Hölder Regularity of the Gradient for Solutions of Fully Nonlinear Equations with Sub Linear First Order Term

Abstract

Using an improvement of flatness Lemma, we prove Hölder regularity of the gradient of solutions with higher order term a uniformly elliptic fully nonlinear operator and with Hamiltonian which is sub-linear. The result is based on some general compactness results.

Dedicated to Ermanno Lanconelli and his imperishable mathematical enthusiasm on the occasion of his 70th birthday

Mathematical Subject Classification: 35J25, 35J60, 35P30

1 Introduction

In this paper we shall establish some regularity results of solutions of a class of fully nonlinear equations, with a first order term which is sub-linear; it is a natural continuation of [5, 12]. Precisely we shall consider the following family of equations

$$\displaystyle{ F(D^{2}u) + b(x)\vert \nabla u\vert ^{\beta } = f(x)\ \mbox{ in}\ \varOmega \subset \mathbb{R}^{N}. }$$

(1)

See also [1] for related recent results.

Theorem 1.1

Suppose that F is uniformly elliptic, that β ∈ (0,1), f and b are in $\mathcal{C}(\overline{\varOmega })$ . For any u, bounded viscosity solution of  (1) and for any r < 1, there exist γ ∈ (0,1) depending on ellipticity constants of F, $\|b\|_{\infty }$ , ω(b) and β and C = C(γ) such that

$$\displaystyle{\|u\|_{\mathcal{C}^{1,\gamma }(B_{r}(x_{o}))} \leq C\left (\|u\|_{\infty } +\| b\|_{\infty }^{ \frac{1} {1-\beta }} +\|\, f\|_{\infty }\right ),}$$

as long as $B_{1}(x_{o}) \subset \varOmega$ .

Answering a question that we raised in [4], Imbert and Silvestre in [12] proved an interior Hölder regularity for the gradient of the solutions of

$$\displaystyle{\vert \nabla u\vert ^{\alpha }F(D^{2}u) = f(x)}$$

when α ≥ 0. Their proof relies on a priori Lipschitz bounds, rescaling and an improvement of flatness Lemma, in this way they are lead to use the classical regularity results of Caffarelli, and Evans [7, 8, 11] for uniformly elliptic equations.

Following their breakthrough, in [5], we proved the same interior regularity when α ≥ 0 in the presence of lower order terms. We also proved C ^1, γ regularity up to the boundary if the boundary datum is sufficiently smooth. Our main motivation to investigate the regularity of these solutions i.e. the simplicity of the first eigenvalue associated to the Dirichlet problem for | ∇u | ^α F(D ² u), required continuity of the gradient up to the boundary.

When α ∈ (−1, 0), in [4] we proved $\mathcal{C}^{1,\gamma }$ regularity for solutions of the Dirichlet problem, using a fixed point argument which required global Dirichlet conditions on the whole boundary. So one of the question left open was: is the local regularity valid for α < 0?

Theorem 1.1 answers to this question since the following holds:

Proposition 1.1

Suppose that, for α ∈ (−1,0), u is a viscosity solution of

$$\displaystyle{\vert \nabla u\vert ^{\alpha }F(D^{2}u) = f(x)\ \mbox{ in}\ \varOmega }$$

then u is a viscosity solution of

$$\displaystyle{F(D^{2}u) - f(x)\vert \nabla u\vert ^{-\alpha } = 0\ \mbox{ in}\ \varOmega.}$$

The proof is postponed to the appendix, but recall that singular equations require a special definition of viscosity solutions.

Theorem 1.1 concerns continuous viscosity solutions of (1); we should point out that in the case of L ^p viscosity solutions (see [9]) it is possible to use a different strategy. Indeed one could prove first, using the argument below, that the solutions are Lipschitz continuous. By Rademacher theorem they are almost everywhere differentiable and hence they will be an L ^p viscosity solution of

$$\displaystyle{F(D^{2}u) = g(x)}$$

with $g \in L^{\infty }$. The classical result of Caffarelli [7] implies that the solution are C ^1, α. But this is a different result from ours, since continuous viscosity solutions are L ^p viscosity solutions only when g is continuous, which somehow is what we want to prove.

