On a strong maximum principle for fully nonlinear subelliptic equations with Hörmander condition

Abstract

We investigate a strong maximum principle of Vázquez type for viscosity solutions of fully nonlinear and degenerate elliptic equations involving Hörmander vector fields. We also give a strong comparison principle for such equations.

1 Introduction and main results

In this paper we study the strong maximum principle for a class of nonlinear, possibly degenerate, subelliptic equations in connected open subsets $\Omega$ of $\mathbb {R}^N,\;N\ge 2$. The differential operators involve a family of smooth vector fields $\mathcal {X}=\{X_1,\cdots , X_m \}$ on $\Omega$ that are required to satisfy the Hörmander rank condition at every point of $\Omega$. To be more precise, our goal is to study a strong maximum principle of Vázquez type for non-negative supersolutions of equations of the type

$$\begin{aligned} F(x, D_\mathcal {X} u, (D^2_\mathcal {X} u)^*)=H(u, |D_\mathcal {X} u|)\;\;\;\text {in}\;\;\Omega , \end{aligned}$$

(1.1)

where F and H satisfy certain conditions, to be specified later in this section.

To better describe the setting, we recall the seminal work of Vàzquez in his 1984 paper [20]. Let $f:[0,\infty )\rightarrow \mathbb {R}$ be a non-decreasing and continuous function such that $f(0)=0$ and

$$\begin{aligned} \int _0^1\frac{dt}{\sqrt{F(t)}}=\infty ,\;\;\;\text {where}\;\;F(t)=\int _0^t f(s)\,ds. \end{aligned}$$

(1.2)

In [20], Vàzquez showed the following strong maximum principle. If $u\in C^2(\Omega )$ is a non-negative solution of $\Delta u\le f(u)$ in a connected open set $\Omega$, then either $u\equiv 0$ in $\Omega$ or $u>0$ in $\Omega$.

The aforementioned type of strong maximum principle was expanded considerably in the influential papers [17, 18] to more general type of quasilinear equations. We direct the reader to the well-written and excellent monograph of P. Pucci and J. Serrin [15]. We also refer to the recent papers [12, 14, 19]. In particular, Felmer, Montenegro, and Quaas [12] study the equation¹$\Delta u=f(u)+g(|Du|)$ under the integral condition

$$\begin{aligned} \int _0^1\frac{dt}{at+f(t)+g(t)}=\infty ,\;\;\;\forall \;a>0, \end{aligned}$$

(1.3)

a condition that turns out to be equivalent to (see Lemma 3.2 below or [14, Lemma 3.5])

$$\begin{aligned}\int _0^1\frac{dt}{f(t)+g(t)}=\infty .\end{aligned}$$

In fact, the authors of [12] study the more general equation $\Delta _p u=f(u)+g(|D u|)$ where $\Delta _p$ is the p-Laplace operator and the nonlinearities f and g satisfy a modified version of (1.3).

In recent years increasing attention has been given to the the study of strong maximum principles for nonlinear subelliptic equations. In this regard let us mention the works [4, 5, 9]. The basic ideas utilized in these works stem from the classic work [8] of Bony, see Sect. 2. This has become a useful tool in the study of strong maximum principles for subelliptic operators. In this paper we will use this tool to investigate strong maximum principles for solutions to (1.1) as well. As pointed out in [4], we should mention that the study of fully nonlinear subelliptic equations began with the papers [1, 13].

Our work is motivated by two recent papers: [10], due to Capogna and Zhou and [4], due to Bardi and Goffi. In the paper [10], the authors prove a strong maximum principle for solutions of a class of quasilinear subelliptic equations of the form

$$\begin{aligned}\sum _{i,j=1}^m X_j^*(A_j(D_{\mathcal {X}}u))=f(x,u) \end{aligned}$$

where the $C^1$ functions $A_j$ satisfy suitable ellipticity and growth structure conditions that include the p-Laplacian for $p>1$, and $D_\mathcal {X}u$ denotes the $\mathcal {X}$-gradient of u (see definition below). The nonhomogeneous term f(x, t) is non-decreasing in t and satisfies $|f(x,t)|\le C(\kappa +|t|^{p-2})|t|$ for all $(x,t)\in \Omega \times \mathbb {R}$, where C and $\kappa$ are appropriate positive constants related to the structure condition.

