Local pointwise second derivative estimates for strong solutions to the $\sigma _k$-Yamabe equation on Euclidean domains

Abstract

We prove local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the $\sigma _k$-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.

1 Introduction

Let $\varOmega \subset {\mathbb {R}}^n$ ($n\ge 3$) be a domain. In this paper, we obtain local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the equations

and

Throughout the paper, $A_u$ denotes the symmetric matrix-valued function

$$\begin{aligned} A_u = \nabla ^2 u - \frac{|\nabla u|^2}{2u}I, \end{aligned}$$

where I is the $n\times n$ identity matrix and $\sigma _k$ is the k’th elementary symmetric polynomial, defined on a symmetric matrix A with eigenvalues $\lambda (A)=(\lambda _1,\dots ,\lambda _n)$ by

$$\begin{aligned} \sigma _k(A) = \sigma _k(\lambda _1,\dots ,\lambda _n) {:}{=}\sum _{1\le {i_1}<\dots <{i_k}\le n} \lambda _{i_1}\cdots \lambda _{i_k}. \end{aligned}$$

Note that $\sigma _1(A)$ is the trace of A and $\sigma _n(A)$ is the determinant of A. We also denote by $\varGamma _k^+$ the open convex cone

$$\begin{aligned} \varGamma _k^+ = \{(\lambda _1,\dots ,\lambda _n)\in {\mathbb {R}}^n:\sigma _j(\lambda _1,\dots ,\lambda _n)>0~\text {for all }1\le j\le k\}. \end{aligned}$$

It is well-known that the equations ($1.1^\pm$) are elliptic. Furthermore, $\sigma _k^{1/k}$ is a concave function on the set of symmetric matrices with eigenvalues in $\varGamma _k^+$.

The motivation behind ($1.1^\pm$) comes from conformal geometry: if $g_{ij}=u^{-2}\delta _{ij}$ is a metric conformal to the flat metric on a domain $\varOmega \subset {\mathbb {R}}^n$, then $uA_u$ is the (1, 1)-Schouten tensor of g, and the $\sigma _k$-Yamabe equation in the so-called positive/negative (±) case is given by

$$\begin{aligned} \sigma _k(\pm uA_u) = 1, \quad \lambda (\pm A_u)\in \varGamma _k^+, \quad u>0. \end{aligned}$$

(1.2)

The Eq. (1.2) and their counterparts on Riemannian manifolds were first studied by Viaclovsky in [63]. Since then, these equations have been addressed by various authors – for a partial list of references, see [1,2,3, 8,9,10,11,12, 14, 16,17,19, 24, 27, 28, 31,32,33, 35, 39,40,41, 43, 44, 46, 47, 53, 54, 56, 64, 65] in the positive case and [13, 23, 25, 29, 30, 42, 45, 55] in the negative case. When $k=1$, these equations reduce to the original Yamabe equation. When $k\ge 2$, they are fully nonlinear and elliptic at a solution (although, a priori, not necessarily uniformly elliptic). Fully nonlinear elliptic equations involving eigenvalues of the Hessian were first considered by Caffarelli et al. in [6].

A priori local first and second derivative estimates play an important role in the study of the $\sigma _k$-Yamabe equation, and were established in the positive case by Chen [14], Guan and Wang [27], Jin, Li and Li [39], Li and Li [40], Li [43] and Wang [65]. In the negative case, an a priori (global) $C^1$ estimate is proven by Gursky and Viaclovsky [30], but it is unknown whether a priori $C^2$ estimates hold. In this paper, we are concerned with the local regularity of positive $W^{2,p}$ solutions to the equations ($1.1^\pm$). More precisely, for $2\le k\le n$ we derive local pointwise boundedness of second derivatives, provided $p>kn/2$ in the positive case and $p>(k+1)n/2$ in the negative case. To simplify the discussion, we do not include the case $k=1$, in which the equations ($1.1^\pm$) are semilinear. We prove:

Theorem 1.1

Let $\varOmega$ be a domain in ${\mathbb {R}}^n$ ($n\ge 3$) and let $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times (0,\infty )\times {\mathbb {R}}^n)$ be a positive function. Suppose that $2\le k\le n$, $p>kn/2$ and $u\in W_{\mathrm{loc}}^{2,p}(\varOmega )$ is a positive solution to (1.1$^+$). Then $u\in C_{\mathrm {loc}}^{1,1}(\varOmega )$, and for any concentric balls $B_{R}\subset B_{2R}\Subset \varOmega$ we have

$$\begin{aligned} \Vert \nabla ^2 u\Vert _{L^\infty (B_R)}\le C, \end{aligned}$$

where C is a constant depending only on n, p, R, f and an upper bound for $\Vert \ln u\Vert _{W^{2,p}(B_{2R})}$.

Theorem 1.2

Let $\varOmega$ be a domain in ${\mathbb {R}}^n$ ($n\ge 3$) and let $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times (0,\infty )\times {\mathbb {R}}^n)$ be a positive function. Suppose that $2\le k\le n$, $p>(k+1)n/2$ and $u\in W_{\mathrm{loc}}^{2,p}(\varOmega )$ is a positive solution to (1.1$^-$). Then $u\in C_{\mathrm {loc}}^{1,1}(\varOmega )$, and for any concentric balls $B_{R}\subset B_{2R}\Subset \varOmega$ we have

$$\begin{aligned} \Vert \nabla ^2 u\Vert _{L^\infty (B_R)}\le C, \end{aligned}$$

where C is a constant depending only on n, p, R, f and an upper bound for $\Vert \ln u\Vert _{W^{2,p}(B_{2R})}$.

Remark 1.3

As noted above, it is unknown whether a priori $C^2$ estimates hold for solutions to the $\sigma _k$-Yamabe equation in the negative case. We also note that for the closely related $\sigma _k$-Loewner-Nirenberg problem, there exist locally Lipschitz but non-differentiable viscosity solutions – see [45]. As far as the authors are aware, Theorem 1.2 currently provides the only available local second derivative estimate for solutions to the $\sigma _k$-Yamabe equation in the negative case.

To put things in perspective, we note that our estimates in Theorem 1.1 are closely related to certain analytical aspects in the work of Chang, Gursky and Yang in [12]. In [12], under natural conformally invariant conditions on a Riemannian 4-manifold $(M^4,g_0)$, the authors established the existence of a metric in the conformal class $[g_0]$ whose Schouten tensor has eigenvalues in $\varGamma _2^+$. An important part of the proof in [12] was to obtain $W^{2,s}$ estimates for $4<s<5$ on smooth solutions to a one-parameter family of regularised $\sigma _2$-equations (see Eq. (A.1) in Appendix A) which are uniform with respect to the parameter. This was achieved by first obtaining a uniform $W^{1,4}$ estimate (see Theorem 3.1 in [12]), and subsequently carrying out an integrability improvement argument (see Sections 5 and 6 in [12]). With the $W^{2,s}$ estimate in hand, the authors then applied a heat flow argument to obtain the desired conformal metric.

Remark 1.4

A natural question to ask is whether the heat flow argument in [12] can be avoided by instead taking the regularisation parameter directly to zero. One application of Theorem 1.1 above and [46, Proposition 5.3] is that this can be achieved when $(M^4,g_0)$ is locally conformally flat. We refer the reader to Appendix A for the details.

Our work is also closely related to the work of Urbas in [60], where local pointwise second derivative estimates for $W^{2,p}$ solutions to the k-Hessian equation

$$\begin{aligned} \sigma _k^{1/k}(\nabla ^2 u(x)) = f(x)>0, \quad \lambda (\nabla ^2 u(x))\in \varGamma _k^+ \end{aligned}$$

were established on domains in ${\mathbb {R}}^n$. At the heart of Urbas’ proof is also an integrability improvement argument, assuming an initial lower bound of $p>kn/2$ (see also [15, 49, 58, 61, 62]). By an application of Moser iteration, the $C^{1,1}_{\mathrm{loc}}$ estimate is then obtained. We note that Moser iteration has previously been utilised in the context of the $\sigma _k$-Yamabe equation to establish local boundedness of solutions, see for instance [21, 22, 33].

We will in fact prove a more general version of Theorems 1.1 and 1.2, and consider an operator of the form

$$\begin{aligned} A_H[u] {:}{=}\nabla ^2 u - H[u] \end{aligned}$$

(1.3)

in place of $\pm A_u$. Here, $H[u](x)=H(x,u(x),\nabla u(x))$ for a given matrix-valued function $H=H(x,z,\xi )\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\, \mathrm{Sym}_n({\mathbb {R}}))$, where $\mathrm{Sym}_n({\mathbb {R}})$ denotes the space of real symmetric $n\times n$ matrices. Rather than ($1.1^\pm$), we consider the equation

$$\begin{aligned} \sigma _k^{1/k}\big (A_H[u](x)\big ) = f(x,u(x),\nabla u(x))>0, \quad \lambda (A_H[u](x)) \in \varGamma _k^+ \quad \mathrm {for~a.e.~}x\in \varOmega ,\nonumber \\ \end{aligned}$$

(1.4)

where $f\in C^{1,1}_{\mathrm{loc}}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n)$.

It is clear that if u satisfies (1.1$^+$) with $u\ge \frac{1}{C}>0$, then u satisfies (1.4) provided $H(x,z,\xi ) = \frac{|\xi |^2}{2z}I$ for $z\ge \frac{1}{C}$. Likewise, if $u\in W^{2,p}_{\mathrm{loc}}(\varOmega )$ is a solution to (1.1$^-$) with right hand side (RHS) f and $u\ge \frac{1}{C}>0$, then $w{:}{=}- u \in W_{\mathrm{loc}}^{2,p}(\varOmega )$ satisfies (1.4) with RHS ${\tilde{f}}(x,z,\xi ) {:}{=}f(x,-z,-\xi )$, provided $H(x,z,\xi ) = \frac{|\xi |^2}{2z}I$ for $z\le -\frac{1}{C}$. Therefore, for the purpose of obtaining Theorems 1.1 and 1.2, it will suffice to consider the case that H is a multiple of the identity matrix:

Theorem 1.5

Let $\varOmega$ be a domain in ${\mathbb {R}}^n$ ($n\ge 3$), $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n)$ a positive function and $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$. Suppose $2\le k\le n$, $p\ge 1$ and $u\in W_{\mathrm{loc}}^{2,p}(\varOmega )$ is a solution to (1.4), and that one of the following conditions holds:

• 1. $H(x,z,\xi ) = H_1(x,z)|\xi |^2 I$ with $H_1\ge 0$ and $p>\frac{kn}{2}$,

• 2. $H(x,z,\xi ) = H_2(x,z,\xi )I$ and $p>\frac{(k+1)n}{2}$.

Then $u\in C^{1,1}_{\mathrm{loc}}(\varOmega )$, and for any concentric balls $B_{R}\subset B_{2R} \Subset \varOmega$ we have

$$\begin{aligned} \Vert \nabla ^2 u\Vert _{L^\infty (B_R)}\le C, \end{aligned}$$

(1.5)

where C is a constant depending only on n, p, R, f, H and an upper bound for $\Vert u\Vert _{W^{2,p}(B_{2R})}$.

Remark 1.6

The constant C in (1.5) depends only on n, p, R and upper bounds for $\Vert u\Vert _{W^{2,p}(B_{2R})}$, $\Vert H\Vert _{C^{1,1}(\varSigma )}$ and $\Vert \ln f\Vert _{C^{1,1}(\varSigma )}$, where $\varSigma {:}{=}{\overline{B}}_{2R}\times [-M,M]\times {\overline{B}}_{M}(0)\subset \varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n$ and $M \ge \Vert u\Vert _{C^1({\overline{B}}_{2R})}$. Note that since $p>n$ in Theorem 1.5, an upper bound for $\Vert u\Vert _{W^{2,p}(B_{2R})}$ implies an upper bound for $\Vert u\Vert _{C^1({\overline{B}}_{2R})}$, in light of the Morrey embedding theorem.

Remark 1.7

When $H\equiv 0$ and $f=f(x)$, Theorem 1.5 was proved in [60, Theorem 1.6].

The matrix $A_H[u]$ introduced in (1.3) is sometimes referred to as an augmented Hessian of u. The corresponding augmented Hessian equations have been extensively studied in recent years – see [36,37,38] and the references therein. In this vein, it is therefore of interest to generalise Theorem 1.5 to arbitrary $H\in C_{\mathrm{loc}}^{1,1}$. As we will see, the proof of Theorem 1.5 uses some favourable divergence structure in the case that H is a multiple of the identity matrix. However, when $k=2$, a similar divergence structure holds for general H and we obtain the following:

Theorem 1.8

Let $\varOmega$ be a domain in ${\mathbb {R}}^n$ ($n\ge 3$), $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n)$ a positive function and $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$. Suppose $p>\frac{3n}{2}$ and $u\in W_{\mathrm{loc}}^{2,p}(\varOmega )$ is a solution to (1.4) with $k=2$. Then $u\in C^{1,1}_{\mathrm {loc}}(\varOmega )$, and for any concentric balls $B_{R}\subset B_{2R} \Subset \varOmega$ we have

$$\begin{aligned} \Vert \nabla ^2 u\Vert _{L^\infty (B_R)}\le C, \end{aligned}$$

where C is a constant depending only on n, p, R, f, H and an upper bound for $\Vert u\Vert _{W^{2,p}(B_{2R})}$.

Remark 1.9

In [36,37,38] and the references therein, it is usually assumed that H satisfies a so-called co-dimension one convexity condition, which is known to be a necessary and sufficient condition to obtain $C^1$ estimates – see [50, 51, 57]. We point out that we do not assume a co-dimension one convexity condition in our treatment of second derivative estimates (the exception is Case 1 of Theorem 1.5, where we have convexity in $\xi$).

Under a stronger assumption on p, we will also obtain an extension of Theorem 1.8 to the case $k\ge 3$—see Sect. 6.

In adapting the methods of [60] to prove Theorems 1.5 and 1.8, we will need to deal with the term H[u] which, whilst being of lower order in the definition of $A_H[u]$, creates new higher order terms in our estimates. Roughly speaking, the two terms which are formally problematic consist of:

(i):

a contraction of the linearised operator

$$\begin{aligned} F[u]^{ij} {:}{=}\frac{\partial \sigma _k(A_H[u])}{\partial (A_H[u])_{ij}} \end{aligned}$$

(1.6)

with double difference quotients of $H[u]_{ij}$ (this arises as a result of taking difference quotients of (1.4) twice), and

(ii):

the divergence of $F[u]^{ij}$ multiplied by a term formally of third order in u (this arises after integrating by parts).

In [60], neither of these terms exist since $F[u]^{ij}$ is divergence-free when $H\equiv 0$. In the more general case that we are considering, it is unclear whether these third order terms have a favourable sign individually. However, we will estimate them so as to show that, when combined, they yield a cancellation phenomenon that ensures the overall higher order contribution is positive. For the estimates of the higher order terms arising from the divergence of $F[u]^{ij}$, see Lemmas 4.4 and 4.5, and for those arising from the double difference quotients of H[u], see Lemma 4.10. For the resulting cancellation phenomena, see Corollaries 4.12, 4.13 and 4.14.

We close the introduction by noting that in Theorems 1.1 and 1.2, we do not know whether our lower bounds on p to obtain $C^{1,1}_{\mathrm{loc}}$ regularity are sharp, and it would be interesting to determine the sharp lower bounds. In the case of the k-Hessian equation for $3\le k\le n$, it is shown by Urbas in [59] that there exist $W^{2,p}$-strong solutions with $p<\frac{k(k-1)}{2}$ which fail to be $C^{1,\alpha }_{\mathrm{loc}}$ for any $\alpha >1-\frac{2}{k}$. Other lower bounds on p leading to $C^{1,1}_{\mathrm{loc}}$ regularity for k-Hessian equations have been studied in [15, 49, 58, 61, 62], for instance.