In turn the C ^1, α regularity implies that g is Hölder continuous, so further regularities can be obtained (see e.g. [6, 14]).

Even for F(D ² u) = Δ u it would be impossible to mention all the work that has been done on equation of the form

$$\displaystyle{F(D^{2}u) + \vert \nabla u\vert ^{p} = f(x).}$$

Interestingly most of the literature is concerned with the case p > 1. In particular the so called natural growth i.e. p = 2 has been much studied in variational contexts and the behaviours are quite different when p ≥ 2 or 1 < p < 2. We will just mention the fundamental papers of Lasry and Lions [13] and Trudinger [15]. And more recently the papers of Capuzzo Dolcetta et al. [10] and Barles et al. [2]. In the latter the Hölder regularity of the solution is proved for non local uniformly elliptic operators, and with lower order terms that may be sublinear.

Remark 1.1

Observe that the operator is not Lipschitz continuous with respect to ∇u. This implies that in general uniqueness of the Dirichlet problem does not hold. For example, when Ω is the ball of radius 1, then $u \equiv 0$ and $u(x) = C(1 -\vert x\vert ^{\gamma })$ with $\gamma = \frac{2-\beta } {1-\beta }$ and $C =\gamma ^{-1}(\gamma +N - 2)^{ \frac{1} {\beta -1} }$ are both solutions of equation

$$\displaystyle{\left \{\begin{array}{lc} \varDelta u + \vert \nabla u\vert ^{\beta } = 0& \mathrm{in}\ \varOmega, \\ u = 0 &\mathrm{on}\ \partial \varOmega. \end{array} \right.}$$

2 Interior Regularity Results

Let S ^N denote the symmetric N × N matrices. In the whole paper F indicates a uniformly elliptic operator i.e. F satisfies F(0) = 0 and, for some $0 <\lambda \leq \varLambda$,

$$\displaystyle{\lambda \mathrm{tr}N \leq F(M + N) - F(M) \leq \varLambda \mathrm{tr}N}$$

for any M ∈ S ^N and any N ∈ S ^N such that N ≥ 0. The constants appearing in the estimates below often depend on $\lambda$ and $\varLambda$, but we will not specify them explicitly when it happens.

We recall that we want to prove

Theorem 2.1

Let f and b continuous in $\overline{B}_{1} \subset \varOmega$ . For any u, bounded viscosity solution of  (1) in B ₁ , and for any r < 1 there exist

$$\displaystyle{\gamma =\gamma (\|f\|_{\infty },\|b\|_{\infty },\beta,\omega _{b}(\delta ))\ \mbox{ and}\ C = C(\gamma )}$$

such that

$$\displaystyle{\|u\|_{\mathcal{C}^{1,\gamma }(B_{r})} \leq C\left (\|u\|_{\infty } +\| b\|_{\infty }^{ \frac{1} {1-\beta }} +\| f\|_{\infty }\right ).}$$

Before proving Theorem 2.1, we shall prove a local Lipschitz continuity result.

Lemma 2.1

Suppose that $H: B_{1} \times \mathbb{R}^{N} \rightarrow \mathbb{R}$ is such that

H(.,0) is bounded in B ₁ and there exist C > 0 such that for all $q \in \mathbb{R}^{N}$ ,

$$\displaystyle{\vert H(x,q) - H(x,0)\vert \leq C(\vert q\vert ^{\beta } + \vert q\vert ).}$$

Then there exists C _o such that if C < C _o , any bounded solution u of

$$\displaystyle{F(D^{2}u) + H(x,\nabla u) = f(x)\ \mbox{ in}\ B_{ 1}}$$

is Lipschitz continuous in B _r , for r < 1 with some Lipschitz constant depending on r, $\|f\|_{\infty }$ , C _o and $\|H(.0)\|_{\infty }$ .

Proof of Lemma 2.1

The proof proceeds as in [5, 12]. We outline it here, in order to indicate the changes that need to be done.