In a recent paper [4], Bardi and Goffi establish the strong maximum principle for fully nonlinear subelliptic equations $G(x,u, D_\mathcal {X}u,(D^2_\mathcal {X}u)^*)=0$, where G(x, r, p, X) is proper in the sense of [11], satisfies appropriate scaling property and the condition

$$\begin{aligned}\sup _{\gamma>0}G(x,0,q,X-\gamma q\otimes q)>0\;\;\;\;\forall \;(x,q,X)\in \Omega \times \mathbb {R}^n\setminus \{o\}\times \mathcal {S}^{m\times m}.\end{aligned}$$

The work [4] extends the techniques in the papers [2, 3] to the subelliptic equations situation. At this point we also wish to direct interested readers to another paper [5] on the strong maximum principle for hypoelliptic second-order equations in divergence form. Aside from the obvious structural differences in the equations we deal with, our work differs in theme from the aforementioned papers in one major aspect. In our work we focus on Vàzquez-type strong maximum principle. We hasten to add that in the fully nonlinear equations we consider we allow gradient degeneracies. Moreover, we wish to stress that the usual monotonicity assumption on H(s, t) in the first variable commonly used in the literature for the strong maximum principle of solutions of (1.1) will be weakened.

Next, as a preparation for stating our results more precisely, we introduce notation and definitions that will be followed throughout the work.

The notation $\partial E$ denotes the boundary of a set $E\subseteq \mathbb {R}^N$ and $\overline{E}$ stands for the topological closure of E. Given subsets A, E of $\mathbb {R}^N$ we write $A\Subset E$ to indicate that $\overline{A}\subseteq E$. The letters $x,\;y$ and z will denote points in $\mathbb {R}^N$. By o, we mean the origin in $\mathbb {R}^N$. We use B(x, r) to represent the Euclidean ball in $\mathbb {R}^N$ of radius $r>0$ and center x. We write $\langle a,b\rangle$ to denote the Euclidean inner product of $a,b\in \mathbb {R}^N.$

Also, set

$$\begin{aligned}\mathbb {R}^+_0=[0,\infty ).\end{aligned}$$

The symbol $\mathcal {S}^{m\times m}$ denotes the class of $m\times m$ real symmetric matrices. The matrix $I_m$ is the $m\times m$ identity matrix and $O_m \in \mathcal {S}^{m\times m}$ is the zero matrix. For $X,\;Y\in \mathcal {S}^{m\times m}$, we write $Y\ge X$ if $Y-X$ is a positive semi-definite matrix.

Let $\mathcal {X}:=\{X_1,X_1,\cdots , X_m\}$ be a family of smooth vector fields in $\Omega$ that satisfy Hörmander’s finite rank condition:

$$\begin{aligned} \text {rank Lie}[X_1,\cdots ,X_m](x)=N,\;\;\;\;\;\;\;\;\forall \,x\in \Omega . \end{aligned}$$

(1.4)

Here, $\text {Lie}[X_1,\cdots ,X_m]$ is the Lie algebra generated by $X_1,\cdots , X_m$. For any smooth function $u:\Omega \rightarrow \mathbb {R}$ we denote by $D_{\mathcal {X}}u$ the $\mathcal {X}$-gradient $(X_1u,\cdots ,X_mu)$ of u and we write $D_{\mathcal {X}}^2u$ to denote the $m\times m$ matrix with ij entry $X_i(X_ju)$, which we call the $\mathcal {X}$-Hessian of u. The $\mathcal {X}$-Hessian of u is not necessarily symmetric. The symmetrized $\mathcal {X}$-Hessian matrix, denoted by $(D_{\mathcal {X}}^2u)^*$, is given by the $m\times m$ matrix

$$\begin{aligned}(D_{\mathcal {X}}^2u)^*:=\frac{1}{2}\left( D_{\mathcal {X}}^2u+(D_{\mathcal {X}}^2u)^T\right) .\end{aligned}$$

Here, $Y^T$ denotes the transpose of matrix Y.

Let $\eta (x):=\left[ \eta ^1(x)\cdots ,\eta ^m(x)\right]$ be the $N\times m$ matrix whose jth column $\eta ^j:\Omega \rightarrow \mathbb {R}$ has the coefficient of the vector field $X_j(x)$ for its entries. For $u\in C^2(\Omega )$ the following relationships hold (see [4]).

$$\begin{aligned}D_\mathcal {X}u=\eta ^TDu\;\;\text {and}\;\;\left( D_\mathcal {X}^2u\right) ^*=\eta ^T(D^2u)\eta +Q(\cdot ,Du)\end{aligned}$$

where

$$\begin{aligned}Q_{ij}(x,p)=\frac{1}{2}\left\langle D\eta ^j\eta ^i+D\eta ^i\eta ^j\,,\,p\right\rangle ,\;\;\;\;1\le i,j\le m,\;\;p\in \mathbb {R}^N.\end{aligned}$$

