have Δh(FG)(x)=F(x+h)ΔhG(x)+G(x)ΔhF(x).

The next result about the finite difference operator is a kind of an integral version of the Lagrange theorem.

Lemma 2.7.

If 0<ρ<R, |h|<R-ρ2, 1<p<+ and F,DFLp(BR), then

Bρ|ΔhF(x)|pdxc(n,p)|h|pBR|DF(x)|pdx.

Moreover,

Bρ|F(x+h)|pdxBR|F(x)|pdx.

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.

Lemma 3.1.

Let A:Ω×RnRn be a Carathéodory map such that (A1)(A3) hold. Assume that (1.3) is satisfied. Then A is locally uniformly in VMO, that is, (2.2) holds.

Proof.

We have

BV(x,B)𝑑x=Bsupξ0|𝒜(x,ξ)-𝒜B(ξ)|(μ2+|ξ|2)p-12dxBsupξ0B|𝒜(x,ξ)-𝒜(y,ξ)|(μ2+|ξ|2)p-12𝑑y𝑑xBsupξ0B(g(x)+g(y))|x-y|α𝑑y𝑑x=BB(g(x)+g(y))|x-y|α𝑑y𝑑x(BB(g(x)+g(y))nαdydx)αn(BB|x-y|nαn-αdydx)n-αn(1|B|Bgnα)αnC(α,n)|B|αn=C(n,α)(Bgnα)αν,

and thus (2.2) holds. ∎

Proof of Theorem 1.1.

Let us fix a ball BR such that B2RΩ, and consider a cut off function ηC0(BR), with η1 on BR/2, such that |η|cR. For small enough |h|, we set φ=Δ-h(η2Δhu) as a test function in equation (1.4). Using Proposition 2.6 (1), we obtain

𝒜(x,Du),Δ-hD(η2Δhu)𝑑x=0,

which is equivalent, by Proposition 2.6 (2), to the following equality:

Δh(𝒜(x,Du)),D(η2Δhu)𝑑x=0.(3.1)

We can write (3.1) as follows:

𝒜(x+h,Du(x+h))-𝒜(x+h,Du(x)),D(η2Δhu)𝑑x=𝒜(x,Du(x))-𝒜(x+h,Du(x)),D(η2Δhu)𝑑x,

and therefore

𝒜(x+h,Du(x+h))-𝒜(x+h,Du(x)),η2D(Δhu)𝑑x=-𝒜(x+h,Du(x+h))-𝒜(x+h,Du(x)),2ηηΔhu𝑑x   +𝒜(x,Du(x))-𝒜(x+h,Du(x)),η2D(Δhu)𝑑x   +𝒜(x,Du(x))-𝒜(x+h,Du(x)),2ηηΔhu𝑑x=:I1+I2+I3.

Using the ellipticity assumption (A1) in the left-hand side and Proposition 2.6 (1), the previous equality yields

ν(μ2+|Du(x)|2+|Du(x+h)|2)p-22|ΔhDu|2η2𝑑x|I1|+|I2|+|I3|.

By virtue of assumption (A2) and Young’s inequality, we have

|I1|2L(μ2+|Du(x)|2+|Du(x+h)|2)p-22|ΔhDu||η||η||Δhu|𝑑xε(μ2+|Du(x)|2+|Du(x+h)|2)p-22|ΔhDu|2η2𝑑x+C(ε,L)(μ2+|Du(x)|2+|Du(x+h)|2)p-22|η|2|Δhu|2𝑑x.

To estimate the integrals I2 and I3, we write p-12=p4+p-24, and then we use assumption (1.3) and Young’s inequality as follows:

|I2||h|α(g(x+h)+g(x))(μ2+|Du(x)|2)p-12|DΔhu|η2𝑑x=|h|α(g(x+h)+g(x))(μ2+|Du(x)|2)p4+p-24|DΔhu|η2𝑑xε(μ2+|Du(x)|2)p-22|DΔhu|2η2𝑑x+cε|h|2α(g(x+h)+g(x))2(μ2+|Du(x)|2)p2η2𝑑x

and

|I3|2|h|α(g(x+h)+g(x))(μ2+|Du(x)|2)p-12|η||η||Δhu|𝑑x=2|h|α(g(x+h)+g(x))(μ2+|Du(x)|2)p4+p-24|η||η||Δhu|𝑑xc(μ2+|Du(x)|2)p-22|η|2|Δhu|2𝑑x+c|h|2α(g(x+h)+g(x))2(μ2+|Du(x)|2)p2η2𝑑x.

