A limiting case in partial regularity for quasiconvex functionals

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $p$-growth of the type $$ w \mapsto \int \left[F(Dw)-f\cdot w\right]dx $$ feature almost everywhere $\mbox{BMO}$-regular gradient provided that $f$ belongs to the borderline Marcinkiewicz space $L(n,\infty)$.


Introduction
In this paper we provide a limiting partial regularity criterion for vector-valued minimizers u : Ω ⊂ R n → R N , n ≥ 2, N > 1, of nonhomogeneous, quasiconvex variational integrals as: This is the content of our main theorem.
We immediately refer to Section 1.1 below for a description of the structural assumptions in force in Theorem 1.1.Let us put our result in the context of the available literature.The notion of quasiconvexity was introduced by Morrey [37] in relation to the delicate issue of semicontinuity of multiple integrals in Sobolev spaces: an integrand F (•) is a quasiconvex whenever − B1(0) viewpoint.In fact, a classical result of Evans [21] states that the gradient of minima is locally Hölder continuous outside a negligible, "singular" set, while a celebrated counterexample due to Müller & Šverák [38] shows that the gradient of critical points may be everywhere discontinuous.After Evans seminal contribution [21], the partial regularity theory was extended by Acerbi & Fusco [2] to possibly degenerate quasiconvex functionals with superquadratic growth, and by Carozza & Fusco & Mingione [8] to subquadratic, nonsingular variational integrals.A unified approach that allows simultaneously handling degenerate/nondegenerate, and singular/nonsingular problems, based on the combination of A-harmonic approximation [20], and p-harmonic approximation [19], was eventually proposed by Duzaar & Mingione [18].Moreover, Kristensen & Mingione [29] proved that the Hausdorff dimension of the singular set of Lipschitz continuous minimizers of quasiconvex multiple integrals is strictly less than the ambient space dimension n, see also [5] for further developments in this direction.We refer to [3,15,24,25,26,27,36,40,41] for an (incomplete) account of classical, and more recent advances in the field.In all the aforementioned papers are considered homogeneous functionals, i.e. f ≡ 0 in (1.1).The first sharp ε-regularity criteria for nonhomogeneous quasiconvex variational integrals guaranteeing almost everywhere gradient continuity under optimal assumptions on f were obtained by De Filippis [12], and De Filippis & Stroffolini [13], by connecting the classical partial regularity theory for quasiconvex functionals with nonlinear potential theory for degenerate/singular elliptic equations, first applied in the context of partial regularity for strongly elliptic systems by Kuusi & Mingione [32].Potential theory for nonlinear PDE originates from the classical problem of determining the best condition on f implying gradient continuity in the Poisson equation −∆u = f , that turns out to be formulated in terms of the uniform decay to zero of the Riesz potential, in turn implied by the membership of f to the Lorentz space L(n, 1), [9,30].In this respect, a breakthrough result due to Kuusi & Mingione [31,33] states that the same is true for the nonhomogeous, degenerate p-Laplace equation -in other words, the regularity theory for the nonhomogeneous p-Laplace PDE coincides with that of the Poisson equation up to the C 1 -level.This important result also holds in the case of singular equations [17,39], for general, uniformly elliptic equations [6], up to the boundary [10,11], and at the level of partial regularity for p-Laplacian type systems without Uhlenbeck structure, [7,32].We conclude by highlighting that our Theorem 1.1 fits this line of research as, it determines for the first time in the literature optimal conditions on the inhomogeneity f assuring partial BMO-regularity for minima of quasiconvex functionals expressed in terms of the limiting function space L(n, ∞).