As a consequence of these relationships, we remark that if $u,v\in C^2(\Omega )$ such that $u-v$ has a local maximum (resp. minimum) at $x\in \Omega$ then

$$\begin{aligned} D_\mathcal {X}u(x)=D_\mathcal {X}v(x)\;\;\;\text {and}\;\;(D^2_\mathcal {X}u)^*(x)\le (\text {resp.}\,\ge \,) \,(D^2_\mathcal {X}v)^*(x). \end{aligned}$$

(1.5)

We now describe the conditions we place on F and H in the Eq. (1.1).

Given $A\in \mathcal {S}^{m\times m}$ let us consider the linear operator $\mathcal {L}_A:\mathcal {S}^{m\times m}\rightarrow \mathbb {R}$ given by

$$\begin{aligned}\mathcal {L}_A(M):=\text {trace}(AM),\;\;\;\forall M\in \mathcal {S}^{m\times m}.\end{aligned}$$

Let $0<\lambda \le \Lambda$ be fixed constants and

$$\begin{aligned} \mathcal {A}=\{ A\in \mathcal {S}^{m\times m}\;:\; \lambda I_m\le A\le \Lambda I_m\}.\end{aligned}$$

We recall the Pucci minimal operator

$$\begin{aligned} \mathcal {M}_{\lambda ,\Lambda }^-(X):=\inf _{A\in \mathcal {A}_{\lambda ,\Lambda }} \mathcal {L}_A(X). \end{aligned}$$

(1.6)

Conditions on F:

In this work, we will consider functions $\varpi :\mathbb {R}^+_0\rightarrow \mathbb {R}^+_0$ that satisfy the following conditions:

$\varvec{\mathcal {W}(i)}$::

$\varpi$ is continuous and non-decreasing on $\mathbb {R}^+_0$;

$\varvec{\mathcal {W}(ii)}$::

$\varpi (s)>0$ for $s>0$.

Let

$$\begin{aligned}\omega (t):=\int _0^t\varpi (s)\,ds,\;\;\;\;t\in \mathbb {R}^+_0.\end{aligned}$$

When needed, we may extend $\omega$ to $\mathbb {R}$ as an odd function.

We now state conditions we place on the function F in (1.1). Let $\Omega \subseteq \mathbb {R}^N$ be an open set. Let $F:\Omega \times \mathbb {R}^N\times \mathcal {S}^{m\times m}\rightarrow \mathbb {R}$ be continuous. Assume that there are $0<\lambda \le \Lambda <\infty$ and a function $\varpi :\mathbb {R}^+_0\rightarrow \mathbb {R}^+_0$, satisfying conditions $\varvec{\mathcal {W}(i)}$ and $\varvec{\mathcal {W}(ii)}$ above, such that for all $X,Y\in \mathcal {S}^{m\times m}$ and all $(x,p)\in \Omega \times \mathbb {R}^N$

(F-1)::

$F(x,p,X)\le F(x,p,Y)$ whenever $X\le Y$, and

(F-2)::

$F(x,p, X)\ge \varpi (|p|) \mathcal {M}_{\lambda , \Lambda }^-(X)-\omega (|p|)$,

where $\mathcal {M}^-_{\lambda ,\Lambda }$ is as in (1.6).

Remark 1.1

It should be noted that, if F satisfies (F-2) then $F(x,0,X)\ge 0$ for all $x\in \Omega$, and $X\ge O_m$.

For the rest of the work, we set

$$\begin{aligned}\mathcal {F}[u](x)=F\left( x,D_\mathcal {X}u(x),(D^2_{\mathcal {X}}u)^*(x)\right) ,\;\;\;x\in \Omega .\end{aligned}$$

In this paper we study the strong maximum principle for supersolutions (see definition below) of the fully nonlinear equation

$$\begin{aligned} \mathcal {F}[u]=H(u,|D_{\mathcal {X}}u|). \end{aligned}$$

(1.7)

Next we describe the conditions we impose on H.