The plan of the paper is as follows. We begin in Sect. 2 with an outline of the proof of Theorems 1.5 and 1.8. This prompts us to consider the divergence structure of the linearised operator, which we address in Sect. 3, and also motivate the estimates established from Sect. 4 onwards. In Sect. 4 we carry out the main body of our integral estimates. In Sect. 5, we use these estimates and the Moser iteration technique to obtain the desired $C^{1,1}_{\mathrm{loc}}$ estimates, completing the proofs of Theorems 1.5 and 1.8. In Sect. 6, we give the aforementioned extension of Theorem 1.8 to the case $k\ge 3$.

2 Outline of the proofs of Theorems 1.5 and 1.8

Our proofs of Theorems 1.5 and 1.8 use an integrability improvement argument, from which the $C^{1,1}_{\mathrm{loc}}$ estimate is obtained by the Moser iteration technique. In Case 1 of Theorem 1.5, we will obtain, for a solution $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )$ to (1.4) with $q>\frac{kn}{2}-k+1$, the estimate

$$\begin{aligned} \bigg (\int _{B_{R+\rho }}(\varDelta u+C_1)^{\beta q}\bigg )^{1/\beta }\le \frac{Cq}{\rho ^2}\int _{B_{R+3\rho }}(\varDelta u+C_1)^{q+k-1}, \end{aligned}$$

(2.1)

where $\rho \in (0,\frac{R}{3}]$, $\beta = \frac{kn}{kn-2k+2}$ and $C_1$ is a positive constant ensuring $\varDelta u + C_1\ge 1$ a.e. (see the paragraph after Remark 2.3 for the justification of the existence of $C_1$). Similarly, in Case 2 of Theorem 1.5 and in Theorem 1.8, we will obtain, for a solution $u\in W_{\mathrm{loc}}^{2,q+k}(\varOmega )$ to (1.4) with $q>\frac{(k+1)n}{2}-k$, the estimate

$$\begin{aligned} \bigg (\int _{B_{R+\rho }}(\varDelta u+C_1)^{\beta q}\bigg )^{1/\beta }\le \frac{Cq}{\rho ^2}\int _{B_{R+3\rho }}(\varDelta u+C_1)^{q+k}, \end{aligned}$$

(2.2)

now with $\beta = \frac{(k+1)n}{(k+1)n - 2(k+1)+2}$. The estimates (2.1) and (2.2) then yield an improvement in integrability under the respective lower bounds on q, which can then be iterated to yield the desired $C^{1,1}_{\mathrm{loc}}$ estimates.¹

In the rest of this section we explain how the estimates (2.1) and (2.2) are obtained. Due to the lack of regularity, we derive our estimates through taking difference quotients of the Eq. (1.4). For an index $l\in \{1,\dots ,n\}$ and increment $h\in {\mathbb {R}}\backslash \{0\}$, we recall the first order difference quotient $\nabla ^h_l u(x) {:}{=}h^{-1}(u(x+he_l)-u(x))$ and the second order difference quotient

$$\begin{aligned} \ \varDelta _{ll}^h u(x) {:}{=}\nabla _l^h(\nabla ^{-h}_l u(x)) = \frac{u(x+he_l)-2u(x) + u(x-he_l)}{h^2}. \end{aligned}$$

(2.3)

We also denote

$$\begin{aligned} v_h(x) {:}{=}\sum _{l=1}^n \varDelta _{ll}^h u(x). \end{aligned}$$

The above expressions are well-defined for $x\in \varOmega _h{:}{=}\{y\in \varOmega :\mathrm{dist}(y,\partial \varOmega )>|h|\}$.

It is well-known (see, for instance, [20, Lemma 7.23]) that

$$\begin{aligned} \Vert \nabla _l^h u\Vert _{L^s(\varOmega ')} \le \Vert \nabla _l u \Vert _{L^s(\varOmega )} \quad \text {for all } s\ge 1 \text { and } \varOmega '\Subset \varOmega \text { s.t. }\mathrm{dist}(\varOmega ',\partial \varOmega )>|h|. \end{aligned}$$

(2.4)

It follows from (2.3) and (2.4) that there exists a constant $C=C(n)$ such that

$$\begin{aligned} \Vert v_h\Vert _{L^s(\varOmega ')} \le C\Vert \nabla ^2 u \Vert _{L^s(\varOmega )}\quad \text {for all }s\ge 1. \end{aligned}$$

(2.5)

We will also use the following fact – see Appendix B for a proof:

Lemma 2.1

Suppose $u \in W^{2,s}(\varOmega )$ for some $s\ge 1$. Then $v_h \rightarrow \varDelta u$ in $L^s_{\mathrm {loc}}(\varOmega )$ as $h\rightarrow 0$.

We assume now that both the increment h and our solution u are fixed, and write v as shorthand for $v_h$. Taking difference quotients of the equation $\sigma _k^{1/k}(A_H[u](x))=f[u](x){:}{=}f(x,u(x),\nabla u(x))$ and appealing to the concavity of $\sigma _k^{1/k}$ in $\varGamma _k^+$, we will derive (at the start of Sect. 4) the pointwise estimate

$$\begin{aligned} \sum _l k(f[u])^{k-1}\varDelta _{ll}^h f[u]&\le F[u]^{ij}\nabla _i\nabla _j v - \sum _l F[u]^{ij}\varDelta _{ll}^h (H[u])_{ij}\quad \mathrm {a.e.~in~}\varOmega _h. \end{aligned}$$

(2.6)

Here, $F[u]^{ij}=\partial \sigma _k(A_H[u])/\partial (A_H[u])_{ij}$ is the linearised operator.

Remark 2.2

In (2.6), and from this point onwards, summation notation is employed only over repeated indices which appear in both upper and lower positions. Positioning of indices is purely to indicate whether summation convention is being utilised; since we are working with the Euclidean metric, we are free to raise and lower indices at will. For instance, $A_{ij}$, $A^i_j$, $A^j_i$ and $A^{ij}$ all denote the (i, j)-entry of a symmetric matrix A. Similarly, we do not distinguish between the derivatives $\nabla ^i$ and $\nabla _i$ when using index notation.

Remark 2.3

Since u is fixed, we write $f[u], H[u], A_H[u], F[u]^{ij}$ etc. to emphasise that these are to be considered as functions of x. If it is clear from the context (e.g. if there are no derivatives involved), we will simply write $f, H, A_H, F^{ij}$ etc.

The estimates (2.1) and (2.2) are derived by testing (2.6) against suitable test functions. First fix a ball $B_{2R}\Subset \varOmega _h$. Since $\lambda (A_H)\in \varGamma _2^+$ is equivalent to $\mathrm{tr}(A_H) = \varDelta u - \mathrm{tr}(H)>0$ and $\sigma _2(A_H)>0$, there exists a constant $C_1\ge 0$ (depending on an upper bound for $\Vert H\Vert _{C^0(\varSigma )}$ – see Remark 1.6) for which $\varDelta u + C_1\ge 1$ and $|\nabla ^2 u| \le \varDelta u + C_1$ a.e. in $B_{2R}$. We define ${\tilde{v}} {:}{=}v+C_1$, and for a small parameter $\delta >0$ (that we eventually take to zero) we denote

$$\begin{aligned} Q_\delta {:}{=}\big (({\tilde{v}}^+)^2 + \delta ^2 \big )^{1/2}. \end{aligned}$$

For $\rho \in (0,\frac{R}{3}]$ we also let $\eta \in C_{\mathrm {c}}^\infty (B_{R+2\rho })$ be a standard non-negative cutoff function. Testing (2.6) against $\eta Q_\delta ^{q-1}$ (where $q>1$) then yields

$$\begin{aligned}&\sum _l\int _{B_{R+2\rho }}k\eta Q_\delta ^{q-1}f^{k-1}\varDelta _{ll}^h f[u] \nonumber \\&\quad \le \int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}F^{ij}\nabla _i\nabla _j {\tilde{v}} - \sum _l\int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}F^{ij}\varDelta _{ll}^h (H[u])_{ij} \end{aligned}$$

(2.7)

for all $q>1$ and $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ solving (1.4).

For ease of outlining our argument, let us suppose that $f=f(x,z)$ (the general case $f=f(x,z,\xi )$ will only require minor changes - see Sect. 5.3). Then the integrand on the left hand side (LHS) of (2.7) is a lower order term, whereas the integrands on the RHS of (2.7) involve higher order terms, formally of fourth and third order in the limit $h\rightarrow 0$, and thus need to be treated.

In Sect. 4, we integrate by parts in the first integral on the RHS of (2.7), using a result of Sect. 3 that tells us $\nabla _iF[u]^{ij}$ is a regular distribution belonging to $L_{\mathrm{loc}}^{(q+k-1)/(k-1)}(\varOmega )$ if $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$. After taking $\delta \rightarrow 0$ and carrying out some further calculations (see Lemmas 4.2 and 4.3), we will obtain the estimate

$$\begin{aligned}&\frac{q-1}{Cq^2}\int _{B_{R+\rho }}f^k\frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} + \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\nabla _i F[u]^{ij}\nabla _j{\tilde{v}} \nonumber \\&\qquad + \sum _l \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1} F^{ij}\varDelta _{ll}^h (H[u])_{ij} \nonumber \\&\quad \le \frac{C}{\rho ^2}\bigg (\int _{B_{R+2\rho }}({\tilde{v}}^+)^{q+k-1} + \int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+k-1}\bigg ), \end{aligned}$$

(2.8)

where C is a constant independent of h, q and $\rho$.

Whilst the first integral on the LHS of (2.8) is a favourable positive higher order term, the other two integrals on the LHS (which we denote by $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$, respectively) involve higher order terms which are, a priori, of unknown sign. Treating $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ is the most technical part of our proof.

Now, if we momentarily assume sufficiently high regularity on u, say $u\in W_{\mathrm{loc}}^{2,q+2k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$), the issue of dealing with $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ is largely simplified. As will be detailed in the proof of Theorem 6.1, one may apply the Cauchy inequality to each of the integrands and absorb the resulting third order terms into the positive term on the LHS of (2.8). Under the stated integrability assumption, this crude estimation is sufficient to show

$$\begin{aligned} \frac{q-1}{q^2}\int _{B_{R+\rho }}f^k\frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} \le \frac{C}{\rho ^2}\bigg (\int _{B_{R+2\rho }}({\tilde{v}}^+)^{q+2k-1} + \int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+2k-1}\bigg ). \end{aligned}$$

An estimate analogous to (2.1) and (2.2) can then be obtained, assuming $q>kn-2k+1$.

The difficulty is to therefore deal with $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ under the weaker integrability assumptions of Theorems 1.5 and 1.8. At this point, we make the distinction between the various cases. In each case, we estimate $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ so as to produce a cancellation phenomenon when combined, leaving only lower order terms; see Lemmas 4.4 and 4.5 for the estimates on $\mathrm{(I_2)}_h$, Lemma 4.10 for the estimates on $\mathrm{(I_3)}_h$, and Corollaries 4.12, 4.13 and 4.14 for the resulting cancellations. It will then follow from (2.8) that, in Case 1 of Theorem 1.5 with the relaxed assumption $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$), we have the estimate

$$\begin{aligned} \frac{q-1}{q^2}\int _{B_{R+\rho }}f^k\frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} \le \frac{C}{\rho ^2}\bigg (\int _{B_{R+2\rho }}({\tilde{v}}^+)^{q+k-1} + \int _{B_{R+3\rho }}(\varDelta u+C_1)^{q+k-1}\bigg ).\nonumber \\ \end{aligned}$$

(2.9)

Similarly, in the remaining cases with $u\in W_{\mathrm{loc}}^{2,q+k}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$), we will obtain

$$\begin{aligned} \frac{q-1}{q^2}\int _{B_{R+\rho }}f^k\frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} \le \frac{C}{\rho ^2}\bigg (\int _{B_{R+2\rho }}({\tilde{v}}^+)^{q+k}+\int _{B_{R+3\rho }}(\varDelta u+C_1)^{q+k}\bigg ).\nonumber \\ \end{aligned}$$

(2.10)

To obtain (2.1) from (2.9) (resp. (2.2) from (2.10)), we proceed as follows (the details can be found in Sect. 5). We first obtain an integral estimate for $\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2$, to which we can apply the Sobolev inequality. We then justify taking the limit $h\rightarrow 0$ and impose the lower bound $q+k-1> \frac{kn}{2}$ (resp. $q+k>\frac{(k+1)n}{2}$), from which we obtain (2.1) (resp. (2.2)).

3 Divergence structure of the linearised operator $F[u]^{ij}$

In this section we derive a divergence formula for the linearised operator $F[u]^{ij}$ (defined in (1.6)), which we will use at various stages of our proof.

We note that in the case that $A_H[u]=\nabla ^2 u$ or $A_H[u] = A_u$, the divergence properties of $F[u]^{ij}$ are well-documented (for smooth u). In the former case, $F[u]^{ij}$ is divergence-free with respect to the flat metric (see [52]), and in the latter case, $u^{1-k}F[u]^{ij}$ is divergence-free with respect to the conformal metric $g_{ij}=u^{-2}\delta _{ij}$ (see [63]). For related discussions, see also [4, 5, 26, 34, 53].

For $A\in \mathrm{Sym}_n({\mathbb {R}})$ and $1\le k\le n$, define the k’th Newton tensor of A inductively by

$$\begin{aligned} T_k(A) {:}{=}\sigma _k(A)I - T_{k-1}(A) A, \quad T_0(A)^{ij} {:}{=}\delta ^{ij}. \end{aligned}$$

(3.1)

It is well-known (see [52]) that

$$\begin{aligned} \frac{\partial \sigma _k(A)}{\partial A_{ij}} = T_{k-1}(A)^{ij} \end{aligned}$$

(3.2)

and

$$\begin{aligned} \mathrm{tr}(T_k(A)) = (n-k)\sigma _k(A), \end{aligned}$$

(3.3)

and moreover $T_{k-1}(A)^{ij}$ is positive definite when $\lambda (A)\in \varGamma _k^+$ (see [6]). In particular, by (1.6) and (3.2), $F[u]^{ij} = T_{k-1}(A_H[u])^{ij}$.