Collecting the estimates of I1, I2 and I3, we obtain

ν(μ2+|Du(x)|2+|Du(x+h)|2)p-22|ΔhDu|2η2𝑑x2ε(μ2+|Du(x)|2+|Du(x+h)|2)p-22|ΔhDu|2η2𝑑x   +c(μ2+|Du(x)|2+|Du(x+h)|2)p-22|η|2|Δhu|2𝑑x   +c|h|2α(g(x+h)+g(x))2(μ2+|Du(x)|2)p2η2𝑑x.

Choosing ε=ν8 and reabsorbing the first integral in the right-hand side by the left-hand side, we obtain

3ν4(μ2+|Du(x)|2+|Du(x+h)|2)p-22|ΔhDu|2η2𝑑xc(μ2+|Du(x)|2+|Du(x+h)|2)p-22|η|2|Δhu|2𝑑x   +c|h|2α(g(x+h)+g(x))2(μ2+|Du(x)|2)p2η2𝑑x.(3.2)

Using Hölder’s inequality and the first estimate of Lemma 2.7 in the first integral on the right-hand side, and the fact that suppηBR, we get

BR(μ2+|Du(x)|2+|Du(x+h)|2)p-22|η|2|Δhu|2𝑑xcR2(BR|Δhu|pdx)2p(BR(μ2+|Du(x)|2+|Du(x+h)|2)p2)p-2pdxc|h|2R2BR+|h|(μ2+|Du(x)|2)p2.

Inserting the previous estimate in (3.2), we obtain

3ν4BR(μ2+|Du(x)|2+|Du(x+h)|2)p-22|ΔhDu|2η2𝑑xc|h|2R2BR+|h|(μ2+|Du(x)|2)p2𝑑x+c|h|2αBR(g(x+h)+g(x))2(μ2+|Du(x)|2)p2𝑑x.

Using Lemma 2.1 in the left-hand side of the previous estimate yields

BR|Δh(Vp(Du))|2η2dxc|h|2R2BR+|h|(μ2+|Du(x)|2)p2dx+c|h|2αBR(g(x+h)+g(x))2(μ2+|Du(x)|2)p2dx.(3.3)

We now divide both sides of inequality (3.3) by |h|2α, and use the fact that ηχBR/2 to obtain

BR/2|Δh(Vp(Du))|h|α|2𝑑xcBR(g(x+h)+g(x))2(μ2+|Du(x)|2)p2𝑑x+c|h|2-2αR2BR+|h|(μ2+|Du|2)p2𝑑x,

where c=c(ν,L,p,n). The homogeneity of the equation together with Theorem 2.5 yields that DuLlocs(Ω) for every finite s>1, and so, in particular, DuLnpn-2α(BR). Therefore, by Hölder’s inequality,

BR(g(x+h)+g(x))2(μ2+|Du(x)|2)p2𝑑x(BR(g(x+h)+g(x))nα)2αn(BR(μ2+|Du(x)|2)np2(n-2α)𝑑x)n-2αnc(BR+|h|gnα)2αn(BR(μ2+|Du(x)|2)np2(n-2α)𝑑x)n-2αn,

and so we conclude

BR/2|Δh(Vp(Du))|h|α|2dxc(BR+|h|gnα)2αn(BR(μ2+|Du|2)np2(n-2α))n-2αn+c|h|2-2αR2BR+|h|(μ2+|Du|2)p2dx.

Since the above inequality holds for every h, we can take suprema over hB(0,δ) for some δ<R and obtain

sup|h|<δBR/2|Δh(Vp(Du))|h|α|2dxc(1+(B2Rgnα)2αn)(B2R(μ2+|Du|2)np2(n-2α))n-2αn.

In particular, this tells us that Vp(Du)B2,α, locally. ∎

4 Proof of Theorems 1.2 and 1.3

We first prove that if 𝒜 satisfies (A1)(A4), then it is locally uniformly in VMO. This result is a straightforward extension to the case of operators 𝒜 with (p-1)-growth of [1, Lemma 18], which refers to operator with linear growth. We report it here for the sake of completeness.

Lemma 4.1.

Let A be such that (A1)(A4) hold. Then A is locally uniformly in VMO, that is, (2.2) holds.

Proof.