Outline of the paper.In Section 2 we recall some well-known results from the study of nonlinear problems also establishing some Caccioppoli and Gehring type lemmas.In Section 3 we prove the excess decay estimates; considering separately the nondegenerate and the degenerate case.Section 4 is devoted to the proof of Theorem 1.1.
for all z ∈ R N ×n , Λ 1 being a positive absolute constant and µ : [0, ∞) → [0, 1] being a concave nondecreasing function with µ(0) = 0.In the rest of the paper we will always assume p ≥ 2. In order to derive meaningful regularity results, we need to update (1.6) to the stronger strict quasiconvexity condition holding for all z ∈ R N ×n and ϕ ∈ W 1,p 0 (B, R N ), with λ being a positive, absolute constant.Furthermore, we allow the integrand F (•) to be degenerate elliptic in the origin.More specifically, we assume that F (•) features degeneracy of p-Laplacian type at the origin, i. e.
which means that we can find a function ω : (0, ∞) → (0, ∞) such that for every z ∈ R N ×n and all s ∈ (0, ∞).Moreover, the right-hand side term f : Ω → R N in (1.1) verifies as minimal integrability condition the following Here it is intended that, when p n, the Sobolev conjugate exponent p * can be chosen as large as needed -in particular it will always be larger than p.By (1.6) see [23,Chapter 5].

Preliminaries
In this section we display our notation and collect some basic results that will be helpful later on.
2.1.Notation.In this paper, Ω ⊂ R n is an open, bounded domain with Lipschitz boundary, and n 2. By c we will always denote a general constant larger than one, possibly depending on the data of the problem.Special occurrences will be denoted by c * , c or likewise.Noteworthy dependencies on parameters will be highlighted by putting them in parentheses.Moreover, to simplify the notation, we shall array the main parameters governing functional (1.1) in the shorthand data := n, N, λ, Λ, p, µ(•), ω(•) .By B r (x 0 ) := {x ∈ R n : |x − x 0 | < r}, we denote the open ball with radius r, centred at x 0 ; when not necessary or clear from the context, we shall omit denoting the center, i.e.B r (x 0 ) ≡ B r -this will happen, for instance, when dealing with concentric balls.
For x 0 ∈ Ω, we abbreviate d x0 := min 1, dist(x 0 , ∂Ω) .Moreover, with B ⊂ R n being a measurable set with bounded positive Lebesgue measure 0 < |B| < ∞, and a : B → R k , k 1, being a measurable map, we denote (a We will often employ the almost minimality property of the average, i.e. for all z ∈ R N ×n and any t 1.Finally, if t > 1 we will indicate its conjugate by t ′ := t/(t − 1) and its Sobolev exponents as t * := nt/(n − t) if t < n or any number larger than one for t n and t * := max nt/(n + t), 1 .
2.2.Tools for nonlinear problems.When dealing with p-Laplacian type problems, we shall often use the auxiliary vector field V s : R N ×n → R N ×n , defined by incorporating the scaling features of the p-Laplacian.If s = 0 we simply write and where the constants implicit in " ", "≈" depend on n, N, p.A relevant property which is relevant for the nonlinear setting is recorded in the following lemma.
with constants implicit in "≈" depending only on n, N, t.
The following iteration lemma will be helpful throughout the rest of the paper; for a proof we refer the reader to [23,Lemma 6

non-negative and bounded function, and let
We will often consider the "quadratic" version of the excess functional defined in (1.5), i. e. (2.4) .

Basic regularity results.
In this section we collect some basic estimates for local minimizers of nonhomogeneous quasiconvex functionals.We start with a variation of the classical Caccioppoli inequality accounting for the presence of a nontrivial righthand side term, coupled with an higher integrability result of Gehring-type.• For every ball B ρ (x 0 ) ⋐ Ω and any , where E(•) is defined in (1.5), ℓ(x) := u 0 + z 0 , x − x 0 and c ≡ c(n, N, λ, Λ, p).
• There exists an higher integrability exponent p 2 ≡ p 2 (n, N, λ, Λ, p) > p such that Du ∈ L p2 loc (Ω, R N ×n ) and the reverse Hölder inequality Proof.For the ease of exposition, we split the proof in two steps, each of them corresponding to the proof of (2.7) and (2.8) respectively.