Lemma 3.1

Let $\varOmega \subset {\mathbb {R}}^n$ be a domain and $u\in C^3(\varOmega )$. Then for $H\in C^1(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ and $2\le k \le n$,

$$\begin{aligned} \nabla _i F[u]^{ij} = \sum _{p=1}^{k-1} (-1)^{p+1}T_{k-p-1}(A_H)^{ab}\Big (\nabla _a (H[u])^c_b - \nabla ^c (H[u])_{ab}\Big )(A_H^{p-1})^j_c {=}{:}V[u]^j.\nonumber \\ \end{aligned}$$

(3.4)

Moreover, if $H(x,z,\xi ) = H_2(x,z,\xi )I$, then

$$\begin{aligned} \nabla _i F[u]^{ij} = -(n-k+1)\nabla _i (H_2[u])\, T_{k-2}(A_H)^{ij}. \end{aligned}$$

(3.5)

Proof

The identity (3.4) will follow once we show that for $1\le k\le n-1$,

$$\begin{aligned} \nabla _i T_k(A_H[u])^{ij} = \sum _{p=1}^k (-1)^{p+1} T_{k-p}(A_H)^{ab}\Big (\nabla _a (H[u])^c_b - \nabla ^c (H[u])_{ab}\Big )(A_H^{p-1})^j_c \quad \mathrm {in~}\varOmega . \end{aligned}$$

(3.6)

Similarly, (3.5) will follow once we show that for $1\le k\le n-1$ and $H(x,z,\xi )=H_2(x,z,\xi )I$,

$$\begin{aligned} \nabla _i T_k(A_H[u])^{ij} = -(n-k)\nabla _i (H_2[u])\,T_{k-1}(A_H)^{ij}\quad \mathrm {in~}\varOmega . \end{aligned}$$

(3.7)

To this end, we take the divergence of both sides in (3.1), which yields

$$\begin{aligned} \nabla _i T_k(A_H[u])^{ij}&= \nabla ^j\sigma _k(A_H[u]) - \nabla _i \big (T_{k-1}(A_H[u])^{il}(A_H[u])^j_l\big ) \nonumber \\&= \frac{\partial \sigma _k(A_H)}{\partial (A_H)_{il}}\nabla ^j (A_H[u])_{il} - \nabla _i(T_{k-1}(A_H[u]))^{il}(A_H)^j_l\nonumber \\&\quad - T_{k-1}(A_H)^{il}\nabla _i (A_H[u])^j_l\nonumber \\&{\mathop {=}\limits ^{(3.2)}} T_{k-1}(A_H)^{il}\big (\nabla ^j(A_H[u])_{il}-\nabla _i(A_H[u])^j_l\big ) - \nabla _i(T_{k-1}(A_H[u]))^{il}(A_H)^j_l \nonumber \\&= T_{k-1}(A_H)^{il}\big (\nabla _i (H[u])_l^j - \nabla ^j (H[u])_{il}\big ) - \nabla _i\big (T_{k-1}(A_H[u])\big )^{il}(A_H)^j_l. \end{aligned}$$

(3.8)

Then (3.6) is readily seen by applying (3.8) iteratively.

We now turn to (3.7), for which we apply an induction argument on k using (3.8). The base case $k=1$ is clear. We suppose that for some $k\ge 2$ we have the identity

$$\begin{aligned} \nabla _i T_{k-1}(A_H[u])^{ij} = -(n-k+1)\nabla _i (H_2[u])\,T_{k-2}(A_H)^{ij}, \end{aligned}$$

(3.9)

and we show that (3.7) then follows. First observe that, by (3.9) and the fact $H_{ij} = H_2\delta _{ij}$, (3.8) simplifies to

$$\begin{aligned} \nabla _i T_k(A_H[u])^{ij}&=\nabla _i (H_2[u]) T_{k-1}(A_H)^{ij} -\nabla ^j (H_2[u]) \mathrm{tr}(T_{k-1}(A_H)) \nonumber \\&\quad + (n-k+1)\nabla _i (H_2[u]) (T_{k-2}(A_H) A_H)^{ij}. \end{aligned}$$

(3.10)

After substituting (3.1) and (3.3) into the last term and the penultimate term in (3.10), respectively, we arrive at (3.7). $\square$

Note that $V[u]^j$ (defined in (3.4)) contains at most second order derivatives of u. As a consequence, $\nabla _i F[u]^{ij}$ is a regular distribution for $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$. More precisely, we have:

Lemma 3.2

Let $\varOmega \subset {\mathbb {R}}^n$ be a domain and $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ with $q>1$ and $2\le k\le n$. Then for $H\in C_{\mathrm{loc}}^{0,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n;\mathrm{Sym}_n({\mathbb {R}}))$ and $\phi \in W_0^{1,s}(\varOmega ;{\mathbb {R}}^n)$, $s{:}{=}\frac{q+k-1}{q}$, we have

$$\begin{aligned} \int _{\varOmega } F[u]^{ij}\nabla _i\phi _j = -\int _{\varOmega }V[u]^j\phi _j, \end{aligned}$$

(3.11)

where $V[u]^j$ is defined in (3.4). In particular, $\nabla _i F[u]^{ij} = V[u]^j\in L_{\mathrm{loc}}^{(q+k-1)/(k-1)}(\varOmega )$ and

$$\begin{aligned} \big |\nabla _i F[u]^{ij}\big | \le C\big (1+ |\nabla ^2 u |^{k-1} \big )\quad \mathrm {a.e.~in~}B_{2R}, \end{aligned}$$

(3.12)

where C is a constant depending on an upper bound for $\Vert H\Vert _{C^{0,1}(\varSigma )}$.

Proof

It is clear that $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ implies $V[u]^j \in L_{\mathrm{loc}}^{(q+k-1)/(k-1)}(\varOmega )$. Since $\frac{1}{s} + \frac{k-1}{q+k-1}=1$, it suffices to prove (3.11) for $\phi \in C_0^\infty (\varOmega ;{\mathbb {R}}^n)$. Let $u_{(m)}\in C^3(\varOmega )$ be such that $u_{(m)}\rightarrow u$ in $W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )$. Then by (3.4), we have for each $m\in {\mathbb {N}}$ the identity $\nabla _i F[u_{(m)}]^{ij} = V[u_{(m)}]^j$, and it follows that

$$\begin{aligned} \int _{\varOmega } F[u_{(m)}]^{ij}\nabla _i\phi _j = -\int _{\varOmega } V[u_{(m)}]^j\phi _j. \end{aligned}$$

(3.13)

Now, since $u_{(m)}\rightarrow u$ in $W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )$, we have both $F^{ij}[u_{(m)}]\rightarrow F^{ij}[u]$ and $V[u_{(m)}]\rightarrow V[u]$ in $L_{\mathrm{loc}}^{(q+k-1)/(k-1)}(\varOmega )$. In particular, we can take $m\rightarrow \infty$ in (3.13) to get (3.11). The estimate (3.12) follows from the definition of $V[u]^j$. $\square$

4 Main estimates

In this section we prove our main estimates, which will then be used in the proof of our main results in Sect. 5. Largely, our estimates will be concerned with terms involving the contraction of the linearised operator $F=(F^{ij})$ and its divergence with various other tensors, such as $\nabla ^2 {\tilde{v}}$, $\nabla {\tilde{v}}$ and $(\varDelta ^h_{ll}H[u]_{ij})$.

4.1 Initial integral estimates: isolating higher order terms

The following lemma provides the starting point for our integral estimates:

Lemma 4.1

Suppose $f\in C^0(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n)$ is positive, $H\in C^0(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ and u is a solution to (1.4). Then for fixed h,

$$\begin{aligned} \sum _l kf^{k-1}\varDelta _{ll}^h f[u]&\le F^{ij}\nabla _i\nabla _j v - \sum _l F^{ij}\varDelta _{ll}^h (H[u])_{ij} \quad \mathrm {a.e.~in~}\varOmega _h. \end{aligned}$$

(4.1)

Proof

The proof follows [60], with some adjustments. For $A\in \mathrm{Sym}_n({\mathbb {R}})$, let $G^{ij}(A) = \partial \sigma _k^{1/k}(A)/\partial A_{ij} = k^{-1}\sigma _k(A)^{(1-k)/k}F^{ij}(A)$, and denote $G^{ij} {:}{=}G^{ij}(A_H[u])$. Fix $l\in \{1,\dots ,n\}$ and $h\in {\mathbb {R}}\backslash \{0\}$. Then there exists a set $S_{h,l}\subset \varOmega _h$ with ${\mathscr {L}}(\varOmega _h\backslash S_{h,l})=0$ (where ${\mathscr {L}}$ is the Lebesgue measure) such that $\lambda (A_H[u](x)), \lambda (A_H[u](x\pm he_l))\in \varGamma _k^+$ for all $x\in S_{h,l}$. By concavity of $\sigma _k^{1/k}$ in $\varGamma _k^+$, it follows that for $x\in S_{h,l}$ we have

$$\begin{aligned} \sigma _k^{1/k}\big (A_H[u](x\pm he_l)\big ) - \sigma _k^{1/k}(A_H[u](x)) \le G^{ij}(x)\big (A_H[u](x\pm he_l)- A_H[u](x)\big )_{ij}.\nonumber \\ \end{aligned}$$

(4.2)

Adding the two equations in (4.2), dividing through by $h^2$ and summing over l, we have

$$\begin{aligned} \sum _l \varDelta _{ll}^h \sigma _k^{1/k}\big (A_H[u](x)\big ) \le \sum _l G^{ij}(x) \varDelta ^h_{ll}\big (A_H[u](x)\big )_{ij}\quad \text {for all }x\in S_{h} {:}{=}\bigcap _{l=1}^n S_{h,l},\nonumber \\ \end{aligned}$$

(4.3)

with $S_h$ clearly satisfying ${\mathscr {L}}(\varOmega _h\backslash S_{h})=0$. Substituting the definition of $G^{ij}$ into (4.3) and recalling that $A_H[u] = \nabla ^2 u - H[u]$, we obtain

$$\begin{aligned} \sum _l k\sigma _k^{\frac{k-1}{k}}(A_H)\varDelta _{ll}^h \sigma _k^{1/k}(A_H[u]) \le \sum _l F^{ij} \varDelta _{ll}^h\big (\nabla ^2 u - H[u]\big )_{ij}\quad \mathrm {in~}S_h. \end{aligned}$$

(4.4)

Substituting the equation $\sigma _k^{1/k}(A_H) = f$ into the LHS of (4.4), and commuting difference quotients with derivatives on the RHS of (4.4), we arrive at (4.1). $\square$

As outlined in Sect. 2, we proceed to derive a series of integral estimates by multiplying (4.1) by suitable test functions and integrating by parts using the divergence structure proved in Lemma 3.2. Recall that for a fixed increment $h>0$, we defined $v(x) = \sum _l \varDelta _{ll}^h u(x)$, and that we fixed a ball $B_{2R}\Subset \varOmega _h$ and a constant $C_1$ (depending on an upper bound for $\Vert H\Vert _{C^0(\varSigma )}$) such that $\varDelta u + C_1 \ge 1$ and $|\nabla ^2 u| \le \varDelta u +C_1$ a.e. in $B_{2R}$. The existence of such a constant is guaranteed by the assumption $\lambda (A_H)\in \varGamma _2^+$. We then defined ${\tilde{v}} = v + C_1$, and for a small parameter $\delta >0$ (that we eventually take to zero) we defined $Q_\delta = \big (({\tilde{v}}^+)^2 + \delta ^2\big )^{1/2}$. For $\rho \in (0,\frac{R}{3}]$, we also fix a cutoff function $\eta \in C_{\mathrm {c}}^\infty (B_{R+2\rho })$ satisfying $0\le \eta \le 1$, $\eta \equiv 1$ on $B_{R+\rho }$ and $|\nabla ^l\eta | \le C(n)\rho ^{-l}$ for $l=1,2$.

Suppose $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$) is a solution to (1.4). Multiplying (4.1) by $\eta Q_\delta ^{q-1}$ and integrating over the domain $B_{R+2\rho }$, we see

$$\begin{aligned}&\sum _l\int _{B_{R+2\rho }}k\eta Q_\delta ^{q-1}f^{k-1}\varDelta _{ll}^h f[u] \nonumber \\&\quad \le \int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}F^{ij}\nabla _i\nabla _j {\tilde{v}} - \sum _l\int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}F^{ij}\varDelta _{ll}^h (H[u])_{ij}, \end{aligned}$$

(4.5)

which is just the estimate (2.7) in Sect. 2, repeated here for convenience.

We are now in a position to prove our first integral estimate. In what follows, let

$$\begin{aligned} J_h^{(s)} {:}{=}\int _{B_{R+2\rho }}({\tilde{v}}^+)^{s} + \int _{B_{R+3\rho }}(\varDelta u + C_1)^{s}. \end{aligned}$$

Roughly speaking, if $u\in W_{\mathrm{loc}}^{2,s}(\varOmega )$ then $J_h^{(s)}$ should be interpreted as a lower order term, and terms bounded by $J_h^{(s)}$ are consequently considered ‘good terms’.

We will first address the case $f=f(x,z)$ for simplicity and postpone the more general case until Sect. 5.3. The relevant equation is therefore

$$\begin{aligned} \sigma _k^{1/k}\big (A_H[u](x)\big ) = f(x,u(x))>0, \quad \lambda (A_H[u](x)) \in \varGamma _k^+ \quad \mathrm {for~a.e.~}x\in \varOmega . \end{aligned}$$

(4.6)

Throughout Sect. 4, unless otherwise stated, C will denote a generic positive constant which may vary from line to line, depending only on n, R, f, H and an upper bound for $\Vert u\Vert _{W^{1,\infty }(B_{2R})}$. In particular, C is independent of h, q and $\rho$, and any norm of $\nabla ^2 u$. In addition, we will often use the inequalities $\varDelta u + C_1 \ge 1$ and $|\nabla ^2 u| \le \varDelta u +C_1$ without explicit reference.

Lemma 4.2

Suppose $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}})$ is positive, $H\in C_{\mathrm{loc}}^{0,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ and $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$) is a solution to (4.6). Then for $R>0$ with $B_{2R}\Subset \varOmega$, $\rho \in (0,\frac{R}{3}]$ and |h| sufficiently small, we have

$$\begin{aligned}&(q-1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-2}F^{ij}\nabla _i{\tilde{v}}\nabla _j{\tilde{v}} + \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\nabla _i F[u]^{ij}\nabla _j{\tilde{v}} \nonumber \\&\quad + \sum _l \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1} F^{ij}\varDelta _{ll}^h (H[u])_{ij} \le C\rho ^{-2} J_h^{(q+k-1)}. \end{aligned}$$

(4.7)

Proof

Appealing to Lemma 3.2 with $\phi _j = \eta Q_\delta ^{q-1}\nabla _j {\tilde{v}}$, and noting that

$$\begin{aligned} \nabla _i\phi _j = Q_\delta ^{q-1}\nabla _i\eta \nabla _j {\tilde{v}} +(q-1){\tilde{v}}^+Q_\delta ^{q-3}\nabla _i {\tilde{v}}\nabla _j {\tilde{v}} + \eta Q_\delta ^{q-1}\nabla _i\nabla _j {\tilde{v}}, \end{aligned}$$

we have

$$\begin{aligned}&\int _{B_{R+2\rho }}F^{ij}\Big ( Q_\delta ^{q-1}\nabla _i\eta \nabla _j {\tilde{v}} +(q-1){\tilde{v}}^+Q_\delta ^{q-3}\nabla _i {\tilde{v}}\nabla _j {\tilde{v}} + \eta Q_\delta ^{q-1}\nabla _i\nabla _j {\tilde{v}}\Big ) \nonumber \\&\quad = -\int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}\nabla _i F[u]^{ij}\nabla _j {\tilde{v}}. \end{aligned}$$

(4.8)

Rearranging (4.8) to get the desired integration by parts formula for $\int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}F^{ij}\nabla _i\nabla _j {\tilde{v}}$, and substituting this back into (4.5), we obtain

$$\begin{aligned}&(q-1)\int _{B_{R+2\rho }}\eta {\tilde{v}}^+Q_\delta ^{q-3}F^{ij}\nabla _i {\tilde{v}}\nabla _j {\tilde{v}} + \int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}\nabla _i F[u]^{ij}\nabla _j {\tilde{v}} \nonumber \\&\quad + \sum _l\int _{B_{R+2\rho }}\eta Q_\delta ^{q-1}F^{ij}\varDelta _{ll}^h (H[u])_{ij} \le - \int _{B_{R+2\rho }} Q_\delta ^{q-1}F^{ij}\nabla _i\eta \nabla _j {\tilde{v}}\nonumber \\&\quad - \sum _l\int _{B_{R+2\rho }}k\eta Q_\delta ^{q-1}f^{k-1}\varDelta _{ll}^h f[u]. \end{aligned}$$