Let r < r′ < 1 and x _o  ∈ B _r , we consider on B _r′ × B _r′ the function

$$\displaystyle{\varPhi (x,y) = u(x) - u(y) - L^{2}\omega (\vert x - y\vert ) - L\vert x - x_{ o}\vert ^{2} - L\vert y - x_{ o}\vert ^{2}}$$

where the continuous function ω is given by $\omega (s) = s - w_{o}s^{\frac{3} {2} }$ for $s \leq (2/3w_{o})^{2}$ and constant elsewhere; here w _o is chosen in order that $(2/3w_{o})^{2} > 1$.

The scope is to prove that, for L independent of x _o , chosen large enough,

$$\displaystyle{ \varPhi (x,y) \leq 0\ \mbox{ on}\ B_{r}^{2}. }$$

(2)

This will imply that u is Lipschitz continuous on B _r by taking x = x _o , and letting x _o vary.

So we begin to choose $L > \frac{8\sup u} {(r^{{\prime}}-r)^{2}}$. Suppose by contradiction that $\varPhi (\bar{x},\bar{y}) =\sup \varPhi (x,y) > 0$. By the hypothesis on L, $(\bar{x},\bar{y})$ is in the interior of $B_{r}^{2}$. Proceeding in the calculations as in [2] (see also [3, 12]) we get that if (2) is not true then there exist X and Y such that

$$\displaystyle{(q_{x},X) \in J^{2,+}u(\bar{x}),(q_{ y},-Y ) \in J^{2,-}u(\bar{y})}$$

where $\overline{J}^{2,+}$, $\overline{J}^{2,-}$ are the standard semi-jets, while $q_{x} = L^{2}\omega ^{{\prime}}(\vert x - y\vert ) \frac{x-y} {\vert x-y\vert } + 2L(x - x_{o})$ and $q_{y} = L^{2}\omega ^{{\prime}}(\vert x - y\vert ) \frac{x-y} {\vert x-y\vert } - 2L(y - x_{o})$.

Then, there exist constant κ ₁, κ ₂ depending only on $\lambda,\varLambda,$ ω _o such that

$$\displaystyle{\mathcal{M}^{+}(X + Y ) \leq -\kappa _{ 1}L^{2}}$$

and | q _x  | , | q _y  | ≤ κ ₂ L ².

Using the equation,

$$\displaystyle\begin{array}{rcl} f(\bar{x})& \leq & H(\bar{x},q_{x}) + F(X) {}\\ & \leq & H(\bar{x},q_{x}) + F(-Y ) + \mathcal{M}^{+}(X + Y ) {}\\ & \leq & f(\bar{y}) -\kappa _{1}L^{2} {}\\ & & +\|H(.,0)\|_{\infty } + C(\vert q_{x}\vert ^{\beta } + \vert q_{y}\vert ^{\beta } + \vert q_{x}\vert + \vert q_{y}\vert ). {}\\ \end{array}$$

The term $\|H(.,0)\|_{\infty }$ is o(L ²), while for $C_{o} \leq \frac{\kappa _{1}} {16\kappa _{2}}$

$$\displaystyle\begin{array}{rcl} C(\vert q_{x}\vert ^{\beta } + \vert q_{y}\vert ^{\beta } + \vert q_{x}\vert + \vert q_{y}\vert )& \leq & \frac{\kappa _{1}L^{2}} {2} + 4C_{o}(1 +\kappa _{2}L^{2}) {}\\ & \leq & \frac{3\kappa _{1}L^{2}} {4} + 4C_{o}. {}\\ \end{array}$$

In conclusion we have obtained that $f(\bar{x}) \leq f(\bar{y}) -\frac{\kappa _{1}L^{2}} {4} + o(L^{2})$. This is a contradiction for L large.