Given a point xΩ, let us write Ak(x)={yΩ:2-k|x-y|<2-k+1}. We have

BV(x,B)𝑑x=Bsupξ0|𝒜(x,ξ)-𝒜B(ξ)|(μ2+|ξ|2)p-12dxBsupξ0B|𝒜(x,ξ)-𝒜(y,ξ)|(μ2+|ξ|2)p-12𝑑y𝑑x=Bsupξ01|B|kBAk(x)|𝒜(x,ξ)-𝒜(y,ξ)|(μ2+|ξ|2)p-12𝑑y𝑑x1|B|2kBBAk(x)|x-y|α(gk(x)+gk(y))dydx.

The last term above is bounded by

(1|B|2kBBAk(x)|x-y|nαn-αdydx)n-αn(1|B|2kBBAk(x)(gk(x)+gk(y))nαdydx)αn=III

The first sum is very easy to handle, since

I=(1|B|2kBBAk(x)|x-y|nαn-αdydx)n-αnC(n,α)|B|αn.

Concerning the second, we see that

IIc(1|B|2k|BAk(x)|Bgk(x)nαdx)αnc(1|B|2k(Bgk(x)nαdx)αqn)αnnαq(1|B|2k|BAk(x)|αqαq-n)αnαq-nαq=c1|B|2q(kgkLnα(B)q)1q1|B|2(αn-1q)(k|BAk(x)|αqαq-n)αnαq-nαq1|B|2q(kgkLnα(B)q)1q1|B|2(αn-1q)C(n,α,q)|B|αn=C(n,α,q)|B|-αn(kgkLnα(B)q)1q,

thus

BV(x,B)dxIIIC(n,α,q)(kgkLnα(B)q)1q.

In order to get the VMO condition, it just remains to prove that

limr0supxK(kgkLnα(B(x,r))q)1q=0

on every compact set KΩ. To this end, we can fix r>0 small enough and observe that the function xgkq(Lnα(B(x,r))) is continuous on the set {xΩ:d(x,Ω)>r}, as a uniformly converging series of continuous functions. As a consequence, there exists a point xrK (at least for small enough r>0) such that

supxKgkq(Lnα(B(x,r)))=gkq(Lnα(B(xr,r))).

Now, from gkLnα(B(x,r))gkLnα(B(xr,r)) and since this belongs to q, we can use dominated convergence to say that

limr0gkq(Lnα(B(xr,r)))=(klimr0(B(xr,r)gknα)qαn)1q.

Each of the limits on the term on the right-hand side are equal to 0, since the points xr cannot escape from the compact set K as r0. This completes the proof. ∎

Proof of Theorem 1.2.

We first assume that αβ. Let us fix a ball BR such that B2RΩ and a cut off function ηC0(BR), with η1 on BR/2, such that |η|cR. For small enough h, we set φ=Δ-h(η2Δhu) as a test function in equation (1.5). Using Proposition 2.6 (1), we obtain

BR𝒜(x,Du),Δ-hD(η2Δhu)𝑑x=BRF,Δ-hD(η2Δhu)𝑑x,

which, by Proposition 2.6 (2), is equivalent to

Δh(𝒜(x,Du)),D(η2Δhu)𝑑x=Δh(F),D(η2Δhu)𝑑x.(4.1)

We can write (4.1) as follows:

𝒜(x+h,Du(x+h))-𝒜(x+h,Du(x)),D(η2Δhu)𝑑x=𝒜(x,Du(x))-𝒜(x+h,Du(x)),D(η2Δhu)𝑑x+Δh(F),D(η2Δhu)dx,

and therefore

𝒜(x+h,Du(x+h))-𝒜(x+h,Du(x)),η2D(Δhu)𝑑x=-𝒜(x+h,Du(x+h))-𝒜(x+h,Du(x)),2ηηΔhu𝑑x   +𝒜(x,Du(x))-𝒜(x+h,Du(x)),η2D(Δhu)𝑑x   +𝒜(x,Du(x))-𝒜(x+h,Du(x)),2ηηΔhu𝑑x   +Δh(F),η2D(Δhu)𝑑x+Δh(F),2ηηΔhu𝑑x=:I1+I2+I3+I4+I5.

Now, using assumption (A1) in the left-hand side and Proposition 2.6 (1), the previous equality yields

νBR(μ2+|Du(x)|2+|Du(x+h)|2)p-22|Δh(Du)|2η2𝑑x|I1|+|I2|+|I3|+|I4|+|I5|.