(4.9)

We now take $\delta \rightarrow 0$ in (4.9), using Fatou’s lemma for the first integral (which is positive) and the dominated convergence theorem elsewhere (which is justified since $q>1$). This yields

$$\begin{aligned}&(q-1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-2}F^{ij}\nabla _i {\tilde{v}}\nabla _j {\tilde{v}} + \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\nabla _i F[u]^{ij}\nabla _j {\tilde{v}} \nonumber \\&\quad + \sum _l\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}F^{ij}\varDelta _{ll}^h (H[u])_{ij} \le - \int _{B_{R+2\rho }} ({\tilde{v}}^+)^{q-1}F^{ij}\nabla _i\eta \nabla _j {\tilde{v}} \nonumber \\&\quad - \sum _l\int _{B_{R+2\rho }}k\eta ({\tilde{v}}^+)^{q-1}f^{k-1}\varDelta _{ll}^h f[u]. \end{aligned}$$

(4.10)

To conclude the proof of Lemma 4.2, we must bound the RHS of (4.10) from above by $C\rho ^{-2}J_h^{(q+k-1)}$. We begin with the first integral on the RHS of (4.10). Appealing again to Lemma 3.2, now with $\phi _j = \frac{1}{q}({\tilde{v}}^+)^q\nabla _j \eta$ and $\nabla _i\phi _j = ({\tilde{v}}^+)^{q-1}\nabla _i {\tilde{v}}\nabla _j \eta + \frac{1}{q}({\tilde{v}}^+)^q\nabla _i\nabla _j \eta$, we have

$$\begin{aligned} \int _{B_{R+2\rho }}F^{ij}\Big (({\tilde{v}}^+)^{q-1}\nabla _i \eta \nabla _j {\tilde{v}} + \frac{1}{q}({\tilde{v}}^+)^q\nabla _i\nabla _j \eta \Big ) = -\frac{1}{q}\int _{B_{R+2\rho }}({\tilde{v}}^+)^q\nabla _i F[u]^{ij}\nabla _j \eta . \end{aligned}$$

Therefore,

$$\begin{aligned}&\bigg |\int _{B_{R+2\rho }} ({\tilde{v}}^+)^{q-1}F^{ij}\nabla _i\eta \nabla _j {\tilde{v}}\bigg | \nonumber \\&\quad \le \bigg |\frac{1}{q}\int _{B_{R+2\rho }}({\tilde{v}}^+)^qF^{ij}\nabla _i\nabla _j\eta \bigg | + \bigg |\frac{1}{q}\int _{B_{R+2\rho }}({\tilde{v}}^+)^q\nabla _i F[u]^{ij}\nabla _j\eta \bigg | \nonumber \\&\quad \le \frac{C}{\rho ^2}\int _{B_{R+2\rho }}({\tilde{v}}^+)^q |F| + \frac{C}{\rho } \int _{B_{R+2\rho }}({\tilde{v}}^+)^q \big | \mathrm{div}F[u]\big |, \end{aligned}$$

(4.11)

where $F = (F^{ij})$. Recalling $|F|\le C(\varDelta u + C_1)^{k-1}$ and applying Hölder’s inequality to the penultimate integral in (4.11), we see that $\int _{B_{R+2\rho }}({\tilde{v}}^+)^q |F| \le CJ_h^{(q+k-1)}$. The final integral in (4.11) satisfies the same estimate, since $|\mathrm{div}F[u]|\le C(\varDelta u + C_1)^{k-1}$ by (3.12).

It remains to estimate the second term on the RHS of (4.10). Keeping in mind that $f=f(x,z)\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}})$, we apply Hölder’s inequality followed by (2.5) to obtain

$$\begin{aligned}&\bigg |\sum _l\int _{B_{R+2\rho }}k\eta ({\tilde{v}}^+)^{q-1}f^{k-1}\varDelta _{ll}^h f[u]\bigg | \nonumber \\&\quad \le C \bigg (\int _{B_{R+2\rho }}({\tilde{v}}^+)^q\bigg )^\frac{q-1}{q}\bigg (\int _{B_{R+2\rho }}\bigg |\sum _l\varDelta _{ll}^h f[u]\bigg |^{q}\bigg )^{\frac{1}{q}} \nonumber \\&\quad {\mathop {\le }\limits ^{(2.5)}} C\bigg (\int _{B_{R+2\rho }}({\tilde{v}}^+)^q\bigg )^\frac{q-1}{q}\bigg (\int _{B_{R+3\rho }}\big |\varDelta f[u]\big |^{q}\bigg )^{\frac{1}{q}} \le CJ_h^{(q)}. \end{aligned}$$

(4.12)

This concludes the proof. $\square$

To clear up notation, we denote the three integrals on the LHS of (4.7) involving higher order terms by

$$\begin{aligned} \mathrm{(I_1)}_h&{:}{=}(q-1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-2}F^{ij}\nabla _i {\tilde{v}}\nabla _j {\tilde{v}}, \\ \mathrm{(I_2)}_h&{:}{=}\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\nabla _i F[u]^{ij}\nabla _j {\tilde{v}}\,\, \text { and} \\ \mathrm{(I_3)}_h&{:}{=}\sum _l\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}F^{ij}\varDelta _{ll}^h (H[u])_{ij}. \end{aligned}$$

The terms $\mathrm{(I_1)}_h$, $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ will be considered in turn. In Sect. 4.2, we prove an estimate for $\mathrm{(I_1)}_h$. In Sect. 4.3.1, we estimate $\mathrm{(I_2)}_h$ in the case that H is a multiple of the identity, and in Sect. 4.3.2 we estimate $\mathrm{(I_2)}_h$ for general H when $k=2$. The estimate for $\mathrm{(I_3)}_h$ in the general case is slightly involved, so for illustrative purposes we first address the simpler case when $H(x,z,\xi )=H_1(x,z)|\xi |^2I$ with $H_1\ge 0$, which includes the $\sigma _k$-Yamabe equation in the positive case. This is done in Sect. 4.4.1. The estimate for $\mathrm{(I_3)}_h$ in the general case is proved in Sect. 4.4.2. In the process, we will prove the cancellation phenomenon between $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ alluded to earlier – see Corollaries 4.12, 4.13 and 4.14.

4.2 A pointwise lower bound for $F[u]^{ij}\nabla _i{\tilde{v}}\nabla _j{\tilde{v}}$

The term $F^{ij}\nabla _i{\tilde{v}}\nabla _j {\tilde{v}}$ in $\mathrm{(I_1)}_h$ can be bounded in the same way as in [60] (see Eq. (3.6) therein). We reproduce the argument here for the reader’s convenience.

Lemma 4.3

Suppose $f\in C^0(\varOmega \times {\mathbb {R}})$ is positive, $H\in C^0(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ and u is a solution to (4.6). Then for $q>0$,

$$\begin{aligned} (v^+)^{q-2}F^{ij}\nabla _i{\tilde{v}}\nabla _j {\tilde{v}} \ge \frac{4f^k}{q^2}\frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)}\quad \mathrm {a.e.~in~}\varOmega _h. \end{aligned}$$

(4.13)

In particular, for $R>0$ with $B_{2R}\Subset \varOmega$, $\rho \in (0,\frac{R}{3}]$, $q>1$ and |h| sufficiently small, we have

$$\begin{aligned} \mathrm{(I_1)}_h \ge \frac{q-1}{Cq^2}\int _{B_{R+\rho }}f^k\frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)}. \end{aligned}$$

(4.14)

Proof

Denote by ${\mathscr {M}}_{k}^+\subset \mathrm{Sym}_n({\mathbb {R}})$ the set of symmetric matrices M with $\lambda (M)\in \varGamma _{k}^+$. For $1\le l\le n$, denote by $F_{(l)}^{ij}(A)$ the matrix with entries $\partial \sigma _l(A)/\partial A_{ij}$. Using the concavity of $\sigma _k(A)/\sigma _{k-1}(A)$ on ${\mathscr {M}}_k^+$, we have

$$\begin{aligned} \frac{F_{(k)}^{ij}(A)}{\sigma _k(A)} \ge \frac{F_{(k-1)}^{ij}(A)}{\sigma _{k-1}(A)}\quad \text {for all }A\in {\mathscr {M}}_k^+ \end{aligned}$$

(4.15)

(see e.g. [48, 60]). Applying (4.15) inductively, it follows that

$$\begin{aligned} \frac{F_{(k)}^{ij}(A)}{\sigma _k(A)} \ge \cdots \ge \frac{F_{(1)}^{ij}(A)}{\sigma _1(A)} = \frac{\delta ^{ij}}{\mathrm{tr}(A)}\quad \text {for all }A\in {\mathscr {M}}_k^+. \end{aligned}$$

(4.16)

Taking $A=A_H[u]$ in (4.16), where u is a solution to (4.6), we obtain

$$\begin{aligned} \frac{F[u]^{ij}(x)}{f^k[u](x)} \ge \frac{\delta ^{ij}}{\varDelta u(x) - \mathrm{tr}(H[u](x))} \quad \text {for a.e. }x\in \varOmega , \end{aligned}$$

from which (4.13) is readily seen. The estimate (4.14) then follows from properties of $\eta$. $\square$

4.3 Integral estimates for $\nabla _i F[u]^{ij}\nabla _j{\tilde{v}}$

In this section we obtain estimates for the term $\mathrm{(I_2)}_h = \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\nabla _i F[u]^{ij}\nabla _j {\tilde{v}}$. The case in which H is a multiple of the identity matrix will be dealt with first, in Sect. 4.3.1. The case for general H when $k=2$ will then be addressed in Sect. 4.3.2.

4.3.1 The case $H=H_2(x,z,\xi )I$

In what follows we denote $\mathrm{tr}(F) = \sum _i F^{ii}$. We prove the following two lemmas which address the case that H is a multiple of the identity:

Lemma 4.4

Suppose $f\in C^0(\varOmega \times {\mathbb {R}})$ is positive, $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ with $H(x,z,\xi )=H_1(x,z)|\xi |^2I$, and that $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$) is a solution to (4.6). Then for $R>0$ with $B_{2R}\Subset \varOmega$, $\rho \in (0,\frac{R}{3}]$ and |h| sufficiently small, we have

$$\begin{aligned} \mathrm{(I_2)}_h&\ge -\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(F) \frac{\partial (H_1|\xi |^2)}{\partial \xi _a}[u]\nabla _a {\tilde{v}} - C\rho ^{-1}J_h^{(q+k-1)}. \end{aligned}$$

(4.17)

Lemma 4.5

Suppose $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ with $H(x,z,\xi ) = H_2(x,z,\xi )I$, and that $u\in W_{\mathrm{loc}}^{2,q+k}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$). Then for $R>0$ with $B_{2R}\Subset \varOmega$, $\rho \in (0,\frac{R}{3}]$ and |h| sufficiently small, we have

$$\begin{aligned} \mathrm{(I_2)}_h&\ge -\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(F) \frac{\partial H_2}{\partial \xi _a}[u]\nabla _a {\tilde{v}} - C\rho ^{-1}J_h^{(q+k)}. \end{aligned}$$

(4.18)

Remark 4.6

Note that in Lemma 4.5, we do not assume that u solves (4.6). In contrast, the fact that Lemma 4.4 holds under a weaker integrability assumption uses both the fact that u solves (4.6) and that $H_2$ depends quadratically on $\nabla u$.

Remark 4.7

The first term on the RHS of (4.17) and (4.18) will later be shown to cancel with a term arising from our estimate for $\mathrm{(I_3)}_h$.

Proof of Lemmas 4.4 and 4.5

The proof consists of three steps. In Step 1, we prove a preliminary estimate assuming only $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ and $H=H_2(x,z,\xi )I$, but we do not assume at this point that u necessarily solves (4.6). Only in Steps 2 and 3 will we appeal to the specific hypotheses of Lemmas 4.4 and 4.5.

Our starting point is the following expression for $\mathrm{(I_2)}_h$, which follows from (3.5):

$$\begin{aligned} \mathrm{(I_2)}_h = -(n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\nabla _j (H_2[u])\, T_{k-2}(A_H)^{ij}\nabla _i {\tilde{v}}. \end{aligned}$$

Step 1: In this step, we show that for every $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$,

$$\begin{aligned} \mathrm{(I_2)}_h&\ge -\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(F) \frac{\partial H_2}{\partial \xi _a}[u]\nabla _a{\tilde{v}} \nonumber \\&\quad -\frac{n-k+1}{q}\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^qF^i_a\frac{\partial ^2 H_2}{\partial \xi _a\partial \xi _b}[u](A_H)_{ib} - C\rho ^{-1}J_h^{(q+k-1)}. \end{aligned}$$

(4.19)

Note that the first integral on the RHS of (4.19) is the desired term seen in (4.17) and (4.18).

First observe that by the chain rule,

$$\begin{aligned} \mathrm{(I_2)}_h&= -(n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial x^j}[u]\,T_{k-2}(A_H)^{ij}\nabla _i {\tilde{v}} \nonumber \\&\quad - (n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial z}[u]\,T_{k-2}(A_H)^{ij}\nabla _j u\nabla _i {\tilde{v}} \nonumber \\&\quad - (n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,T_{k-2}(A_H)^{ij}\nabla _j\nabla _a u\, \nabla _i {\tilde{v}}. \end{aligned}$$

(4.20)

Denote the top two lines of the RHS of (4.20) collectively by $L_1$, and the bottom line by $L_2$. Recalling that $\nabla _j\nabla _a u = H_2\delta _{ja} + (A_H)_{ja}$ and, in view of (3.1) and (3.3), that

$$\begin{aligned} \big (T_{k-2}(A_H) A_H\big )_{ia} = - F_{ia} + \frac{1}{n-k+1}\mathrm{tr}(F)\delta _{ia}, \end{aligned}$$

(4.21)

we have

$$\begin{aligned} L_2&= - (n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,T_{k-2}(A_H)_{ia}H_2\nabla ^i {\tilde{v}} \\&\quad \,\, -(n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,\big (T_{k-2}(A_H) A_H\big )_{ia}\nabla ^i {\tilde{v}} \\&{=}{\mathop {-}\limits ^{(4.21)}} (n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,T_{k-2}(A_H)_{ia}H_2\nabla ^i {\tilde{v}} \\&\quad \,\,\,+ (n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,F_{ia}\nabla ^i {\tilde{v}} \\&\quad \,\,\, -\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(F)\frac{\partial H_2}{\partial \xi _a}[u]\,\nabla _a {\tilde{v}}. \end{aligned}$$

Substituting this identity for $L_2$ into (4.20) yields

$$\begin{aligned} \mathrm{(I_2)}_h&= L_1 - (n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,T_{k-2}(A_H)_{ia}H_2\nabla ^i {\tilde{v}} \nonumber \\&\quad + (n-k+1)\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,F_{ia}\nabla ^i {\tilde{v}} \nonumber \\&\quad -\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(F)\frac{\partial H_2}{\partial \xi _a}[u]\,\nabla _a {\tilde{v}}. \end{aligned}$$

(4.22)

We claim that the terms on the top line of the RHS of (4.22) are bounded from below by $- C\rho ^{-1}J_h^{(q+k-1)}$. Indeed, as $T_{k-2}(A_H)^{ij} = \partial \sigma _{k-1}(A_H)/\partial A_{ij}$, by Lemma 3.2 we have $|\nabla _i T_{k-2}(A_H[u])^{ij}| \le C(\varDelta u + C_1)^{k-2}$. It is also clear that $|T_{k-2}(A_H)^{ij}| \le C(\varDelta u + C_1)^{k-2}$. Thus, after integrating by parts using Lemma 3.2 and applying Hölder’s inequality, the lower bound for these terms follows.