Step 1: proof of (2.7) and (3.22).We choose parameters ρ where we have used the simple relation Dϕ 1 + Dϕ 2 = Du − z 0 .Terms I 1 and I 3 can be controlled as done in [18, Proposition 2]; indeed we have ) and apply Sobolev-Poincaré inequality to get where c ≡ c(n, N, m) and we also used that ρ/2 τ 2 ρ.Merging the content of the two above displays, recalling that η ≡ 1 on B τ1 (x 0 ) and choosing ε > 0 sufficiently small, we obtain , with c ≡ c(n, N, λ, Λ, p).At this stage, the classical hole-filling technique, Lemma 2.2 and (2.3) yield (2.7) and the first bound in the statement is proven.
Step 2: proof of (2.8).To show the validity of (2.8), we follow [32, proof of Proposition 3.2] and first observe that if u is a local minimizer of functional F (•) on B ρ (x 0 ), setting f ρ (x) := ρf (x 0 + ρx), the map u ρ (x) := ρ −1 u(x 0 + ρx) is a local minimizer on B 1 (0) of an integral with the same integrand appearing in (1.1) satisfying (1.7) 1,2,3 and f ρ replacing f .This means that (2.10) still holds for all balls , with x ∈ B 1 (0) being any point -in particular it remains true if |z 0 | = 0, while condition |z 0 | = 0 was needed only in the estimate of term I 2 in (2.11), that now requires some change.So, in the definition of the affine map ℓ we choose z 0 = 0, u 0 = (u ρ ) Bσ (x) and rearrange estimates (2.10)-(2.11)as: and, recalling that ϕ 1 ∈ W 1,p 0 (B τ2 (x), R N ), via Sobolev Poincaré, Hölder and Young inequalities and (1.12) 2 , we estimate with c ≡ c(n, N, p).Plugging the content of the two previous displays in (2.9), reabsorbing terms and applying Lemma 2.2, we obtain with equality holding when p < n, while for p n any value of p * > 1 will do.We then manipulate the second term on the right-hand side of (2.12) as where we set . Plugging the content of the previous display in (2.12) and applying Sobolev-Poincaré inequality we get with c ≡ c(n, N, Λ, λ, p).Now we can apply a variant of Gehring lemma [23, Corollary 6.1] to determine a higher integrability exponent s ≡ s(n, N, Λ, λ, p) such that 1 < s m/(p * ) ′ and for c ≡ c(n, N, Λ, λ, p).Next, notice that , so plugging this last inequality in (2.14) and recalling that s(p * ) ′ m, we obtain .
Setting p 2 := sp > p above and recalling that x ∈ B 1 (0) is arbitrary, we can fix x = 0, scale back to B ρ (x 0 ) and apply (2.1) to get (2.8) and the proof is complete.

Excess decay estimate
In this section we prove some excess decay estimates considering separately two cases: when a smallness condition on the excess functional of our local minimizer u is satisfied and when such an estimate does not hold true.
Proof of Proposition 3.1.For the sake of readability, since all balls considered here are concentric to B ρ (x 0 ), we will omit denoting the center.Moreover, we will adopt the following notation (Du) Bς (x0) ≡ (Du) ς and, for all ϕ ∈ C ∞ c (B ρ ; R N ), we will denote Dϕ L ∞ (Bρ) ≡ Dϕ ∞ .We spilt the proof in two steps.
We begin proving that condition (3.1) implies that and Let us note that we have Since |(Du) ρ | > 0 we have that the hypothesis of [12, Lemma 3.2] are satisfied with (3.8) Then, Fix ε > 0 and let δ ≡ δ(data, ε) > 0 be the one given by [32, Lemma 2.4] and choose ε 0 and ε 1 sufficiently small such that (3.9) With this choice of ε 0 and ε 1 it follows that u 0 is almost A -harmonic on B ρ , in the sense that with A as in (3.8).Hence, by [32,Lemma 2.4] we obtain that there exists We choose now τ 0 ∈ (0, 2 −10 ), which will be fixed later on, and estimate where c ≡ c(data) > 0 and where we have used the following property of A -harmonic functions with γ > 1 and c depending on n, N , and on the ellipticity constants of A .Now, choosing we have that this together with (3.9) gives that ε 0 ≡ ε 0 (data, τ 0 ) and Recalling the definition of u 0 in (3.7) and (3.12) we eventually arrive at for c ≡ c(data) > 0. By a similar computation, always using (3.13), (3.10) and (3.11), we obtain that In this way, as for (3.14), by the definition of u 0 in (3.7), we eventually arrive at with c ≡ c(data).