Conditions on H:

We assume that $H: \mathbb {R}\times \mathbb {R}_0^+\rightarrow \mathbb {R}^+_0$ is continuous. In addition, for a given $\varpi$ that satisfies $\varvec{\mathcal {W}(i)}$ and $\varvec{\mathcal {W}(ii)}$, we require that there is a constant $a_0\ge 1$ such that for all $a\ge a_0$

(H-i)::

$s\mapsto a\omega (s)+H(as,t)$ and $t\mapsto a\omega (t)+H(s,at)$ are non-decreasing on $\mathbb {R}^+_0$;

(H-d)::

$\displaystyle {\int _0^1\frac{dt}{H(a\omega ^{-1}(t),a\omega ^{-1}(t))}=\infty }$.

We make a few remarks about the above conditions.

Remark 1.2

 

• (i) Condition (H-i) implies that for each $a\ge a_0$, the function $t\mapsto 2a\omega (t)+H(at,at)$ is non-decreasing on $\mathbb {R}^+_0$.

• (ii) Suppose $\varpi (0)=0$, and $t>0$. Then $s\mapsto a\omega (s)+H(as,t)$ is non-decreasing in $\mathbb {R}^+_0$ for all $a\ge a_0\ge 1$, implies $s\mapsto H(s,t)$ is non-decreasing in $\mathbb {R}^+_0$. Obviously similar comment applies when $t\mapsto a\omega (t)+H(s,t)$ is non-decreasing in $\mathbb {R}^+_0$ for all $a\ge a_0\ge 1$. For the reader’s convenience we include a short proof of this statement in the “Appendix”.

• (iii) In (H-d), we allow the possibility that $H(t,t)=0$ in $(0,\delta )$ for some $\delta >0$.

• (iv) A change of variable shows that condition (H-d) is equivalent to requiring that

  $$\begin{aligned}\int _0^1\frac{\varpi (s/a)}{H(s,s)}\,ds=\infty \end{aligned}$$

  holds for all $a\ge a_0$.

• (v) If $\varpi (0)>0$, then we note that (H-d) holds for all $a\ge a_0\ge 1$ if and only if

  $$\begin{aligned}\int _0^1\frac{dt}{H(t,t)}=\infty .\end{aligned}$$

Viscosity subsolutions and supersolutions:

We study (1.7) in the context of viscosity solutions. We recall the relevant definitions. By USC(A) we mean the set of upper semicontinuous functions on a set $A\subseteq \mathbb {R}^N$ and LSC(A) is the set of lower semicontinuous functions on A.

Let $G:\mathbb {R}\times \mathbb {R}^+_0\rightarrow \mathbb {R}$ and consider the equation

$$\begin{aligned} \mathcal {F}[u]=G(u, |D_{\mathcal {X}} u|) \;\;\;\text {in }\Omega . \end{aligned}$$

(1.8)

A function $v\in \text {USC}(\Omega )$ is called a viscosity subsolution (resp. strict viscosity subsolution) of (1.8) if for any pair $(x_0,\psi )\in \Omega \times C^2(\Omega )$ such that $x_0$ is a local maximum of $v-\psi$ we have

$$\begin{aligned}\mathcal {F}[\psi ](x_0)\ge (\text {resp.} >)\, G(v(x_0),|D_{\mathcal {X}}\psi (x_0)|). \end{aligned}$$

Similarly, we say $w\in \text {LSC}(\Omega )$ is a viscosity supersolution (resp. strict viscosity supersolution) of (1.8) if for any $(x_0,\psi )\in \Omega \times C^2(\Omega )$ such that $x_0$ is a local minimum of $w-\psi$ we have

$$\begin{aligned}\mathcal {F}[\psi ](x_0)\le (\text {resp.} <)\, G(w(x_0),|D_{\mathcal {X}}\psi (x_0)|). \end{aligned}$$

We write

$$\begin{aligned} \mathcal {F}[u]\ge (\,\text {resp.}\,\le )\, G(u, |D_{\mathcal {X}}u|)\;\;\;\text {in}\;\Omega \end{aligned}$$

(1.9)

to indicate that u is a viscosity subsolution (resp. viscosity supersolution) of (1.8) in $\Omega$. Likewise, strict inequalities will be used in (1.9) to indicate strict viscosity subsolution or strict viscosity supersolution.

If $u\in C(\Omega )$ is a viscosity subsolution and a viscosity supersolution of (1.8), then we say u is a viscosity solution of (1.8) and we write

$$\begin{aligned}\mathcal {F}[u]=G(u,|D_{\mathcal {X}}u|)\;\;\; \text {in}\;\Omega . \end{aligned}$$

Remark 1.3

Assume that F satisfies (F-1). If u is a $C^2$ classical subsolution (resp., classical supersolution) of (1.8), then u is also a viscosity subsolution (resp., viscosity supersolution) of (1.8). This follows from (1.5).