To estimate the penultimate integral in (4.22), we integrate by parts using Lemma 3.2 and apply the identity

$$\begin{aligned}&\nabla _i\bigg (\frac{\partial H_2}{\partial \xi _a}[u](x)\bigg ) \\&\quad = \bigg (\frac{\partial ^2 H_2}{\partial \xi _a\partial \xi _b}[u](x)\bigg )\big ((A_H)_{ib} + H_{ib}\big ) + \bigg (\frac{\partial ^2 H_2}{\partial z\partial \xi _a}[u](x)\bigg )\nabla _i u(x) + \frac{\partial ^2 H_2}{\partial x^i\partial \xi _a} [u](x). \end{aligned}$$

After an application of Hölder’s inequality, this gives

$$\begin{aligned}&\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H_2}{\partial \xi _a}[u]\,F_{ia}\nabla ^i {\tilde{v}} \\&\quad \ge -\frac{1}{q}\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^qF^i_a\frac{\partial ^2 H_2}{\partial \xi _a\partial \xi _b}[u]\,(A_H)_{ib} - C\rho ^{-1}J_h^{(q+k-1)}, \end{aligned}$$

from which (4.19) follows.

Step 2: In this step we prove Lemma 4.5. Indeed, for $u\in W_{\mathrm{loc}}^{2,q+k}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ (not necessarily solving (4.6)) we have the estimate

$$\begin{aligned} -\frac{n-k+1}{q}\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^qF^i_a\frac{\partial ^2 H_2}{\partial \xi _a\partial \xi _b}[u]\,(A_H)_{ib}&\ge -C\int _{B_{R+2\rho }}({\tilde{v}}^+)^q|F||A_H| \\&\ge - CJ_h^{(q+k)}, \end{aligned}$$

where $F=(F^{ij})$ and the last inequality follows once again from the estimate $|F|\le C(\varDelta u + C_1)^{k-1}$ and Hölder’s inequality. Substituting this into (4.19) then yields the desired estimate (4.18).

Step 3: In this step we prove Lemma 4.4. Since we assume in this case that $H_2(x,z,\xi ) = H_1(x,z)|\xi |^2$ and that u solves (4.6), rather than estimating as in Step 2 we observe

$$\begin{aligned} F^i_a\frac{\partial ^2 H_2}{\partial \xi _a \partial \xi _b}[u](A_H)_{ib} =2H_1F^i_a\delta ^{ab}(A_H)_{ib} = 2H_1F^i_a (A_H)^a_i = 2H_1k\sigma _k(A_H) = 2H_1kf^k.\nonumber \\ \end{aligned}$$

(4.23)

Substituting (4.23) into the second integral in (4.19), we arrive at (4.17). $\square$

4.3.2 The case $k=2$ for general H

In this section we obtain an estimate in the case $k=2$ analogous to (4.17) and (4.18). We do not assume that H is a multiple of the identity and, as in Lemma 4.5, we do not assume that u solves (4.6):

Lemma 4.8

Suppose $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$, $k=2$ and $u\in W_{\mathrm{loc}}^{2,q+2}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$). Then for $R>0$ with $B_{2R}\Subset \varOmega$, $\rho \in (0,\frac{R}{3}]$ and |h| sufficiently small, we have

$$\begin{aligned} \mathrm{(I_2)}_h&\ge \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial H^{ij}}{\partial \xi _a}[u]\,\nabla _i\nabla _a u\,\nabla _j {\tilde{v}} - \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\frac{\partial \mathrm{tr}(H)}{\partial \xi _a}[u]\,\mathrm{tr}(A_H)\nabla _a {\tilde{v}} \nonumber \\&\quad - C\rho ^{-1}J_h^{(q+2)}. \end{aligned}$$

(4.24)

Remark 4.9

The first two terms on the RHS of (4.24) will later be shown to cancel with a term arising from our estimate for $\mathrm{(I_3)}_h$ (cf. Remark 4.7).

Proof of Lemma 4.8

As $k=2$, we have $\nabla _i F[u]^{ij} = \nabla _i H[u]^{ij} - \nabla ^j \mathrm{tr}(H[u])$ (by (3.4)) and $\nabla ^j\nabla ^a u = \mathrm{tr}(A_H)\delta ^{ja} - F^{ja} - H^{ja}$. It follows that

$$\begin{aligned}&\mathrm{(I_2)}_h = \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\big (\nabla _i H[u]^{ij}-\nabla ^j\mathrm{tr}(H[u]) \big )\nabla _j {\tilde{v}} \nonumber \\&\quad = \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\bigg (\frac{\partial H^{ij}}{\partial \xi _a}[u]\,\nabla _i\nabla _a u - \frac{\partial \mathrm{tr}(H)}{\partial \xi ^a}[u]\,\nabla ^j\nabla ^a u\bigg )\nabla _j{\tilde{v}} \nonumber \\&\qquad + \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\bigg (\frac{\partial H^{ij}}{\partial x^i}[u] + \frac{\partial H^{ij}}{\partial z}[u]\,\nabla _i u - \frac{\partial \mathrm{tr}(H)}{\partial x_j}[u] - \frac{\partial \mathrm{tr}(H)}{\partial z}[u]\,\nabla ^j u\bigg )\nabla _j{\tilde{v}}\nonumber \\&\quad = \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\bigg (\frac{\partial H^{ij}}{\partial \xi _a}[u]\,\nabla _i\nabla _a u - \frac{\partial \mathrm{tr}(H)}{\partial \xi ^a}[u]\,\mathrm{tr}(A_H)\delta ^{ja}\bigg )\nabla _j{\tilde{v}} \nonumber \\&\qquad + \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\bigg (\frac{\partial H^{ij}}{\partial x^i}[u] + \frac{\partial H^{ij}}{\partial z}[u]\,\nabla _i u - \frac{\partial \mathrm{tr}(H)}{\partial x_j}[u] - \frac{\partial \mathrm{tr}(H)}{\partial z}[u]\,\nabla ^j u \nonumber \\&\qquad + \frac{\partial \mathrm{tr}(H)}{\partial \xi ^a}[u](F^{ja}+H^{ja}) \bigg )\nabla _j{\tilde{v}}. \end{aligned}$$

(4.25)

The integral on the last two lines of (4.25) can be bounded from below by $- C\rho ^{-1}J_h^{(q+2)}$ in exactly the same way as in the proof of Lemmas 4.4 and 4.5: we integrate by parts using Lemma 3.2, estimate the relevant quantities in terms of $\varDelta u + C_1$ and apply Hölder’s inequality. The estimate (4.24) then follows. $\square$

4.4 Integral estimates for $F[u]^{ij}\varDelta _{ll}^h H[u]_{ij}$

In this section we obtain estimates for the quantity $\mathrm{(I_3)}_h =$ $\sum _l\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}F^{ij}\varDelta _{ll}^h (H[u])_{ij}$. More precisely, we prove the following lemma:

Lemma 4.10

Suppose $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$, $R>0$ is such that $B_{2R}\Subset \varOmega$ and $\rho \in (0,\frac{R}{3}]$.

• a) If $u\in W_{\mathrm{loc}}^{2,q+k}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$), then for |h| sufficiently small, we have

  $$\begin{aligned} \mathrm{(I_3)}_h \ge \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}F^{ij}\frac{\partial H_{ij}}{\partial \xi _a}[u]\,\nabla _a {\tilde{v}} - CJ_h^{(q+k)}. \end{aligned}$$

  (4.26)

• b) If $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$) and $H(x,z,\xi )=H_1(x,z)|\xi |^2 I$ with $H_1 \ge 0$, then for |h| sufficiently small, we have

  $$\begin{aligned} \mathrm{(I_3)}_h \ge \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}F^{ij}\frac{\partial H_{ij}}{\partial \xi _a}[u]\,\nabla _a {\tilde{v}} - CJ_h^{(q+k-1)}. \end{aligned}$$

  (4.27)

Remark 4.11

Neither estimate in Lemma 4.10 requires u to be a solution to (4.6).

Before proving Lemma 4.10 we first discuss its consequences, namely the resulting cancellations between $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$. First consider the case $H=H_1(x,z)|\xi |^2I$ with $H_1\ge 0$:

Corollary 4.12

Suppose $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}})$ is positive, $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ with $H = H_1(x,z)|\xi |^2I$ and $H_1 \ge 0$, and that $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$) is a solution to (4.6). Then for $R>0$ with $B_{2R}\Subset \varOmega$, $\rho \in (0,\frac{R}{3}]$ and |h| sufficiently small, we have

$$\begin{aligned} \mathrm{(I_2)}_h + \mathrm{(I_3)}_h \ge - C\rho ^{-1}J_h^{(q+k-1)}. \end{aligned}$$

(4.28)

In particular,

$$\begin{aligned} \begin{aligned} \frac{q-1}{q^2} \int _{B_{R+\rho }} {f}^k \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)} \le C\rho ^{-2}J_h^{(q+k-1)}. \end{aligned} \end{aligned}$$

(4.29)

Proof

The estimate (4.28) follows from combining the estimates (4.17) and (4.27). The estimate (4.29) is then obtained by substituting (4.14) and (4.28) into (4.7). $\square$

Similarly, we obtain the following in the case that $H=H_2(x,z,\xi )I$:

Corollary 4.13

Suppose $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$ with $H=H_2(x,z,\xi )I$, and $u\in W_{\mathrm{loc}}^{2,q+k}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$). Then for $R>0$ with $B_{2R}\Subset \varOmega$, $\rho \in (0,\frac{R}{3}]$ and |h| sufficiently small, we have

$$\begin{aligned} \mathrm{(I_2)}_h + \mathrm{(I_3)}_h \ge - C\rho ^{-1}J_h^{(q+k)}. \end{aligned}$$

(4.30)

If, in addition, u solves (4.6) for some positive $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}})$, then

$$\begin{aligned} \begin{aligned} \frac{q-1}{q^2} \int _{B_{R+\rho }} {f}^k \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)} \le C\rho ^{-2}J_h^{(q+k)}. \end{aligned} \end{aligned}$$

(4.31)

Proof

The estimate (4.30) follows from combining the estimates (4.18) and (4.26). The estimate (4.31) is then obtained by substituting (4.14) and (4.30) into (4.7). $\square$

A similar cancellation also holds in the setting of Theorem 1.8, although this requires a little more work:

Corollary 4.14

Suppose $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$, $k=2$ and $u\in W_{\mathrm{loc}}^{2,q+2}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$). Then for |h| sufficiently small, we have

$$\begin{aligned} \mathrm{(I_2)}_h + \mathrm{(I_3)}_h \ge - C\rho ^{-2}J_h^{(q+2)}. \end{aligned}$$

(4.32)

If, in addition, u solves (4.6) for some positive $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}})$, then

$$\begin{aligned} \begin{aligned} \frac{q-1}{q^2} \int _{B_{R+\rho }} {f}^2 \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)} \le C\rho ^{-1}J_h^{(q+2)}. \end{aligned} \end{aligned}$$

(4.33)

Proof

The estimate (4.33) will immediately follow once (4.32) is established, by substituting (4.14) and (4.32) into (4.7).

Taking $k=2$ in Lemma 4.10 a) and using $F^{ij} = \mathrm{tr}(A_H)\delta ^{ij} - \nabla ^i\nabla ^j u - H^{ij}$, we see

$$\begin{aligned} \mathrm{(I_3)}_h&\ge \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(A_H)\frac{\partial \mathrm{tr}(H)}{\partial \xi _a}[u]\,\nabla _a{\tilde{v}} - \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\nabla ^i\nabla ^j u\frac{\partial H_{ij}}{\partial \xi _a}[u]\,\nabla _a{\tilde{v}} \nonumber \\&\quad - \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}H^{ij}\frac{\partial H_{ij}}{\partial \xi _a}[u]\,\nabla _a{\tilde{v}} - CJ_h^{(q+2)}. \end{aligned}$$

(4.34)

Now, the first term on the RHS of (4.34) cancels with the second term on the RHS of (4.24), and the first term on the last line of (4.34) can be estimated by $-C\rho ^{-1}J_h^{(q+2)}$, after integrating by parts and applying Hölder’s inequality. Therefore, combining (4.24) and (4.34), we obtain

$$\begin{aligned} \mathrm{(I_2)}_h + \mathrm{(I_3)}_h \ge \frac{1}{q}\int _{B_{R+2\rho }}\!\!\!\eta \frac{\partial H_{ij}}{\partial \xi _a}[u]\,\bigg (\!\nabla ^i\nabla _a u\,\nabla ^j ({\tilde{v}}^+)^q - \nabla ^i\nabla ^j u\,\nabla _a ({\tilde{v}}^+)^q \bigg )\!- C\rho ^{-1}J_h^{(q+2)}. \end{aligned}$$

Now, if u were to have enough regularity, we could integrate by parts here, observe that the third derivatives of u cancel, and obtain (4.32) by estimating the remaining terms in the usual way. To circumvent the lack of regularity, we instead apply the following lemma:

Lemma 4.15

Let $U\subset {\mathbb {R}}^n$ be a smooth bounded domain and let $B\in L^\infty (U;{\mathbb {R}}^{n\times n})$ be an antisymmetric matrix with $\mathrm{supp}(B)\Subset U$. For $1\le p<\infty$ and $p'{:}{=}\frac{p}{p-1}$, consider the bilinear form ${\mathscr {B}}:W^{1,p}(U)\times W^{1,p'}(U)\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} {\mathscr {B}}(g,h) = \int _U B_j^a\,\nabla _a g\,\nabla ^j h. \end{aligned}$$

(4.35)

If $\mathrm{div}B\in L^q(U;{\mathbb {R}}^n)$ with $\frac{1}{p} + \frac{1}{q} = 1 - \frac{1}{r}$ for some $1\le q,r\le \infty$, then we have the estimate

$$\begin{aligned} |{\mathscr {B}}(g,h)| \le \int _U |\mathrm{div}B||\nabla g||h| \end{aligned}$$

(4.36)

for all $g\in W^{1,p}(U)$ and $h \in W^{1,p'}(U)\cap L^r(U)$.

