When are weak solutions of the H-system locally bounded under bounded distortion?

Bounded distortion on the large-|u| set gives local boundedness

The paper studies weak solutions u ∈ W^{1,n}(U, R^{n+1}) of the H-system −div(|Du|^{n-2}Du) = (H∘u)Ju, where n ≥ 2, U is the open unit ball in R^n, and H is bounded. The main boundedness statement (Theorem 4.7) assumes 0 ≤ R_0 < ∞, n^{n/2} ≤ κ < ∞, and |Du(x)|^n ≤ κ|Ju(x)| for L^n almost every x ∈ U with |u(x)| > R_0. Under those hypotheses, u ∈ L^∞(U(0,ρ), R^{n+1}) for every 0 < ρ < 1.

Two conditions in that statement are easy to conflate. The distortion bound is required only where |u| is large, not everywhere on U; and the conclusion is local boundedness on compactly contained subballs, not a global L^∞ bound on U. The constant κ is bounded below by n^{n/2}, the value attained exactly when A*∘A is a multiple of the identity, i.e. when the linear map A is conformal.

The companion regularity statement (Theorem 1.1, proved as Corollary 4.8) adds Hölder continuity of H: if |H(y) − H(z)| ≤ L|y − z|^α for some 0 < α ≤ 1 and 0 ≤ L < ∞, then u has a representative that is Hölder continuous on every compact subset of U with some positive exponent. The paper does not claim a specific numerical value for that exponent.

Conformality is what the earlier argument silently used

For n = 2 the regularity theory is classical: Wente proved continuity for constant H with suitable boundary data, Grüter for conformal solutions with bounded H, Bethuel for all weak solutions with bounded Lipschitz H, and Rivière for every bounded H. For n ≥ 3 the picture is thinner. Duzaar and Fuchs proved continuity of bounded weak solutions for bounded Lipschitz H; Mou and Yang proved continuity of conformal solutions for bounded H; Wang obtained C^{1,α} under a decay condition on H and its gradient; Duzaar and Grotowski obtained Hölder continuous minimisers of the Dirichlet problem. Without additional hypotheses, continuity of weak solutions for n ≥ 3 is open even for bounded Lipschitz H.

The paper explains why the conformal case was tractable. Mou and Yang tested the equation with u − a multiplied by a cut-off in |u − a| and integrated the resulting differential inequality for ρ ↦ μ(U(a,ρ)), where μ is the push-forward of the energy measure |Du|^n L^n by u. That differential inequality uses conformality only through |Du(x)*(v)|^2 ≤ n^{-1}|Du(x)|^2|v|^2 for v ∈ R^{n+1}.

For a general map the corresponding object is the push-forward of |Du|^{n-2}Du∘Du*, whose density X with respect to μ is a weighted average over the fibres of u of the maps |A|^{-2}A∘A* with A = Du(x). If A is conformal this map equals n^{-1}(im A)^♮; otherwise its largest eigenvalue exceeds n^{-1}, and the inequality used by Mou and Yang fails. Averaging over fibres may compensate, but the paper notes it need not.

The push-forward functional replaces a varifold first variation

The proof pushes |Du|^{n-2}Du∘Du* forward by u to obtain a linear functional Σ on continuous maps of R^{n+1} into Hom(R^{n+1}, R^{n+1}) with compact support. The equation controls the values of Σ on derivatives, which is the structural input the rest of the argument uses.

The density X of Σ with respect to μ, defined by Σ(F) = ∫ F • X dμ, is a weighted average over the fibres of u of |A|^{-2}A∘A* with A = Du(x). The paper is explicit that Σ need not be a multiple of the functional F ↦ ∫ F(y) • S^♮ dV(y,S) for a varifold V, whose values on derivatives give the first variation of V. So the classical varifold first-variation route is not available here.

The substitute is an inequality of Michael–Simon type for linear functionals ζ(F) = ∫ F • X dμ such that g ↦ ζ(Dg) has finite total variation, proved by De Philippis, Gennaioli, Pigati, and Rindler. In that inequality μ is carried by a set of finite H^m measure and X takes values in a compact convex set on which the m-th singular value is bounded from below; neither an integrand nor a monotonicity identity is involved. The paper shows Σ has this form with m = n whenever the distortion |Du|^n/|Ju| is bounded.

Bounded distortion turns the density into a compact convex set

The bridge between the distortion hypothesis and the Michael–Simon-type inequality is Lemma 2.8. For n^{n/2} ≤ κ < ∞, let Ψ_κ be the closure in Hom(R^{n+1}, R^{n+1}) of the convex hull of {Y(A) : A ∈ Hom(R^n, R^{n+1}), 0 < |A|^n ≤ κ|J(A)|}, where Y(A) = |A|^{-2}A∘A*. Then Ψ_κ is a non-empty compact convex set, and every X ∈ Ψ_κ is symmetric and satisfies Xw • w ≥ 0 for w ∈ R^{n+1}, trace X = 1, σ_1(X) ≤ 1, and σ_n(X) ≥ (2κ^2)^{-1}.

The lower bound σ_n(X) ≥ (2κ^2)^{-1} is the quantitative content that matters: it is exactly the kind of lower bound on the n-th singular value that the De Philippis–Gennaioli–Pigati–Rindler inequality requires. The bound degrades as κ grows, so a larger allowed distortion weakens the constant but does not destroy the argument.