Corollary 2.1

Suppose that $(f_{n})_{n}$ and $(H_{n}(\cdot,0))_{n}$ are sequences converging uniformly respectively to $f_{\infty }$ and $H_{\infty }$ on any compact subset of B ₁ , such that for all $q \in \mathbb{R}^{N}$ ,

$$\displaystyle{ \left \vert H_{n}(x,q) - H_{n}(x,0)\right \vert \leq \epsilon _{n}(\vert q\vert ^{\beta } + \vert q\vert ) }$$

(3)

with $\epsilon _{n} \rightarrow 0$ . Let u _n be a sequence of solutions of

$$\displaystyle{F(D^{2}u_{ n}) + H_{n}(x,\nabla u_{n}) = f_{n}(x)\ \mbox{ in }\ B_{1}.}$$

If $\|u_{n}\|_{\infty }$ is a bounded sequence, then up to subsequences, u _n converges, in any compact subset of B ₁ , to $u_{\infty }$ a solution of the limit equation

$$\displaystyle{F(D^{2}u_{ \infty }) + H_{\infty }(x) = f_{\infty }(x)\ \mbox{ in }\ B_{1}.}$$

2.1 Holder Regularity of the Gradient: Main Ingredients

We will follow the line of proof in [5, 12]. The modulus of continuity of a function g is defined by $\omega _{g}(\delta ) =\sup _{[x-y\vert \leq \delta }\vert g(x) - g(y)\vert$. In the following, ω will denote some continuous increasing function on [0, δ _o ] such that ω(0) = 0.

Lemma 2.2 (Improvement of Flatness)

There exist ε _o ∈ (0,1) and there exists ρ ∈ (0,1) depending on $(\beta,N,\lambda,\varLambda,\omega )$ such that: for any ε < ε _o , for any $p \in \mathbb{R}^{N}$ and for any f and b such that $\|f\|_{\infty }\leq \epsilon$ , $\|b\|_{\infty }\leq \epsilon$ and such that $\omega _{b}(\delta ) \leq \| b\|_{\infty }\omega (\delta )$ , if u is a solution of

$$\displaystyle{F(D^{2}u) + b(x)\vert \nabla u + p\vert ^{\beta } = f(x)\ \mbox{ in}\ B_{ 1}}$$

with $\mathop{\mathrm{osc}}\limits _{B_{1}}u \leq 1$ , then there exists $q^{\star } \in \mathbb{R}^{N}$ such that

$$\displaystyle{\mathop{\mathrm{osc}}\limits _{B_{\rho }}(u - q^{\star } \cdot x) \leq \frac{1} {2}\rho.}$$

Proof of Lemma 2.2

We argue by contradiction i.e. we suppose that, for any $n \in \mathbb{N}$, there exist $p_{n} \in \mathbb{R}^{N}$, and u _n a solution of

$$\displaystyle{F(D^{2}u_{ n}) + b_{n}(x)\vert \nabla u_{n} + p_{n}\vert ^{\beta } = f_{ n}(x)\ \mbox{ in}\ B_{1}}$$

with $\mathop{\mathrm{osc}}\limits _{B_{1}}u_{n} \leq 1$ and such that, for any ρ ∈ (0, 1) and any $q^{\star } \in \mathbb{R}^{N}$,

$$\displaystyle{\mathop{\mathrm{osc}}\limits _{B_{\rho }}(u_{n} - q^{\star } \cdot x) \geq \frac{1} {2}\rho.}$$

Observe that u _n − u _n (0) satisfies the same equation as u _n , it has oscillation 1 and it is bounded, we can then suppose that the sequence (u _n ) is bounded. Suppose first that | p _n  | is bounded, so it converges, up to subsequences. Let $v_{n}(x) = u_{n}(x) + p_{n} \cdot x$, which is a solution of

$$\displaystyle{F(D^{2}v_{ n}) + b_{n}(x)\vert \nabla v_{n}\vert ^{\beta } = f_{ n}(x).}$$

We can apply Corollary 2.1 with $H_{n}(x,q) = b_{n}(x)\vert q\vert ^{\beta }$, since (3) holds.

Hence v _n converges uniformly to $v_{\infty }$, a solution of the limit equation

$$\displaystyle{F(D^{2}v_{ \infty }) = 0\ \mbox{ in}\ B_{1}.}$$

Furthermore $v_{\infty }$ satisfies, for any ρ ∈ (0, 1) and any $q^{\star } \in \mathbb{R}^{N}$,

$$\displaystyle{ \mathop{\mathrm{osc}}\limits _{B_{\rho }}(v_{\infty }- q^{\star } \cdot x) \geq \frac{1} {2}\rho. }$$

(4)

This contradicts the classical $\mathcal{C}^{1,\alpha }$ regularity results, see Evans [11] and Caffarelli [7].