Before proving Lemma 4.15, we use it to complete the proof of (4.32): for each $i\in \{1,\dots ,n\}$, taking $B^a_j = \eta \frac{\partial H_{ij}}{\partial \xi _a}[u] - \eta \frac{\partial H_i^a}{\partial \xi ^j}[u]$, $g= \nabla _i u$ and $h = ({\tilde{v}}^+)^q$ in Lemma 4.15 we obtain

$$\begin{aligned}&\int _{B_{R+2\rho }}\eta \frac{\partial H_{ij}}{\partial \xi _a}[u]\,\bigg (\nabla ^i \nabla _a u \nabla ^j({\tilde{v}}^+)^q-\nabla ^i\nabla ^j u\nabla _a({\tilde{v}}^+)^q\bigg ) \\&\quad {\mathop {\le }\limits ^{(4.36)}} C\rho ^{-1}\int _{B_{R+2\rho }}(\varDelta u +C_1)^{2}({\tilde{v}}^+)^q \le C\rho ^{-1}J_h^{(q+2)}. \end{aligned}$$

It remains to prove Lemma 4.15. By a standard approximation argument, it suffices to prove (4.36) for $g,h\in C^\infty (U)$. We are then justified in integrating by parts in (4.35), giving

$$\begin{aligned} |{\mathscr {B}}(g,h)| = \bigg |\int _U \bigg (\nabla ^j B_j^a\, \nabla _a g + \underbrace{B^a_j\,\nabla _a\nabla ^jg}_{=0}\bigg )h\bigg | \le \int _U |\mathrm{div}B||\nabla g||h|, \end{aligned}$$

where we have used antisymmetry of B to assert that $B^a_j\,\nabla _a\nabla ^jg = 0$. $\square$

4.4.1 Proof of Lemma 4.10 b)

We now turn our attention back to the proof of Lemma 4.10. Whilst the two estimates (4.26) and (4.27) can be dealt with simultaneously (see the proof of Lemma 4.10 in Sect. 4.4.2), for illustrative purposes we first provide a more direct proof of (4.27), which includes the $\sigma _k$-Yamabe equation in the positive case. Indeed, when $H=H_1(x,z)|\xi |^2I$ we are able to calculate $\varDelta _{ll}^h (H[u])_{ij}$ explicitly by deriving the following discrete version of the Bochner identity, avoiding the more involved estimates required for the general case. In what follows, we denote

$$\begin{aligned} u_l^h(x){:}{=}u(x+he_l). \end{aligned}$$

Lemma 4.16

(Discrete Bochner identity) Suppose $H_1\in C^0(\varOmega \times {\mathbb {R}})$ and $l\in \{1,\dots ,n\}$. Then

$$\begin{aligned} \varDelta _{ll}^h\big (H_1[u]|\nabla u|^2\big )&= 2H_1\nabla ^i u\nabla _i\varDelta _{ll}^h u + (H_1[u])^{-h}_l\big |\nabla \nabla _l^{-h} u\big |^2 + (H_1[u])^h_l\big |\nabla \nabla _l^h u\big |^2 \nonumber \\&\quad + \nabla _l^{-h}\nabla _i u \nabla ^i u \nabla _l^{-h}H_1[u] + \nabla _l^h\nabla ^i u \nabla _i u \nabla _l^h H_1[u] \nonumber \\&\quad + \nabla _l^h\Big (\nabla _i u(\nabla ^i u)_l^{-h}\nabla _l^{-h}H_1[u]\Big ). \end{aligned}$$

(4.37)

Assuming the validity of Lemma 4.16, the proof of (4.27) in Lemma 4.10 b) is then straightforward:

Proof of Lemma 4.10 b)

Substituting the discrete Bochner identity (4.37) into the definition of $\mathrm{(I_3)}_h$ and dropping the two positive terms, we obtain

$$\begin{aligned} \mathrm{(I_3)}_h&\ge 2\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(F)H_1\nabla ^i u \nabla _i {\tilde{v}} \nonumber \\&\quad + \sum _l \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}\mathrm{tr}(F)\bigg (\nabla _l^{-h}\nabla _i u \nabla ^i u \nabla _l^{-h}H_1[u] + \nabla _l^h\nabla ^i u \nabla _i u \nabla _l^h H_1[u] \nonumber \\&\quad + \nabla _l^h\Big (\nabla _i u(\nabla ^i u)_l^{-h}\nabla _l^{-h}H_1[u]\Big )\bigg ). \end{aligned}$$

(4.38)

After applying the difference quotient product rule

$$\begin{aligned} \nabla ^h_l (uv)(x) = u^h_l(x)\nabla _l^h v(x) + v(x)\nabla _l^h u(x) \end{aligned}$$

(4.39)

to the integrand in the last line of (4.38), we may then estimate the last two lines of (4.38) in the usual way. Namely, after applying the bound $\mathrm{tr}(F)\le C(\varDelta u + C_1)^{k-1}$, using Hölder’s inequality and appealing to (2.4), we see that the last two lines of (4.38) are collectively bounded from below by $-CJ_h^{(q+k-1)}$. The estimate (4.27) then follows. $\square$

Proof of the discrete Bochner identity (Lemma 4.16)

Using the product rule (4.39) to first calculate $\nabla ^{-h}_l(H_1[u]|\nabla u|^2)$, we see

$$\begin{aligned}&\varDelta _{ll}^h \big (H_1[u]|\nabla u|^2\big ) = \nabla ^h_l\Big (\nabla _l^{-h}\big (H_1[u]\nabla ^i u \nabla _i u\big )\Big ) \\&\quad = \nabla ^h_l \Big (\!(H_1[u]\nabla ^i u)^{-h}_l \nabla _l^{-h}\nabla _i u\!\Big )\!+ \nabla _l^h \Big (\!H_1[u]\nabla _i u\nabla _l^{-h}\nabla ^i u\!\Big )\!+ \nabla _l^h\Big (\!\nabla _i u(\nabla ^i u)_l^{-h}\nabla _l^{-h}H_1[u]\Big ). \end{aligned}$$

On the other hand, noting that $\nabla _l^h u_l^{-h} u(x) = \nabla _l^{-h} u(x)$ and $(\nabla ^{-h}_l u)^h_l(x) = \nabla ^h_l u(x)$, we also have by (4.39) the identities

$$\begin{aligned} \nabla ^h_l \Big ((H_1[u]\nabla ^i u)^{-h}_l \nabla _l^{-h} \nabla _i u \Big )&= H_1\nabla ^i u \nabla ^h_l\nabla _l^{-h}\nabla _i u + \nabla _l^{-h}\nabla _i u \nabla _l^{-h}(H_1[u]\nabla ^i u) \\&= H_1\nabla ^i u \nabla _i\varDelta _{ll}^h u + (H_1[u])^{-h}_l\big |\nabla \nabla _l^{-h} u\big |^2 \\&\quad + \nabla _l^{-h}\nabla _i u \nabla ^i u \nabla _l^{-h}H_1[u] \end{aligned}$$

and

$$\begin{aligned} \nabla _l^h \Big (H_1[u]\nabla _i u\nabla _l^{-h}\nabla ^i u\Big )&= \nabla ^h_l \nabla ^i u \nabla _l^h\Big (H_1[u]\nabla _i u \Big ) + H_1\nabla _i u \nabla _l^h\nabla _l^{-h}\nabla ^i u \\&= (H_1[u])^h_l\big |\nabla \nabla _l^h u\big |^2 + \nabla _l^h\nabla ^i u \nabla _i u \nabla _l^h H_1[u] + H_1\nabla _i u \nabla ^i\varDelta _{ll}^h u. \end{aligned}$$

Putting these three identities together, we arrive at (4.37). $\square$

4.4.2 Proof of Lemma 4.10 in the general case

We now prove Lemma 4.10 in the general case. To simplify our analysis, we will make use of the following semi-convexity property of $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n;\mathrm{Sym}_n({\mathbb {R}}))$: there exists a constant $C_\varSigma >0$ such that the mapping $\xi \longmapsto H(x,z,\xi ) + C_\varSigma |\xi |^2 I$ is convex for all $(x,z,\xi )\in \varSigma$ (this is an immediate consequence of the $C_{\mathrm{loc}}^{1,1}$ regularity of H). We will make use of this property in the form

$$\begin{aligned} H_{ij}(x,z,\xi ) \ge H_{ij}(x,z,\zeta )+ \frac{\partial H_{ij}}{\partial \xi _a}(x,z,\zeta )(\xi -\zeta )_a - C_\varSigma \delta _{ij}|\xi -\zeta |^2 \end{aligned}$$

(4.40)

for all $(x,z,\xi ),(x,z,\zeta )\in \varSigma$. Note that in Case 1 of Theorem 1.5, we may take $C_\varSigma =0$ in (4.40), as $H(x,z,\xi )=H_1(x,z)|\xi |^2I$ is convex with respect to $\xi$ when $H_1\ge 0$. The inequality (4.40) will play a role similar to that of the discrete Bochner identity used in the previous section (see Lemma 4.16).

Proof of Lemma 4.10

We first prove Lemma 4.10 a). It suffices to show that

$$\begin{aligned} F^{ij}\varDelta _{ll}^h (H[u])_{ij} \ge F^{ij}\frac{\partial H_{ij}}{\partial \xi _a}[u]\,\nabla _a\varDelta _{ll}^h u + \text {error terms} \quad \forall l\in \{1,\dots ,n\}, \end{aligned}$$

(4.41)

where the error terms satisfy

$$\begin{aligned} \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}|\text {error terms}| \le CJ_h^{(q+k)}. \end{aligned}$$

(4.42)

To keep notation succinct, we denote $x^\pm = x\pm he_l$ and $F=(F^{ij})$ in what follows.

Step 1: We first prove a lower bound for $F^{ij}(x) \varDelta _{ll}^h (H[u](x))_{ij}$, identifying the error terms in (4.41). Observe that by (4.40) and the fact that $F^{ij}$ is positive definite in $\varGamma _k^+$, we have

$$\begin{aligned}&\frac{F^{ij}(x)}{h^2}\bigg [(H[u](x^\pm ))_{ij} - H(x^\pm , u(x^\pm ), \nabla u(x))_{ij}\bigg ] \\&\quad \, \ge \frac{F^{ij}(x)}{h^2}\frac{\partial H_{ij}}{\partial \xi _a}\big (x^\pm ,u(x^\pm ),\nabla u(x)\big )\big (\nabla _a u(x^\pm )-\nabla _a u(x)\big ) - \frac{C_\varSigma |F|}{h^2}\big |\nabla u(x^\pm )-\nabla u(x)\big |^2 \\&\quad \ge \frac{F^{ij}(x)}{h^2}\frac{\partial H_{ij}}{\partial \xi _a}[u](x)\big (\nabla _a u(x^\pm )-\nabla _a u(x)\big ) - \frac{C_\varSigma |F|}{h^2}\big |\nabla u(x^\pm )-\nabla u(x)\big |^2 \\&\qquad - \frac{C|F|}{|h|}\big |\nabla u(x^\pm )- \nabla u(x)\big | \quad \text {for a.e. }x\in B_{R+2\rho }, \end{aligned}$$

where to obtain the second inequality we have estimated

$$\begin{aligned} \bigg |\frac{\partial H_{ij}}{\partial \xi _a}\big (x^\pm ,u(x^\pm ),\nabla u(x)\big ) - \frac{\partial H_{ij}}{\partial \xi _a}[u](x)\bigg |&\le \Vert H\Vert _{C^{1,1}(\varSigma )}\big (|x^\pm -x| + |u(x^\pm )-u(x)|\big ) \\&\le C|h|. \end{aligned}$$

Recalling the definition of $\varDelta _{ll}^h (H[u](x))_{ij}$, we therefore see that for a.e. $x\in B_{R+2\rho }$,

$$\begin{aligned}&F^{ij}(x) \varDelta _{ll}^h (H[u](x))_{ij} \nonumber \\&\quad \, \ge F^{ij}(x)\frac{\partial H_{ij}}{\partial \xi _a}[u](x)\nabla _a \varDelta _{ll}^h u(x) \nonumber \\&\qquad + \frac{F^{ij}(x)}{h^2}\bigg (H(x^+,u(x^+),\nabla u(x))_{ij} - 2(H[u](x))_{ij}+ H(x^-,u(x^-),\nabla u(x))_{ij}\bigg ) \nonumber \\&\qquad - C_\varSigma |F||\nabla _l^h\nabla u|^2 - C_\varSigma |F||\nabla _l^{-h}\nabla u|^2 - C|F||\nabla _l^h\nabla u| - C|F||\nabla _l^{-h}\nabla u|. \end{aligned}$$

(4.43)

Step 2: To prove (4.26), we need to show that the error terms in last two lines of (4.43) satisfy (4.42). Formally, these terms behave like $|F|(|\nabla ^2 u|^2 + |\nabla ^2 u|)$, and so by the estimate $|F| \le C(\varDelta u + C_1)^{k-1}$, the bound (4.42) is then conceivable. We now give the details.

Denote the terms on the penultimate line of (4.43) collectively by $E_1$, and the terms on the last line of (4.43) collectively by $E_2$. The error terms in $E_2$ are easier to deal with. Indeed, by the bound $|F| \le C(\varDelta u + C_1)^{k-1}$, Hölder’s inequality and (2.4), we have

$$\begin{aligned}&\int _{B_{R+2\rho }}\!\eta ({\tilde{v}}^+)^{q-1}|F||\nabla _l^{\pm h}\nabla u|^2 \le C (J_h^{(q+k)})^\frac{q+k-2}{q+k} \bigg (\int _{B_{R+2\rho }}|\nabla _l^{\pm h}\nabla u|^{q+k}\bigg )^\frac{2}{q+k} \le CJ_h^{(q+k)}. \end{aligned}$$

(4.44)

In exactly the same way, one can show $\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}|F||\nabla _l^{\pm h} \nabla u|\le C J_h^{(q+k-1)}$, and combining these estimates we obtain $\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}|E_2| \le CJ_h^{(q+k)}$.

We now treat the error terms in $E_1$. We first observe that by the fundamental theorem of calculus followed by the chain rule, we have the identities

$$\begin{aligned}&H(x^\pm ,u(x^\pm ),\xi )_{ij} - H(x,u(x),\xi )_{ij} \\&\quad = \int _0^1 \frac{d}{dt}H(x\pm the_l,u(x^\pm ),\xi )_{ij}\,dt + \int _0^1 \frac{d}{dt}H(x,u(x\pm the_l),\xi )_{ij}\,dt \\&\quad = \pm h\int _0^1 \frac{\partial H_{ij}}{\partial x^l}(x\pm the_l,u(x^\pm ),\xi )\,dt \pm h\int _0^1 \frac{\partial H_{ij}}{\partial z}(x,u(x\pm the_l), \xi )\nabla _l u(x\pm the_l)\,dt, \end{aligned}$$

and therefore

$$\begin{aligned}&H(x^+,u(x^+),\xi )_{ij} - 2H(x,u(x),\xi )_{ij} + H(x^-,u(x^-),\xi )_{ij} \nonumber \\&\quad = h\int _0^1\bigg (\frac{\partial H_{ij}}{\partial z}(x,u(x+the_l), \xi )\nabla _l u(x+the_l) \nonumber \\&\qquad - \frac{\partial H_{ij}}{\partial z}(x,u(x-the_l),\xi )\nabla _l u(x-the_l) \bigg )\,dt \nonumber \\&\qquad + h\int _0^1\bigg (\frac{\partial H_{ij}}{\partial x^l}(x+the_l,u(x^+),\xi ) - \frac{\partial H_{ij}}{\partial x^l}(x-the_l,u(x^-),\xi ) \bigg )\,dt. \end{aligned}$$

(4.45)

Now, by the $C_{\mathrm{loc}}^{1,1}$ regularity of H and the Lipschitz regularity of the mapping (x, z, p) $\mapsto \frac{\partial H_{ij}}{\partial z}(x,z,\xi )p_l$ for fixed $\xi$ and each $l\in \{1,\dots ,n\}$, we can estimate the last line of (4.45) from above by $Ch^2$ and the middle line of (4.45) from above by

$$\begin{aligned} Ch^2 + Ch^2\int _0^1 \frac{1}{t|h|}\Big |\nabla _l u(x+the_l) - \nabla _l u(x-the_l)\Big |\,dt. \end{aligned}$$

Applying these estimates in (4.45) and taking $\xi =\nabla u(x)$, we therefore see that

$$\begin{aligned} |E_1|&\le C|F| + C|F|\int _0^1|\nabla _l^{th}\nabla _l u(x)|\,dt + C|F|\int _0^1 |\nabla _l^{-th}\nabla _l u(x)|\,dt. \end{aligned}$$

(4.46)

Using (4.46), one readily obtains the estimate $\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}|E_1| \le CJ_h^{(q+k-1)}$, applying the same line of argument as seen above for $E_2$. For example, by Fubini’s theorem and Young’s inequality, we have

$$\begin{aligned}&\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}|F|\bigg (\int _0^1 |\nabla ^{\pm th}_l\nabla _l u(x)|\,dt\bigg )\,dx \\&\quad = \int _0^1\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}|F||\nabla _l^{\pm th}\nabla _l u(x)|\,dx\,dt \\&\quad \le C J_h^{(q+k-1)} + C \int _0^1\int _{B_{R+2\rho }}|\nabla ^{\pm th}_l \nabla _l u|^{q+k-1}\,dx\,dt \\&{\mathop {\le }\limits ^{(2.4)}} C J_h^{(q+k-1)}. \end{aligned}$$

This completes the proof of Lemma 4.10 a).