The integrals I1, I2 and I3 can be estimated exactly as we did in the proof of Theorem 1.1. After doing this for 2-k|h|<2-k+1, the above inequality reads as

3ν4(μ2+|Du(x)|2+|Du(x+h)|2)p-22|Δh(Du)|2η2𝑑xc|h|2R2B2R(μ2+|Du(x)|2)p2𝑑x+c|h|2αBR(gk(x+h)+gk(x))2(μ2+|Du(x)|2)p2𝑑x+|I4|+|I5|.(4.2)

Now we estimate I4 and I5. By using Young’s inequality, we get

|I4|BRη2|Δh(F)||Δh(Du)|𝑑xcεBR|Δh(F)|2dx+εBR|Δh(Du)|2η2dx=cε|h|2βBR|Δh(F)|h|β|2dx+εμp-2BR|Δh(Du)|2μp-2η2dxcε|h|2βBR|Δh(F)|h|β|2dx+εμp-2BRη2|ΔhDu|2(μ2+|Du(x)|2+|Du(x+h)|2)p-22dx,

since μ>0, where ε>0 will be chosen later. Similarly,

|I5|cRBR|Δh(F)||Δhu|dxc|h|2R2B2R|Du|2+c|h|2βBR|Δh(F)|h|β|2dx,

where we used the first estimate in Lemma 2.7. Inserting the estimates of I4 and I5 in (4.2), we have

3ν4BR(μ2+|Du(x)|2+|Du(x+h)|2)p-22|Δh(Du)|2η2𝑑xc|h|2R2B2R(μp+|Du(x)|2+|Du(x)|p)𝑑x+c|h|2αBR(gk(x+h)+gk(x))2(μ2+|Du(x)|2)p2𝑑x   +cε|h|2βBR|Δh(F)|h|β|2dx+εμp-2BRη2|Δh(Du)|2(μ2+|Du(x)|2+|Du(x+h)|2)p-22dx.

Choosing ε=3μp-2ν8, reabsorbing the last integral in the right-hand side of the previous estimate by the left-hand side, using Lemma 2.1, the fact that η1 on BR/2, and dividing both side by |h|2γ, γ=min{α,β}=α, we conclude that

BR/2|Δh(Vp(Du))|h|γ|2dxc|h|2-2γR2B2R(μp+|Du(x)|2+|Du(x)|p)dx+c|h|2(α-γ)BR(gk(x+h)+gk(x))2(μ2+|Du(x)|2)p2dx+c|h|2(β-γ)μp-2BR|Δh(F)|h|β|2dx,

and so

(BR/2|Δh(Vp(Du))|h|γ|2dx)12c|h|1-γR(B2R(μp+|Du(x)|2+|Du|p)dx)12+c|h|β-γμp-22(BR|Δh(F)|h|β|2dx)12+c|h|α-γ(BR(gk(x+h)+gk(x))2(μ2+|Du(x)|2)p2𝑑x)12,

where c=c(ν,L,p,n). Taking the Lq norm with the measure dh|h|n restricted to the ball B(0,δ) on the h-space, we obtain that

(Bδ(BR/2|Δh(Vp(Du))|h|γ|2dx)q2dh|h|n)1qc(Bδ|h|q(1-γ)(B2R(μp+|Du(x)|2+|Du(x)|p)dx)q2dh|h|n)1q+cμp-22(Bδ(BR|Δh(F)|h|β|2dx)q2dh|h|n)1q+c(Bδ(BR(gk(x+h)+gk(x))2(μ2+|Du(x)|2)p2dx)q2dh|h|n)1q=:J1+J2+J3,

where c=c(ν,L,R,p,n,δ). For the estimate of J1, one can easily check that

J1=c(B2R(μp+|Du(x)|2+|Du(x)|p)dx)12(Bδ|h|(1-γ)q-ndh)1qc(n)(B2R(μp+|Du(x)|2+|Du(x)|p)𝑑x)12(0δρ(1-γ)q-1𝑑ρ)1q=c(γ,n,q,δ)(B2R(μ2+|Du(x)|2)p2𝑑x)12,

since γ<1. The term J2 can be controlled by the B2,qβ-seminorm of F, which is finite thanks to our assumption. Before estimating J3, recall that, by virtue of assumption (A4), for every k, one has gk2Ln2α. Also, since q2β*, by Lemma 2.2, we have that FLloc2nn-2β, and so, by Theorem 2.5, we have that DuLloc2n(p-1)n-2β. Now, from p2 and αβ, we easily see that

2n(p-1)n-2βnpn-2α,(4.3)

and so we can proceed as follows. We write the Lq norm in the integral J3 in polar coordinates, assuming without loss of generality that δ=1, so hB(0,1) if and only if h=rξ for some 0r<1 and some ξ in the unit sphere Sn-1 on n. We denote by dσ(ξ) the surface measure on Sn-1. We bound the term J3 by