We suppose now that | p _n  | goes to infinity. There are two cases, suppose first that $\vert p_{n}\vert ^{\beta }\|b_{n}\|_{\infty }$ is bounded. Let $H_{n}(x,q) = b_{n}(x)\vert q + p_{n}\vert ^{\beta }$. Since $\omega _{\vert p_{n}\vert ^{\beta }b_{n}}(\delta ) \leq \vert p_{n}\vert ^{\beta }\|b_{n}\|_{\infty }\omega (\delta )$, H _n (x, 0) is equicontinuous and up to a subsequence, it converges uniformly to some function $H_{\infty }(x)$, while (u _n ) _n is a uniformly bounded sequence of solutions of

$$\displaystyle{F(D^{2}u_{ n}) + H_{n}(x,\nabla u_{n}) = f_{n}(x).}$$

We can apply Corollary 2.1 and up to a subsequence, u _n converges to $u_{\infty }$ which is a solution of

$$\displaystyle{F(D^{2}u_{ \infty }) + H_{\infty }(x) = 0.}$$

Furthermore $u_{\infty }$ satisfies (4), for any ρ ∈ (0, 1) and any $q^{\star } \in \mathbb{R}^{N}$. As in the case p _n bounded, this contradicts the classical $\mathcal{C}^{1,\gamma }$ regularity results cited above.

We are left to treat the case where $a_{n} = \vert p_{n}\vert ^{\beta }\|b_{n}\|_{\infty }$ is unbounded. Hence, up to a subsequence, it goes to $+\infty$. We divide the equation by a _n , so $v_{n}:= \frac{u_{n}} {a_{n}}$ satisfies

$$\displaystyle{F(D^{2}v_{ n}) + \frac{b_{n}(x)} {a_{n}} \vert a_{n}\nabla v_{n} + p_{n}\vert ^{\beta } = \frac{f_{n}(x)} {a_{n}}.}$$

We can apply Corollary 2.1 with

$$\displaystyle{H_{n}(x,q) = b_{n}(x)a_{n}^{\beta -1}\vert q + a_{ n}^{-1}p_{ n}\vert ^{\beta }.}$$

Observe that, $H_{n}(x,0) = b_{n}(x)a_{n}^{-1}\vert p_{n}\vert ^{\beta }$ is equicontinuous, of $L^{\infty }$ norm 1 and up to a subsequence, it converges uniformly to some function $H_{\infty }(x)$.

Passing to the limit one gets that the limit equation is

$$\displaystyle{F(0) + H_{\infty }(x) = 0.}$$

This yields a contradiction, since $H_{\infty }$ has norm 1 and it ends the proof of Lemma 2.2.

The next step is an iteration process which is needed in order to prove Theorem 2.1.

Lemma 2.3

Given ε _o , ω and ρ as in Lemma  2.2 . Let b and f be such that $\|f\|_{\infty },\|b\|_{\infty }\leq \epsilon _{o}$ and such that $\omega _{b}(\delta ) \leq \| b\|_{\infty }\omega (\delta )$ . Suppose that u is a viscosity solution of

$$\displaystyle{ F(D^{2}u) + b(x)\vert \nabla u\vert ^{\beta } = f(x)\ \mbox{ in}\ B_{ 1} }$$

(5)

and, $\mathop{\mathrm{osc}}\limits _{B_{1}}u \leq 1$ . Then, there exists γ ∈ (0,1), such that for all k > 1, $k \in \mathbb{N}$ there exists $p_{k} \in \mathbb{R}^{N}$ such that

$$\displaystyle{ \mathop{\mathrm{osc}}\limits _{B_{r_{ k}}}(u(x) - p_{k} \cdot x) \leq r_{k}^{1+\gamma } }$$

(6)

where r _k := ρ ^k .