Denote now with ℓ 2τ0ρ the unique affine function such that Hence, by (3.14) and (3.15), we conclude that (3.16) Notice that we have also used the property that Recalling the definition of the excess functional E(•), in (1.5), we can estimate the following quantity as follows where we have used the following property of the affine function ℓ 2τ0ρ for a constant c ≡ c(n, p) > 0; see for example [32,Lemma 2.2].Now, starting from (3.1) and (3.9), we further reduce the size of ε 0 such that where c ≡ c(n) is the same constant appearing in (3.17).Thus, combining (3.17) and (3.18), we get The information provided by (3.18) combined with (3.16) allow us to conclude that (3.20) By triangular inequality and (3.19) we also get which, therefore, implies that where c ≡ c(data) > 0. By triangular inequality, we can further estimate where c ≡ c(p) > 0. We now separately estimate the previous integrals.We begin considering I 1 .By Young and triangular inequalities we get , with c ≡ c(data) > 0. In a similar fashion, we can treat the integral I 2 , where we have used the following property of the affine function ℓ 2τ0ρ for a given constant c ≡ c(n, p) > 0; see [32,Lemma 2.2].Finally, the last integral I 3 can be treated recalling (3.21) and (2.1), i. e. .
All in all, combining the previous estimate Step 2: proof of (3.5).The proof follows by [12,Lemma 2.4] which yields .
Multiplying both sides by E(u; B τ0ρ ) we get the desired estimate.
Proof.The proof is analogous to estimate (2.7), up to treating in a different way the term I 2 in (2.9), taking in consideration the eventuality z 0 = 0. Exploiting (1.11) and fact that ϕ 1 ∈ W 1,p 0 (B τ2 (x 0 ), R N ), an application of the Sobolev-Poincaré inequality yields , (3.23) where c ≡ c(n, N, m) and we also used that ρ/2 τ 2 ρ.Hence, proceeding as in the proof of (2.7), we obtain that , with c ≡ c(n, N, λ, Λ, p).Concluding as in the proof of (2.7), we eventually arrive at (3.22).
We will also need the following result.Proof.Given the regularity properties of the integrand F , we have that a local minimizer u of (1.1) solves weakly the following integral identity (see [41,Lemma 7 We begin estimating the first integral I 1 .For s ∈ (0, ∞) we get |Du| p dx.
On the other hand, the integral I 2 can be estimated as follows Combining the inequalities above we obtain (3.24).
In this setting the analogous result of Proposition 3.1 is the following one.
Proof.We adopt the same notations used in the proof of Proposition and v 0 := u κ , for ε 3 ∈ (0, 1], which will be fixed later on.Applying (3.24) to the function v 0 yields For any ε > 0 and ϑ ∈ (0, 1) and let δ be the one given by [16, Lemma 1.1].Then, up to choosing s, ε 2 and ε 3 sufficiently small, we arrive at Then, Lemma 1.1 in [16] implies up to taking ε as small as needed.Now, denoting with h 0 := hκ, we have that Now, we choose ϑ := (s) ′ /2, with s being the exponent given by (2.8).Note that by the proof of (2.8) it actually follows that ϑ < 1.Thus, choosing εε p 2 κ p τ 2n+4α 1 (where α ∈ (0, 1) is given by (3.34)) we arrive at Indeed, choose λ which will be fixed later on.Then, we have that By absolute continuity of the functional E(