01Sn-1(BR(gk(x+rξ)+gk(x))2(μ2+|Du(x)|2)p2dx)q2dσ(ξ)drr=k=0rk+1rkSn-1(BR(gk(x+rξ)+gk(x))2(μ2+|Du(x)|2)p2𝑑x)q2𝑑σ(ξ)drrk=0rk+1rkSn-1(τrξgk+gk)(μ2+|Du|2)p4L2(BR)qdσ(ξ)drr,

where we set rk=12k. Now, from (4.3), we see that DuLlocnpn-2α. This, together with the assumption gkLnα, gives us that

(τrξgk+gk)(μ2+|Du|2)p4L2(BR)(μ2+|Du|2)12Lnpn-2α(BR)p2(τrξgk+gk)Lnα(BR).

On the other hand, we note that for each ξSn-1 and rk+1rrk,

(τrξgk+gk)Lnα(BR)gkLnα(BR-rkξ)+gkLnα(BR)2gkLnα(λB),

where λ=2+1R. Hence,

J3C(n,α,q)(μ2+|Du|2)12Lnpn-2α(BR)p2{gk}kq(Lnα(λB)).

Summarizing, we have

Δh(Vp(Du))|h|γLq(dh|h|n;L2(BR/2))C(1+DuL2(BR)+DuLp(BR)p2)+Cμp-2Δh(F)|h|βLq(dh|h|n;L2(2B))+CDuLnpn-2α(2B)p2{gk}kq(Lnα(λB)),

with C=C(μ,α,β,p,q,n,ν,L). Lemma 2.3 now guarantees that Vp,μ(Du)B2,qγ, locally, and this concludes the proof.

When α>β, we have the embedding Bn/α,qαBn/β,qβVMO. Thus, we can assume that (A4) holds with α replaced by β, and then repeat the previous proof. The claim follows. ∎

Proof of Theorem 1.3.

The proof goes exactly as that of Theorem 1.2 until estimate (4.2). We proceed now with the estimates of the integrals I4 and I5. Assume first that αβp2. By using Hölder’s and Young’s inequalities, we get

|I4|BRη2|Δh(F)||Δh(Du)|c(R)(BR|Δh(F)|2)12(BRη2|Δh(Du)|p)1pcε(BR|Δh(F)|2dx)p2+εBRη2|Δh(Du)|pdxcε|h|βp(BR|Δh(F)|h|β|2dx)p2+εBRη2|Δh(Du)|pdx,

where ε>0 will be chosen later. Similarly, using also Lemma 2.7,

|I5|cRBR|Δh(F)|η|Δhu|dxc|h|pRpBR|Du|p+c|h|βp(BR|Δh(F)|h|β|2dx)p2.

Inserting the estimates of I4 and I5 in (4.2) and recalling that μ=0, we have

3ν4BR(|Du(x)|p-2+|Du(x+h)|p-2)|Δh(Du)|2η2𝑑xc(|h|2R2+|h|pRp)BR|Du|pdx+c|h|2αBR(gk(x+h)+gk(x))2|Du(x)|pdx   +c|h|βp(BR|Δh(F)|h|β|2dx)p2+εBRη2|Δh(Du)|pdx

and, by the elementary inequality (2.1), we get

BR(|Du(x)|2+|Du(x+h)|2)p-22|Δh(Du)|2η2𝑑xc(|h|2R2+|h|pRp)BR|Du(x)|pdx+c|h|2αBR(gk(x+h)+gk(x))2|Du(x)|pdx   +c|h|βp(BR|Δh(F)|h|β|2dx)p2+c(p,ν)εBR(|Du(x)|2+|Du(x+h)|2)p-22Δh(Du)|2η2dx.