The proof is by induction and rescaling. For k = 0 just take p _k  = 0. Suppose now that, for a fixe k, (6) holds with some p _k . Choose γ ∈ (0, 1) such that $\rho ^{\gamma } > \frac{1} {2}$.

Define the function $u_{k}(x) = r_{k}^{-1-\gamma }\left (u(r_{k}x) - p_{k} \cdot (r_{k}x)\right ).$ By the induction hypothesis, p _k is such that $\mathop{\mathrm{osc}}\limits _{B_{1}}u_{k} \leq 1$ and u _k is a solution of

$$\displaystyle{F(D^{2}u_{ k}) + r_{k}^{1-\gamma }b(r_{ k}x)\vert r_{k}^{\gamma }(\nabla u_{ k} + p_{k}r_{k}^{-\gamma })\vert ^{\beta } = r_{ k}^{1-\gamma }f(r_{ k}x).}$$

Denoting by b _k the function $b_{k}(x) = r_{k}^{1-\gamma (1-\beta )}b(r_{k}x)$ which satisfies $\omega _{b_{k}}(\delta ) = r_{k}^{1-\gamma (1-\beta )}\omega _{b}(r_{k}\delta ) \leq r_{k}^{1-\gamma (1-\beta )}\|b\|_{\infty }\omega (r_{k}\delta ) \leq \| b_{k}\|_{\infty }\omega (\delta )$, the equation above can be written as

$$\displaystyle{F(D^{2}u_{ k}) + b_{k}(x)\vert \nabla u_{k} + p_{k}r_{k}^{-\gamma }\vert ^{\beta } = r_{ k}^{1-\gamma }f(r_{ k}x).}$$

Since the $L^{\infty }$ norm of $f_{k} = r_{k}^{1-\gamma }f(r_{k}\cdot )$ is less than ε, we can conclude that there exists q _k such that

$$\displaystyle{\mathop{\mathrm{osc}}\limits _{B_{\rho }}(u_{k}(x) - q_{k} \cdot x) \leq \frac{1} {2}\rho.}$$

So that, for $p_{k+1} = p_{k} + q_{k}r_{k}^{\gamma +1}$,

$$\displaystyle{\mathop{\mathrm{osc}}\limits _{B_{r_{ k+1}}}\left (u(x) - p_{k+1} \cdot x\right ) \leq \frac{\rho } {2}r_{k}^{1+\gamma } \leq r_{ k+1}^{1+\gamma }.}$$

This ends the proof of Lemma 2.3.

2.2 Holder Regularity of the Gradient: Conclusion

Lemma 2.4

Suppose that for any r, there exists p _r such that

$$\displaystyle{\mathop{\mathrm{osc}}\limits _{B_{r}}(u(x) - p_{r} \cdot x) \leq Cr^{1+\gamma }}$$

then u is $\mathcal{C}^{1,\gamma }$ in 0.

Proof

It is clear that it is sufficient to prove that p _r converges when r goes to 0.

We will prove that the sequence $p_{2^{-k}}$ converges and then conclude for the whole sequence. Let $r_{k} = \frac{1} {2^{k}}$, since r _k+1 < r _k for x, y in $\overline{B_{r_{k+1}}}$

$$\displaystyle{\vert u(x) - u(y) - p_{k+1} \cdot (x - y)\vert \leq Cr_{k+1}^{1+\gamma }}$$

and

$$\displaystyle{\vert u(x) - u(y) - p_{k} \cdot (x - y)\vert \leq Cr_{k}^{1+\gamma }.}$$

Subtracting

$$\displaystyle{\vert (p_{k+1} - p_{k} \cdot x - y)\vert \leq C(r_{k+1}^{1+\gamma } + r_{ k}^{1+\gamma }).}$$

Then, choosing $x = \frac{p_{k+1}-p_{k}} {\vert p_{k+1}-p_{k}\vert }r_{k+1} = -y$, one gets

$$\displaystyle{2\vert p_{k+1} - p_{k}\vert r_{k+1} \leq C(r_{k+1}^{1+\gamma } + r_{ k}^{1+\gamma })}$$

which implies

$$\displaystyle{\vert p_{k+1} - p_{k}\vert \leq C2r_{k}^{\gamma }.}$$

This proves that the series of general term $(p_{k+1} - p_{k})$ converges; hence so does the sequence p _k .