Step 3: It remains to prove Lemma 4.10 b) (see Sect. 4.4.1 for an alternative proof which is independent of calculations in Steps 1 and 2 above). Note that in this case, we may take $C_\varSigma =0$ in (4.43) and so the error terms on the last two lines of (4.43) formally behave like $|F||\nabla ^2 u|$. By the same argument as in Step 2, the error terms $E_1$ and $E_2$ considered in Step 2 therefore satisfy $\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}|E_i| \le CJ_h^{(q+k-1)}$, and the conclusion follows. $\square$

5 Proof of main results

In this section we use Corollaries 4.12, 4.13 and 4.14 to prove Theorems 1.5 and 1.8, as outlined at the end of Sect. 2. We will first give a detailed proof of Case 1 of Theorem 1.5 when $f=f(x,z)$ in Sect. 5.1, and then indicate the necessary adjustments for remaining cases, still when $f=f(x,z)$, in Sect. 5.2. In Sect. 5.3, we extend these results to the case $f=f(x,z,\xi )$, completing the proofs of Theorems 1.5 and 1.8.

5.1 Proof of Case 1 of Theorem 1.5 when $f=f(x,z)$

In this case, we recall that by Corollary 4.12 we have the estimate

$$\begin{aligned} \begin{aligned} \int _{B_{R+\rho }} {f}^k \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)} \le \frac{Cq}{\rho ^2}J_h^{(q+k-1)}, \end{aligned} \end{aligned}$$

(5.1)

where $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$). Let $\theta \in (0,1)$ be such that $\frac{2-\theta }{\theta }\le q+k-1$ (we will eventually take $\theta = \frac{4}{kn+2}$). Also denote by $(2-\theta )^* {:}{=}n(2-\theta )/(n-2+\theta )$ the Sobolev conjugate of $2-\theta$. We first obtain from (5.1) the following:

Lemma 5.1

Suppose $f\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}})$ is positive, $H = H_1(x,z)|\xi |^2 I$ with $H_1\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}})$ and $H_1\ge 0$, and that $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$) is a solution to (4.6). Then

$$\begin{aligned}&\bigg (\int _{B_{R+\rho }}(\varDelta u + C_1)^{\frac{q(2-\theta )^*}{2}}\bigg )^{\frac{2}{(2-\theta )^*}} \nonumber \\&\quad \le \frac{Cq}{\rho ^2}\bigg (\int _{B_{R+3\rho }}(\varDelta u + C_1)^{\frac{2-\theta }{\theta }}\bigg )^{\frac{\theta }{2-\theta }}\int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+k-1}. \end{aligned}$$

(5.2)

Proof

The estimate (5.2) will follow immediately once we establish the estimate

$$\begin{aligned} \bigg ( \int _{B_{R+\rho }}({\tilde{v}}^+)^{\frac{q(2-\theta )^*}{2}}\bigg )^{\frac{2}{(2-\theta )^*}} \le \frac{Cq}{\rho ^2}\big | J_h^{(\frac{2-\theta }{\theta })}\big |^{\frac{\theta }{2-\theta }} J_h^{(q+k-1)}, \end{aligned}$$

(5.3)

since we can then apply Fatou’s lemma and the fact that ${\tilde{v}}^+\rightarrow \varDelta u + C_1$ a.e. as $h\rightarrow 0$ to the term on the LHS of (5.3), and Lemma 2.1 to the terms on the RHS of (5.3).

Keeping in mind the lower bound $\inf _{B_{2R}} f>\frac{1}{C} >0$, we first observe that by Hölder’s inequality and (5.1), we have

$$\begin{aligned}&\bigg (\int _{B_{R+\rho }}\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^{2-\theta }\bigg )^{\frac{2}{2-\theta }} \nonumber \\&\quad \le C\bigg (\int _{B_{R+\rho }}(\varDelta u - \mathrm{tr}(H))^{\frac{2-\theta }{\theta }}\bigg )^{\frac{\theta }{2-\theta }}\int _{B_{R+\rho }}{f}^k\frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)} \nonumber \\&\quad {\mathop {\le }\limits ^{(5.1)}}\frac{Cq}{\rho ^2}\big | J_h^{(\frac{2-\theta }{\theta })}\big |^{\frac{\theta }{2-\theta }} J_h^{(q+k-1)}. \end{aligned}$$

(5.4)

On the other hand, since $\frac{q(2-\theta )}{2}\le q+k-1$, Hölder’s inequality gives

$$\begin{aligned} \bigg (\int _{B_{R+\rho }}({\tilde{v}}^+)^{\frac{q(2-\theta )}{2}}\bigg )^{\frac{2}{2-\theta }}&\le \bigg (\int _{B_{R+\rho }}({\tilde{v}}^+)^{\frac{2-\theta }{\theta }}\bigg )^{\frac{\theta }{2-\theta }}\int _{B_{R+\rho }}({\tilde{v}}^+)^{q-1} \nonumber \\&\le \big | J_h^{(\frac{2-\theta }{\theta })}\big |^{\frac{\theta }{2-\theta }} J_h^{(q+k-1)}. \end{aligned}$$

(5.5)

Applying the Sobolev inequality to $({\tilde{v}}^+)^{q/2}\in W^{1,2-\theta }$, and appealing to (5.4) and (5.5), we arrive at (5.3). $\square$

The inequality (5.2) is of reverse Hölder-type if $\theta$ satisfies

$$\begin{aligned} \frac{2-\theta }{\theta }<q+k-1< \frac{q(2-\theta )^*}{2}. \end{aligned}$$

For example, if we fix $\theta = \frac{4}{kn+2}$ and finally impose the assumption $q+k-1>\frac{kn}{2}$, we see that $(2-\theta )/\theta = kn/2 <q+k-1$ and

$$\begin{aligned} \frac{q(2-\theta )^*}{2} - (q + k - 1 ) > \bigg (\frac{kn}{2}-k+1\bigg )\bigg (\frac{kn}{2+kn-2k}-1\bigg ) - k + 1 = 0. \end{aligned}$$

In what follows, we denote

$$\begin{aligned} \beta {:}{=}\frac{(2-\theta )^*}{2} = \frac{kn}{kn+2-2k}>1. \end{aligned}$$

Proof of Case 1 of Theorem 1.5 when f = f (x, z)

With $\theta = \frac{4}{kn+2}$, we obtain from (5.2) the estimate

$$\begin{aligned} \bigg (\int _{B_{R+\rho }}(\varDelta u + C_1)^{\beta q}\bigg )^{1/\beta } \le \frac{Cq}{\rho ^2}\int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+k-1} \end{aligned}$$

(5.6)

for all $q>\frac{kn}{2}-k+1$ and $\rho \in (0,\frac{R}{3}]$. The constant C in (5.6) and below now depends on $\int _{B_{R+3\rho }}(\varDelta u + C_1)^{kn/2}$, which is finite due to our hypotheses.

We now carry out the Moser iteration argument. Let $p>\frac{kn}{2}$ be as in the statement of Theorem 1.5, and define a sequence $q_j$ inductively by

$$\begin{aligned} q_0 = p-k+1, \quad q_j = \beta q_{j-1} - k+1 \mathrm {~for~}j\ge 1. \end{aligned}$$

Then $q_j = \beta q_{j-1}-(k-1) = \beta ^j q_0 - (k-1)(\beta ^{j-1}+\cdots + \beta + 1)$, which implies

$$\begin{aligned} \frac{q_j}{\beta ^j} = q_0-(k-1)\bigg (\frac{1-\beta ^{-j}}{\beta -1}\bigg ) {\mathop {\longrightarrow }\limits ^{j\rightarrow \infty }} q_0 - \frac{k-1}{\beta -1} >0. \end{aligned}$$

(5.7)

Note that the limit in (5.7) is positive by definition of $\beta$ and the fact that $q_0>\frac{kn}{2} -k+1$. In particular, $q_j\rightarrow \infty$ as $j\rightarrow \infty$.

Applying (5.6) iteratively with $q=q_j$ and $\rho =3^{-j-1}R$, we have for each $j\ge 0$

$$\begin{aligned} \bigg (\int _{B_{(1+3^{-j-1})R}}(\varDelta u + C_1)^{\beta q_j}\bigg )^{\beta ^{-j-1}}&\le \bigg (9^{j}Cq_j \int _{B_{(1+3^{-j})R}}(\varDelta u + C_1)^{\beta q_{j-1}}\bigg )^{\beta ^{-j}} \\&{\mathop {\le }\limits ^{(5.7)}} \prod _{i=0}^j \big ((9\beta )^i C\big )^{\beta ^{-i}} \int _{B_{2R}}(\varDelta u + C_1)^{p} \\&\le (9\beta )^{\sum _{i=0}^\infty i\beta ^{-i}} C^{\sum _{i=0}^\infty \beta ^{-i}} \int _{B_{2R}}(\varDelta u + C_1)^{p}. \end{aligned}$$

Letting $j\rightarrow \infty$ and appealing once again to (5.7), we arrive at

$$\begin{aligned} \Vert \varDelta u + C_1\Vert _{L^\infty (B_R)} \le C\bigg (\int _{B_{2R}}(\varDelta u+C_1)^p\bigg )^{\big (q_0-\frac{k-1}{\beta -1}\big )^{-1}}, \end{aligned}$$

which implies the desired bound on $\Vert \nabla ^2 u\Vert _{L^\infty (B_R)}$ by the choice of $C_1$. $\square$

5.2 Proof of Case 2 of Theorems 1.5 and 1.8 when $f=f(x,z)$

In these cases, we recall that by Corollaries 4.13 and 4.14 we have the estimate

$$\begin{aligned} \begin{aligned} \int _{B_{R+\rho }} {f}^k \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\varDelta u - \mathrm{tr}(H)} \le \frac{Cq}{\rho ^2}J_h^{(q+k)}, \end{aligned} \end{aligned}$$

(5.8)

where $u\in W_{\mathrm{loc}}^{2,q+k}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$).

Proof of Case 2 of Theorems 1.5 and 1.8 when f = f (x, z)

We let $\theta \in (0,1)$ be such that $\frac{2-\theta }{\theta }\le q+k$. Following the same arguments as in Sect. 5.1, one readily obtains the following counterpart to the estimate (5.2):

$$\begin{aligned}&\bigg (\int _{B_{R+\rho }}(\varDelta u + C_1)^{\frac{q(2-\theta )^*}{2}}\bigg )^{\frac{2}{(2-\theta )^*}} \nonumber \\&\quad \le \frac{Cq}{\rho ^2}\bigg (\int _{B_{R+3\rho }}(\varDelta u + C_1)^{\frac{2-\theta }{\theta }}\bigg )^{\frac{\theta }{2-\theta }}\int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+k}. \end{aligned}$$

(5.9)

Taking $\theta = \frac{4}{(k+1)n+2}$ and imposing $q+k>\frac{(k+1)n}{2}$, we see

$$\begin{aligned} \frac{2-\theta }{\theta } = \frac{(k+1)n}{2}< q+k < \frac{q(2-\theta )^*}{2}. \end{aligned}$$

We thus obtain from (5.9) the estimate

$$\begin{aligned} \bigg (\int _{B_{R+\rho }}(\varDelta u + C_1)^{\beta q}\bigg )^{1/\beta } \le \frac{Cq}{\rho ^2}\int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+k}, \end{aligned}$$

where

$$\begin{aligned} \beta {:}{=}\frac{(k+1)n}{(k+1)n + 2 - 2(k+1)}> 1 \end{aligned}$$

and C now depends on $\int _{B_{R+3\rho }}(\varDelta u + C_1)^{(k+1)n/2}$. The Moser iteration argument then follows through as before, using $p>\frac{(k+1)n}{2}$ and defining $q_j$ inductively by $q_0 = p - k$ and $q_j = \beta q_{j-1} -k \text { for }j\ge 1$. $\square$

5.3 Proof of Theorems 1.5 and 1.8 for $f=f(x,z,\xi )$

In this section we explain how the preceding arguments may be adjusted to treat the general case $f=f(x,z,\xi )$, thus completing the proofs of Theorems 1.5 and 1.8:

Proof of Theorems 1.5 and 1.8

The arguments up until (4.12) remain valid for $f=f(x,z,\xi )$, but the last term in (4.10) can no longer be estimated as in (4.12). Consequently, under otherwise the same hypotheses, the conclusion of Lemma 4.2 now reads

$$\begin{aligned} \mathrm{(I_1)}_h + \mathrm{(I_2)}_h + \mathrm{(I_3)}_h + \mathrm{(I_4)}_h \le C\rho ^{-2}J_h^{(q+k-1)}, \end{aligned}$$

where $\mathrm{(I_1)}_h$, $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ are as before and

$$\begin{aligned} \mathrm{(I_4)}_h {:}{=}\sum _l \int _{B_{R+2\rho }}k\eta ({\tilde{v}}^+)^{q-1}f^{k-1}\varDelta _{ll}^h f[u]. \end{aligned}$$

The estimates for $\mathrm{(I_1)}_h$, $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ are unchanged (see Lemmas 4.3, 4.4, 4.5, 4.8 and 4.10), since they do not involve differentiating f. The integrand of $\mathrm{(I_4)}_h$ was previously a lower order term, but is now formally of third order in u. However, this can be treated using some of the ideas already seen in the proof of Lemma 4.10. Indeed, by the same argument leading to (4.43), we have for each $l\in \{1,\dots ,n\}$ and a.e. $x\in B_{R+2\rho }$ the estimate

$$\begin{aligned}&\varDelta _{ll}^h f[u](x) \nonumber \\&\quad \ge \frac{\partial f}{\partial \xi _a}[u](x)\nabla _a \varDelta _{ll}^h u(x) - C_\varSigma |\nabla _l^h\nabla u|^2 - C_\varSigma |\nabla _l^{-h}\nabla u|^2 - C|\nabla _l^h\nabla u| - C|\nabla _l^{-h}\nabla u| \nonumber \\&\qquad + \frac{1}{h^2}\bigg (f(x^+,u(x^+),\nabla u(x)) - 2f[u](x) + f(x^-,u(x^-),\nabla u(x))\bigg ). \end{aligned}$$

(5.10)

As before, the constant $C_\varSigma >0$ is such that the mapping $\xi \longmapsto f(x,z,\xi )+C_\varSigma |\xi |^2$ is convex for all $(x,z,\xi )\in \varSigma$. Denoting all but the first term on the RHS of (5.10) as error terms, it follows from (5.10) that

$$\begin{aligned} \mathrm{(I_4)}_h \ge \int _{B_{R+2\rho }}k\eta ({\tilde{v}}^+)^{q-1}f^{k-1}\frac{\partial f}{\partial \xi _a}[u]\nabla _a{\tilde{v}} - \int _{B_{R+2\rho }}k\eta ({\tilde{v}}^+)^{q-1}f^{k-1}|\text {error terms}|. \end{aligned}$$

(5.11)