The proof of Lemma 2.8 also records the conformal endpoint. The inclusion A of R^n into R^{n+1} satisfies |A|^n = n^{n/2} = n^{n/2}|J(A)|, so Y(A) ∈ Ψ_κ and Ψ_κ is non-empty. This is the same n^{n/2} that appears as the lower bound on κ in the main theorems.

Lower mass ratios on the push-forward force a bounded image

The Michael–Simon-type inequality is applied to the functional F ↦ Σ_ρ(φF), where φ is a radial cut-off centred at a point a of the support of μ_ρ. Lemma 4.6 concludes that if 0 ≤ R_0 < ∞, n^{n/2} ≤ κ < ∞, the distortion condition holds, ρ is regular, a ∈ spt μ_ρ, and 0 < r ≤ min{dist(a, K_ρ), |a| − R_0}, then μ_ρ(U(a,r)) ≥ (4n^{1/2}Γ_κ)^{-n} min{r, h^{-1}}^n, with h^{-1} = ∞ when h = 0.

That is a lower bound on the mass ratio of the push-forward of |Du|^n L^n by u, with the radius r capped by the distance to K_ρ and by |a| − R_0. The constant depends on n, on Γ_κ, and on the sup norm h of H through the term min{r, h^{-1}}^n.

The step from mass ratios to boundedness is a counting argument. Since μ(R^{n+1}) = ∫ |Du|^n dL^n < ∞, the support of μ_ρ can contain only finitely many points at mutual distance at least 2r far from the image of the boundary. In the proof of Theorem 4.7, points a_1, …, a_N in spt μ_{ρ'} ∼ W with |a_i − a_j| ≥ 2 give disjoint balls U(a_i, 1) and N c_0 ≤ μ_{ρ'}(R^{n+1}) < ∞, so N is finite and spt μ_{ρ'} is contained in some B(0, R_1). A cut-off ψ(y) = γ(|y|)y that vanishes for |y| ≤ R_1 then makes w = ψ∘u have Dw = 0 almost everywhere on U(0, ρ'), and the Poincaré inequality forces w to be constant there, giving |u(x)| ≤ max{R_1 + 1, |c|} almost everywhere on U(0, ρ').

Hölder continuity of H upgrades boundedness to Hölder continuity

Corollary 4.8 adds the assumption |H(y) − H(z)| ≤ L|y − z|^α for 0 < α ≤ 1 and 0 ≤ L < ∞. Because |H(y) − H(z)| ≤ min{2h, L|y − z|^α} ≤ (2hL)^{1/2}|y − z|^{α/2}, the proof may assume α < 1. Rescaling u_ρ(x) = u(ρx) for 0 < ρ < 1 and changing variables in the weak formulation shows u_ρ is a bounded weak solution of the H-system, so the known result that locally bounded weak solutions are locally Hölder continuous for continuous H applies to u_ρ; since ρ is arbitrary, the conclusion follows.

This is why the paper frames the boundedness theorem as the substantive step. The paper cites Fusco, Kristensen, Leone, and Verde for the fact that locally bounded weak solutions are locally Hölder continuous when H is continuous, and locally C^{1,α} for some α ∈ (0,1) when H is locally Lipschitz. For continuous H, it therefore suffices to prove local boundedness.

The paper also notes what its result does not settle. Martino and Schikorra and Schlagenhauf constructed bounded discontinuous maps U → R^{n+2} for n ≥ 3 solving n-Laplace systems with antisymmetric potentials in L^n, but those examples concern a larger class of equations and give no counterexample for H-systems. Without additional hypotheses, continuity of weak solutions for n ≥ 3 remains open even for bounded Lipschitz H.

The distortion threshold is the paper's real dividing line

The paper states that for conformal u the distortion equals n^{n/2}, and continuity in that case is due to Mou and Yang. It also states that it does not know an earlier reference for bounded distortion greater than n^{n/2}. That is the paper's own framing of its increment: the new hypothesis is a bounded distortion strictly above the conformal value, imposed only where |u| is large.

Two boundaries follow directly from the statements. First, the boundedness conclusion is local, on U(0, ρ) for 0 < ρ < 1, and the distortion hypothesis is only required on the set where |u| exceeds R_0; the theorem does not assert a global bound on U or a bound that holds without the large-|u| restriction. Second, the Hölder conclusion requires H to be Hölder continuous in addition to bounded, and the paper does not specify the exponent it obtains.

The most decision-relevant comparison in the paper is therefore between the conformal case, where the distortion is pinned at n^{n/2} and the Mou–Yang argument applies, and the bounded-distortion case above n^{n/2}, where the Mou–Yang inequality fails and the push-forward functional Σ is needed instead. The quantitative cost of moving above n^{n/2} is visible in the constant: σ_n(X) ≥ (2κ^2)^{-1} in Lemma 2.8 and μ_ρ(U(a,r)) ≥ (4n^{1/2}Γ_κ)^{-n} min{r, h^{-1}}^n in Lemma 4.6 both weaken as κ grows. Readers weighing whether a given solution falls inside the theorem should check the distortion bound on the large-|u| region, the value of κ relative to n^{n/2}, and whether H is Hölder continuous, since those three conditions separate the boundedness conclusion from the Hölder conclusion.

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