We deduce the convergence of the whole sequence p _ρ when ρ goes to zero. Let k be such that r ^k+1 ≤ ρ ≤ r ^k. Then for all x ∈ B _ρ

$$\displaystyle{(u(x) - p_{\rho } \cdot x) \leq C\rho ^{1+\gamma } \leq Cr_{ k}^{1+\gamma }}$$

and also, since $x \in B_{r^{k}}$,

$$\displaystyle{(u(x) - p_{r^{k}} \cdot x) \leq Cr_{k}^{1+\gamma }.}$$

Hence, by subtracting, $(p_{\rho } - p_{r^{k}}) \cdot x \leq 2Cr_{k}^{1+\gamma }$. Then, taking $x = \frac{p_{\rho }-p_{r^{k}}} {\vert p_{\rho }-p_{r^{k}}\vert }\rho$, we get $\vert p_{\rho } - p_{r^{k}}\vert \leq C\frac{r_{k}^{1+\gamma }} {\rho } \leq C\frac{r_{k}^{1+\gamma }} {r_{k+1}} = 2Cr_{k}^{\gamma }.$ This implies that p _ρ has the same limit as p _k . This ends the proof of Lemma 2.4.

Suppose now that u is a bounded solution of (5), for general f bounded in $L^{\infty }$, and b continuous. The function v(x) = ε u(x) with $\epsilon ^{-1} =\mathop{ \mathrm{osc}}\limits u + \frac{1} {\epsilon _{o}}(\|f\|_{\infty } +\| b\|_{\infty }^{ \frac{1} {1-\beta }})$ satisfies the equation

$$\displaystyle{F(D^{2}v) + b(x)\epsilon ^{1-\beta }\vert \nabla v\vert ^{\beta } =\epsilon f(x).}$$

Our choice of ε implies that we are under the conditions of Lemma 2.3, so v is in $\mathcal{C}^{1,\gamma }$, by Lemma 2.4, and so is u.

Appendix

Proof of Proposition 1.1

We assume that α ∈ (−1, 0) and that u is a supersolution of

$$\displaystyle{ \vert \nabla u\vert ^{\alpha }F(D^{2}u) = f(x)\ \mbox{ in}\ \varOmega }$$

(7)

i.e. we suppose that for any x _o  ∈ Ω either u is locally constant in a neighbourhood of x _o and then 0 ≤ f in that neighbourhood, or, if it is not constant, for any $\varphi$ test function that touches u by below at x _o and such that $\nabla \varphi (x_{o})\neq 0$, we require that

$$\displaystyle{\vert \nabla \varphi (x_{o})\vert ^{\alpha }F(D^{2}\varphi (x_{ o})) \leq f(x_{o}).}$$

We need to prove that this implies that u is a supersolution of

$$\displaystyle{ F(D^{2}u) - f(x)\vert \nabla u\vert ^{-\alpha } = 0\ \mbox{ in}\ \varOmega, }$$

(8)

in the usual viscosity sense. Without loss of generality we let x _o  = 0. If u is constant around 0, D ² u(0) = 0 and Du(0) = 0, so the conclusion is immediate. If $\varphi$ is some test function by below at zero such that $\nabla \varphi (0)\neq 0$, the conclusion is also immediate. We then suppose that there exists M ∈ S such that

$$\displaystyle{ u(x) \geq u(0) + \frac{1} {2}\langle Mx,x\rangle + o(\vert x\vert ^{2}). }$$

(9)

We want to prove that

$$\displaystyle{F(M) \leq 0.}$$

Let us observe first that one can suppose that M is invertible, since if it is not, it can be replaced by $M_{n} = M - \frac{1} {n}I$ which satisfies (9) and tends to M.