Now, in the same way that we dealt with the error terms in Step 2 of the proof of Lemma 4.10, one readily obtains $\int _{B_{R+2\rho }}k\eta ({\tilde{v}}^+)^{q-1}f^{k-1}|\text {error terms}| \le CJ_h^{(q+1)}$. For the first integral on the RHS of (5.11), we integrate by parts and apply Hölder’s inequality to obtain

$$\begin{aligned} \bigg |\int _{B_{R+2\rho }}k\eta ({\tilde{v}}^+)^{q-1}f^{k-1}\frac{\partial f}{\partial \xi _a}[u]\nabla _a{\tilde{v}} \bigg | \le C\rho ^{-1}J_h^{(q+1)}. \end{aligned}$$

Returning to (5.11), we therefore obtain $\mathrm{(I_4)}_h \ge - C\rho ^{-1}J_h^{(q+1)}$. As a consequence, the estimates (5.1) and (5.8) hold, and the arguments of Sect. 5 therefore apply without any changes. $\square$

6 The case $k\ge 3$ for general H

In this final section we consider a minor extension of Theorem 1.8. Recall that our proof of Theorems 1.5 and 1.8 exploited a cancellation phenomenon between higher order terms arising from $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$, where the divergence structure of $F^{ij}$ played a role in estimating $\mathrm{(I_2)}_h$. When $3\le k\le n$ and H is not necessarily a multiple of the identity, the divergence structure given in (3.4) is more involved and the resulting arguments fall outside the scope of the present paper. That said, if one assumes higher integrability on $\nabla ^2 u$ from the outset, the terms $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ may be estimated by using Cauchy’s inequality and absorbing the resulting negative higher order terms into the positive term $\mathrm{(I_1)}_h$. This avoids the need to prove any cancellation between $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$. We establish:

Theorem 6.1

Let $\varOmega$ be a domain in ${\mathbb {R}}^n$ ($n\ge 3$), $f=f(x,z,\xi )\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n)$ a positive function and $H\in C_{\mathrm{loc}}^{1,1}(\varOmega \times {\mathbb {R}}\times {\mathbb {R}}^n\,;\,\mathrm{Sym}_n({\mathbb {R}}))$. Suppose $3\le k\le n$, $p>kn$ and $u\in W_{\mathrm{loc}}^{2,p}(\varOmega )$ is a solution to (1.4). Then $u\in C^{1,1}_{\mathrm{loc}}(\varOmega )$, and for any concentric balls $B_{R}\subset B_{2R} \Subset \varOmega$ we have

$$\begin{aligned} \Vert \nabla ^2 u \Vert _{L^\infty (B_R)}\le C, \end{aligned}$$

where C is a constant depending only on n, p, R, f, H and an upper bound for $\Vert u\Vert _{W^{2,p}(B_{2R})}$.

Proof

Following the proof of Theorem 1.8 in Sect. 5.3 but leaving the terms $\mathrm{(I_2)}_h$ and $\mathrm{(I_3)}_h$ untreated, we have for $u\in W_{\mathrm{loc}}^{2,q+k-1}(\varOmega )\cap W_{\mathrm{loc}}^{1,\infty }(\varOmega )$ ($q>1$)

$$\begin{aligned} \frac{q-1}{Cq^2}\int _{B_{R+2\rho }}\eta \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} + \mathrm{(I_2)}_h + \mathrm{(I_3)}_h \le C\rho ^{-2}J_h^{(q+k-1)}. \end{aligned}$$

(6.1)

We now suppose further that $\nabla ^2 u \in L_{\mathrm{loc}}^{q+2k-1}(\varOmega )$ ($q>1$). By Cauchy’s inequality and the bound $|\mathrm{div}F[u]| \le C(\varDelta u + C_1)^{k-1}$ in Eq. (3.12), we see that for all $\delta >0$

$$\begin{aligned} \mathrm{(I_2)}_h&= \frac{2}{q}\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q/2}\nabla _i F[u]^{ij}\nabla _j({\tilde{v}}^+)^{q/2} \nonumber \\&\ge -\frac{\delta (q-1)}{q^2}\int _{B_{R+2\rho }}\eta \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} - \frac{1}{\delta (q-1)}\int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^q\mathrm{tr}(A_H)\big |\mathrm{div}F[u]\big |^2 \nonumber \\&\ge -\frac{\delta (q-1)}{q^2}\int _{B_{R+2\rho }}\eta \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} - \frac{C}{\delta (q-1)}J_h^{(q+2k-1)}. \end{aligned}$$

(6.2)

By similar reasoning, it also holds that

$$\begin{aligned} \mathrm{(I_3)}_h&{\mathop {\ge }\limits ^{(4.26)}} \int _{B_{R+2\rho }}\eta ({\tilde{v}}^+)^{q-1}F^{ij}\frac{\partial H_{ij}}{\partial \xi _a}[u]\,\nabla _a {\tilde{v}} -C J_h^{(q+k)} \nonumber \\&\ge -\frac{\delta (q-1)}{q^2}\int _{B_{R+2\rho }}\eta \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} - \frac{C}{\delta (q-1)}J_h^{(q+2k-1)}. \end{aligned}$$

(6.3)

Taking $\delta$ sufficiently small in (6.2) and (6.3), and then substituting these estimates into (6.1), we obtain

$$\begin{aligned} \frac{q-1}{q^2}&\int _{B_{R+2\rho }}\eta \frac{\big |\nabla \big (({\tilde{v}}^+)^{q/2}\big )\big |^2}{\mathrm{tr}(A_H)} \le C\rho ^{-2} J_h^{(q+2k-1)}. \end{aligned}$$

(6.4)

The argument then proceeds as in Sect. 5.1: we let $\theta \in (0,1)$ be such that $\frac{2-\theta }{\theta }\le q+2k-1$ and obtain from (6.4) the estimate

$$\begin{aligned}&\bigg (\int _{B_{R+\rho }}(\varDelta u + C_1)^{\frac{q(2-\theta )^*}{2}}\bigg )^{\frac{2}{(2-\theta )^*}} \nonumber \\&\quad \le \frac{Cq}{\rho ^2}\bigg (\int _{B_{R+3\rho }}(\varDelta u + C_1)^{\frac{2-\theta }{\theta }}\bigg )^{\frac{\theta }{2-\theta }}\int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+2k-1}. \end{aligned}$$

(6.5)

Taking $\theta = \frac{2}{kn+1}$ and imposing $q+2k-1>kn$, we see that $\frac{2-\theta }{\theta } = kn<q+2k-1 < \frac{q(2-\theta )^*}{2}$, and we therefore obtain from (6.5) the estimate

$$\begin{aligned} \bigg (\int _{B_{R+\rho }}(\varDelta u + C_1)^{\beta q}\bigg )^{1/\beta } \le \frac{Cq}{\rho ^2}\int _{B_{R+3\rho }}(\varDelta u + C_1)^{q+2k-1}, \end{aligned}$$

where $\beta {:}{=}kn/(kn+1-2k) > 1$ and C now depends on $\int _{B_{R+3\rho }}(\varDelta u + C_1)^{kn}$. The Moser iteration argument then goes through as before, giving the desired conclusion. $\square$

Appendices

A A remark on the regularity of solutions to the $\sigma _2$-Yamabe equation obtained by vanishing viscosity

Let $(M^4,g_0)$ be a 4-manifold with scalar curvature $R_0>0$ and Schouten tensor $A_0$ satisfying $\int _{M^4}\sigma _2(A_0)\,dv_0>0$. In [12], the existence of smooth solutions $g_{w_\delta }=e^{2w_{\delta }}g_0$ with positive scalar curvature to the fourth order equation

$$\begin{aligned} \sigma _2(A_{g_{w_\delta }}) = \frac{\delta }{4}\varDelta _{g_{w_\delta }} R_{g_{w_\delta }} - 2\gamma _1|\eta |_{g_{w_\delta }}^2 \end{aligned}$$

(A.1)

is established for each $\delta \in (0,1]$, where $\eta$ is any fixed non-vanishing (0, 2)-tensor and $\gamma _1<0$ is the conformal invariant obtained by integrating both sides of (A.1). Moreover, solutions are shown to satisfy the uniform estimates

$$\begin{aligned} \Vert w_\delta \Vert _{W^{2,s}(M^4,g_0)}\le C\quad \mathrm {for~all~}\delta \in (0,1], \quad 1\le s<5, \end{aligned}$$

(A.2)

where the constant $C=C(s)$ is independent of $\delta$. A heat flow argument is then applied to obtain a conformal metric g with $\lambda (A_g)\in \varGamma _2^+$. In this appendix, we show that in the case that $(M^4,g_0)$ is locally conformally flat, we may take the limit $\delta \rightarrow 0$ more directly in (A.1) to obtain the desired conformal metric with $\lambda (A_g)\in \varGamma _2^+$. More precisely, using Theorem 1.1 and a result of [46], we show that, along a subsequence, the solutions $w_\delta$ converge weakly to a smooth solution of the equation $\sigma _2(A_{g_{w_\delta }})=-2\gamma _1|\eta |_{g_{w_\delta }}^2>0$.

To this end, fix $4<s<5$. We first observe that by (A.2), we can find a sequence $\delta _i\rightarrow 0$ for which $w_i{:}{=}w_{\delta _i}$ converges weakly in $W^{2,s}(M^4,g_0)$, say to $w\in W^{2,s}(M^4,g_0)$. By the Morrey embedding $W^{2,s}(M^4,g_0)\hookrightarrow C^{1,1-\frac{4}{s}}(M^4,g_0)$, we may assume $w_i\rightarrow w$ in $C^{1,\alpha }(M^4,g_0)$ for some $\alpha >0$. It then follows from [46, Proposition 5.3] and the estimate (A.2) that for all $\phi \in C^0(M^4)$, we have

$$\begin{aligned} \lim _{i\rightarrow \infty } \int _{M^4}\sigma _2(A_{g_{w_i}})\phi \,dv_0= \int _{M^4}\sigma _2(A_{g_{w}})\phi \,dv_0. \end{aligned}$$

(A.3)

Substituting the Eq. (A.1) into (A.3) and integrating by parts, we therefore see that

$$\begin{aligned} \int _{M^4} \sigma _2(A_{g_w})\phi \,dv_0&= \lim _{i\rightarrow \infty }\int _{M^4} \bigg (\frac{\delta _i}{4}R_{g_{w_i}}\varDelta _{g_{w_i}}\phi -2\gamma _1|\eta |_{g_{w_i}}^2\phi \bigg )\,dv_0 \\&= -\int _{M^4}2\gamma _1|\eta |_{g_w}^2\phi \,dv_0 \end{aligned}$$

for all $\phi \in C^2(M^4,g_0)$. It follows that $w\in W^{2,s}(M^4,g_0)$ solves

$$\begin{aligned} \sigma _2(A_{g_w}) = -2\gamma _1|\eta |_{g_w}^2>0\quad \mathrm {a.e.~in~}M^4. \end{aligned}$$

(A.4)

Moreover, as $R_{g_{w_i}}>0$ for each i, it follows that $R_{g_w}\ge 0$, and by (A.4) we therefore have $R_{g_w}>0$ a.e. If $(M^4,g_0)$ is locally conformally flat, we therefore obtain from Theorem 1.1 that $u{:}{=}e^{-w}\in C^{1,1}(M^4,g_0)$, and consequently (A.4) is uniformly elliptic at w.

At this point, we apply the Evans-Krylov theorem to obtain $u\in C^{2,\alpha }(M^4,g_0)$. Indeed, by the proof of [7, Theorem 6.6], it suffices to observe that, by Lemmas 4.1 and 4.16, $v=\sum _l \varDelta _{ll}^h u$ is a subsolution to a uniformly elliptic linear equation, namely

$$\begin{aligned} F^{ij}\nabla _i\nabla _j v + B^iD_iv \ge C, \end{aligned}$$

where $F^{ij}$ is uniformly elliptic and $F^{ij}$, $B^i$ and C are bounded. Furthermore, since $f(x,u) {:}{=}-2\gamma _1|\eta (x)|_{u^{-2}g_0}^2 = -2\gamma _1 u^4|\eta (x)|_{g_0}^2$ is smooth, standard elliptic regularity ensures that u (and hence w) belongs to $C^\infty (M^4,g_0)$.

B Proof of Lemma 2.1

The proof is a standard argument using Taylor’s theorem. We claim that

$$\begin{aligned} \Vert v_h-\varDelta u\Vert _{L^s(\varOmega ')}&\le \sum _{l=1}^n\int _0^1\Big \Vert \nabla _l\nabla _lu(x+the_l)-\nabla _l\nabla _l u(x)\Big \Vert _{L^s(\varOmega ')} \,dt \nonumber \\&\quad + \sum _{l=1}^n\int _0^1\Big \Vert \nabla _l\nabla _lu(x-the_l)-\nabla _l\nabla _l u(x)\Big \Vert _{L^s(\varOmega ')} \,dt \end{aligned}$$

(B.1)

for all $u\in W^{2,s}(\varOmega )$ and $\varOmega '\Subset \varOmega$ satisfying $|h|<\mathrm{dist}(\varOmega ',\partial \varOmega )$, from which the conclusion follows by the continuity of the translation operator in $L^s(\varOmega )$. By density it suffices to prove (B.1) for $u\in C^2(\varOmega )$. Let $\varOmega '$ be as above. Then for each $x\in \varOmega '$ and $l\in \{1,\dots ,n\}$, we have by Taylor’s theorem

$$\begin{aligned} u(x\pm he_l) = u(x) \pm h\nabla _l u(x) + h^2\int _0^1 (1-t)\nabla _l\nabla _l u(x\pm the_l)\,dt, \end{aligned}$$

and thus

$$\begin{aligned} v_h(x) - \varDelta u(x)&= \sum _{l=1}^n \int _0^1 (1-t)\bigg (\nabla _l\nabla _l u(x+the_l) - \nabla _l\nabla _l u(x)\bigg )\,dt \nonumber \\&\quad + \sum _{l=1}^n\int _0^1(1-t)\bigg (\nabla _l\nabla _l u(x-the_l) - \nabla _l\nabla _l u(x)\bigg )\,dt. \end{aligned}$$

(B.2)

Let $s'$ be such that $\frac{1}{s}+\frac{1}{s'}=1$. It follows from (1.2) and Hölder’s inequality that for all $g\in L^{s'}(\varOmega ')$ satisfying $\Vert g\Vert _{L^{s'}(\varOmega ')}\le 1$, we have

$$\begin{aligned}&\int _{\varOmega '}\big (v_h(x)-\varDelta u(x)\big )g(x)\,dx \nonumber \\&\quad = \sum _{l=1}^n\int _0^1(1-t)\int _{\varOmega '}\bigg (\nabla _l\nabla _l u(x+the_l) - \nabla _l\nabla _l u(x)\bigg )g(x)\,dx\,dt \nonumber \\&\qquad + \sum _{l=1}^n\int _0^1(1-t)\int _{\varOmega '}\bigg (\nabla _l\nabla _l u(x-the_l) - \nabla _l\nabla _l u(x)\bigg )g(x)\,dx\,dt \nonumber \\&\quad \le \sum _{l=1}^n\int _0^1\Big \Vert \nabla _l\nabla _lu(x+the_l)-\nabla _l\nabla _l u(x)\Big \Vert _{L^s(\varOmega ')} \,dt\nonumber \\&\qquad + \sum _{l=1}^n\int _0^1\Big \Vert \nabla _l\nabla _lu(x-the_l)-\nabla _l\nabla _l u(x)\Big \Vert _{L^s(\varOmega ')} \,dt. \end{aligned}$$

(B.3)

Taking the supremum over such g in (B.3), we obtain (B.1). $\square$