We now choose ε=12c(p,ν), use Lemma 2.1, reabsorb the last integral in the right-hand side of the previous estimate by the left-hand side, and divide both side by |h|2α. We conclude that

BR/2|Δh(Vp(Du))|h|α|2dxc(|h|2-2αR2+|h|p-2αRp)BR|Du(x)|pdx+c|h|βp-2α(BR|Δh(F)|h|β|2dx)p2+cBR(gk(x+h)+gk(x))2|Du(x)|p𝑑x,

and so

(BR/2|Δh(Vp(Du))|h|α|2dx)12c(|h|1-αR+|h|p2-αRp2)(BR|Du(x)|pdx)12+c|h|βp2-α(BR|Δh(F)|h|β|2dx)p4+c(Br(gk(x+h)+gk(x))2|Du(x)|p𝑑x)12c|h|1-α(BR|Du(x)|pdx)12+c(BR|Δh(F)|h|β|2dx)p4+c(Br(gk(x+h)+gk(x))2|Du(x)|p𝑑x)12,

for a constant c that depends also on R and δ, and where we used that p2. Taking the Lq norm with the measure dh|h|n restricted to the ball B(0,δ) on the h-space, we obtain that

(Bδ(BR/2|Δh(Vp(Du))|h|α|2dx)q2dh|h|n)1qc(Bδ|h|(1-α)q(BR|Du(x)|pdx)q2dh|h|n)1q+c(Bδ(BR|Δh(F)|h|β|2dx)pq4dh|h|n)1q+c(Bδ(BR(gk(x+h)+gk(x))2|Du(x)|pdx)q2dh|h|n)1q=:J1+J2+J3,

where c=c(ν,L,R,p). For the estimate of J1, one can easily check that

J1=c(BR|Du(x)|pdx)12(Bδ|h|(1-α)q-ndh)1qc(n)(BR|Du(x)|pdx)12(0δρ(1-α)q-1dρ)1q=c(α,n,p,q,δ)(BR|Du(x)|pdx)12,

since α<1. The term J2 can be controlled by the B2,qp/2β-seminorm of F, which is finite thanks to our assumption. In order to estimate J3, we use that gk2Ln2α, by our assumption. Also, we have that |Du(x)|pLlocnn-2α. To see this, use Lemma 2.2, with qp22β=2nn-2β, to deduce that FL2β. Since

2n(p-1)n-2βnpn-2α,

Theorem 2.5 implies that DuLlocnpn-2α.

We now write the Lq norm in the integral J3 in polar coordinates, assuming without loss of generality that δ=1, so hB(0,1) if and only if h=rξ for some 0r<1 and some ξ in the unit sphere Sn-1 on n. We denote by dσ(ξ) the surface measure on Sn-1. We bound the term J3 by

01Sn-1(BR(gk(x+rξ)+gk(x))2|Du(x)|pdx)q2dσ(ξ)drr=k=0rk+1rkSn-1(BR(gk(x+rξ)+gk(x))2|Du(x)|p𝑑x)q2𝑑σ(ξ)drr=k=0rk+1rkSn-1(τrξgk+gk)2pDuLp(BR)qp2dσ(ξ)drr,

where we set rk=12k. Now, since DuLlocnpn-2α and gkLnα, Hölder’s inequality implies

(τrξgk+gk)2pDuLp(BR)DuLnpn-2α(BR)(τrξgk+gk)Lnα(BR)2p.

On the other hand, we note that for each ξSn-1 and rk+1rrk,

(τrξgk+gk)Lnα(BR)gkLnα(BR-rkξ)+gkLnα(BR)2gkLnα(λB),

where λ=2+1R. Hence,

J3C(n,α,q)DuLnpn-2α(BR)p2{gk}kq(Lnα(λB)),

where C(n,α,q)=21-αlog2σ(Sn-1)1q. Summarizing,

Δh(Vp(Du))|h|αLq(dh|h|n;L2(BR/2))CDuLp(BR)+ΔhF|h|βLq(dh|h|n;L2(B2R))+C(n,α,q)DuLnpn-2α(BR)p2{gk}kq(Lnα(λB)).

Lemma 2.3 now yields that Vp(Du)B2,qα, locally.

When α>βp2, we have the embedding Bn/α,qαB2n/βp,qβp2VMO. Thus, we can assume that (A4) holds with α replaced by βp2, and then repeat the previous proof. The claim follows. ∎

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About the article

Received: 2017-02-15

Revised: 2017-07-03

Accepted: 2017-07-29

Published Online: 2017-09-15


Albert Clop is partially supported by projects MTM2013-44699 (Spanish Ministry of Science), 2014-SGR-75 (Generalitat de Catalunya) and FP7-607647 (MAnET- European Union). Raffaella Giova and Antonia Passarelli di Napoli are supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Raffaella Giova has been partially supported by Project Legge 5/2007 Regione Campania “Spazi pesati e applicazioni al Calcolo delle Variazioni” and by Università degli Studi di Napoli Parthenope through the project “Sostegno alla ricerca individuale Triennio 2015-2017”.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 762–778, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0030.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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