Let k > 2 and R > 0 such that

$$\displaystyle{\inf _{\vert x\vert <R}\left (u(x) -\frac{1} {2}\langle Mx,x\rangle + \vert x\vert ^{k}\right ) = u(0)}$$

where the infimum is strict. We choose δ < R such that $k(2\delta )^{k-2} < \frac{1} {2}\inf _{i}\vert \lambda _{i}(M)\vert$. Let ε be such that

$$\displaystyle{\inf _{\delta <\vert x\vert <R}\left (u(x) -\frac{1} {2}\langle Mx,x\rangle + \vert x\vert ^{k}\right ) = u(0)+\epsilon }$$

and let δ ₂ < δ and such that $k(2\delta )^{k-1}\delta _{2} +\| M\|_{\infty }(\delta _{2}^{2} + 2\delta _{2}\delta ) < \frac{\epsilon } {4}$. Then, for x such that | x |  < δ ₂,

$$\displaystyle\begin{array}{rcl} \inf _{\vert y\vert \leq \delta }\{u(y) -\frac{1} {2}\langle M(y - x),y - x\rangle + \vert y - x\vert ^{k}\}& \leq & \inf _{ \vert y\vert \leq \delta }\{u(y) -\frac{1} {2}\langle My,y\rangle + \vert y\vert ^{k}\} + \frac{\epsilon } {4} {}\\ & =& u(0) + \frac{\epsilon } {4} {}\\ \end{array}$$

and on the opposite

$$\displaystyle\begin{array}{rcl} & & \inf _{R>\vert y\vert >\delta }\{u(y) -\frac{1} {2}\langle M(y - x),y - x\rangle + \vert y - x\vert ^{k}\} {}\\ & \geq & \inf _{\vert y\vert >\delta }\{u(y) -\frac{1} {2}\langle My,y\rangle + \vert y\vert ^{k}\} - \frac{\epsilon } {4} > u(0) + 3 \frac{\epsilon } {4}. {}\\ \end{array}$$

Since the function u is supposed to be non locally constant, there exist x _δ and y _δ in B(0, δ ₂) such that

$$\displaystyle{u(x_{\delta }) > u(y_{\delta }) -\frac{1} {2}\langle M(x_{\delta } - y_{\delta }),x_{\delta } - y_{\delta }\rangle + \vert x_{\delta } - y_{\delta }\vert ^{k}}$$

and then the infimum $\inf _{y,\vert y\vert \leq \delta }\{u(y) -\frac{1} {2}\langle M(x_{\delta } - y),x_{\delta } - y\rangle + \vert x_{\delta } - y\vert ^{k}\}$ is achieved on some point z _δ different from x _δ . This implies that the function

$$\displaystyle{\varphi (z):= u(z_{\delta }) + \frac{1} {2}\langle M(x_{\delta } - z),x_{\delta } - z\rangle -\vert x_{\delta } - z\vert ^{k} + \frac{1} {2}\langle M(x_{\delta } - z_{\delta }),x_{\delta } - z_{\delta }\rangle + \vert x_{\delta } - z_{\delta }\vert ^{k}}$$

touches u by below at the point z _δ . But

$$\displaystyle{\nabla \varphi (z_{\delta }) = M(z_{\delta } - x_{\delta }) - k\vert x_{\delta } - z_{\delta }\vert ^{k-2}(z_{\delta } - x_{\delta })\neq 0,}$$

indeed, if it was equal to zero, z _δ − x _δ would be an eigenvector corresponding to the eigenvalue $k\vert x_{\delta } - z_{\delta }\vert ^{k-2}$ which is supposed to be strictly less than any eigenvalue of M.

Since u is a super-solution of (7), multiplying by $\vert \nabla \varphi (z_{\delta })\vert ^{-\alpha }$, we get

$$\displaystyle\begin{array}{rcl} F\left (M - \frac{d^{2}} {dz^{2}}(\vert x_{\delta } - z\vert ^{k})(z_{\delta })\right )& \leq & f(z_{\delta })\vert \nabla \varphi (z_{\delta })\vert ^{-\alpha }. {}\\ \end{array}$$

By passing to the limit for $\delta \rightarrow 0$ we obtain the desired conclusion i.e. F(M) ≤ 0.

We would argue in the same manner for sub-solutions.