Throughout the remainder of this paper we will always assume the following.

(i):

The set $\mathcal {X}:=\{X_1,X_2,\cdots ,X_m\}$ on $\Omega$ of vector fields satisfies the Hörmander’s finite rank condition (1.4).

(ii):

Any reference we make to a solution, subsolution or supersolution is to be understood in the viscosity sense.

(i):

F satisfies condition (F-1).

Statements of the Main Results

In preparation for the statements of the main results, let us first recall the notion of normal vector to the boundary $\partial E$ of a closed set $E\subseteq \mathbb {R}^N$ (see [7, Definition 5.13.9]).

Definition 1.4

Let E be a relatively closed subset of $\Omega$. A vector ${\varvec{v}}\in \mathbb {R}^N\setminus \{0\}$ is said to be normal or orthogonal to E at a point $y\in \Omega \cap \partial E$, written ${\varvec{v}}\perp E$ at y, if and only if

$$\begin{aligned}\overline{B(y+{\varvec{v}},|{\varvec{v}}|)}\subseteq (\Omega \setminus E)\cup \{y\}.\end{aligned}$$

We also write

$$\begin{aligned}E^*:=\{y\in \Omega \cap \partial E: \text {there exists }{\varvec{v}}\text { such that }{\varvec{v}}\perp E\text { at }y\}.\end{aligned}$$

As noted in [7, p.297], when $\Omega$ is connected and $E\ne \emptyset$ and $E\ne \Omega$ we have $E^*\not =\emptyset$.

Definition 1.5

Let $E\subseteq \Omega$ be a relatively closed subset. A vector field X in $\Omega$ is said to be tangent to E if and only if for all $y\in E^*$ and all vectors ${\varvec{v}}$ with ${\varvec{v}}\perp E$ at y the Euclidean inner product $\langle X(y),{\varvec{v}}\rangle$ vanishes; that is

$$\begin{aligned}\langle X(y),{\varvec{v}}\rangle =0.\end{aligned}$$

A family $\mathfrak {X}$ of vector fields in $\Omega$ is said to be tangent to E if and only if X is tangent to E for all $X\in \mathfrak {X}$.

We are now ready to state the main results. In all cases $\mathcal {X}=\{X_1,\cdots ,X_m\}$ satisfies the Hörmander finite rank condition (1.4).

For our next three results, namely Theorems 1.6, 1.7 and 1.8, we assume that H is continuous and that conditions (F-2), (H-i) and (H-d) hold.

Our first result shows that the vector fields $X_i$ are tangent to the set of points of minimum of any non-negative viscosity supersolution of (1.7) in $\Omega$. More, precisely we have the following.

Theorem 1.6

Let $\Omega \subseteq \mathbb {R}^N$ be an open connected set. Suppose $u\in LSU(\Omega )$ is a non-negative supersolution of (1.7) in $\Omega$. Let $E=\{x\in \Omega : u(x)=0\}$. If $E\not =\emptyset$ and $E\not =\Omega$, then for every $y\in E^*$ and every ${\varvec{v}}\perp E$ at y, it follows that

$$\begin{aligned}\langle X_i(y),{\varvec{v}}\rangle =0\;\;\;\;\;\forall \;i=1,\cdots ,m.\end{aligned}$$

The proof of Theorem 1.6 leads to the following.

Theorem 1.7

(Strong Boundary Maximum Principle) Let $\Omega \subseteq \mathbb {R}^N$ be an open set. Suppose $y\in \partial \Omega$ and $u\in LSU(\Omega \cup \{y\})$ is a supersolution of (1.7) such that

(i):

$0=u(y)<u(x)$ for all $x\in \Omega$;

(ii):

there is a ball $B:=B(z,r)\subseteq \Omega$ with $\overline{B}\cap \partial \Omega =\{y\}$;

(iii):

$\langle X_\ell (y),y-z\rangle \not =0$ for some $1\le \ell \le m$.

Then for any ${\varvec{\eta }}\in \mathbb {R}^N$ such that $\langle {\varvec{\eta }},z-y\rangle >0$ we have

$$\begin{aligned} \liminf _{t\rightarrow 0^+}\frac{u(y+t{\varvec{\eta }})-u(y)}{t}>0. \end{aligned}$$

(1.10)

If $s\mapsto H(s,t)$ is non-decreasing for each $t\ge 0$, and $u(y)\le 0$ in (i) above, then (1.10) holds.

The following strong maximum principle is a direct consequence of Theorem 1.6 and Corollary A in Sect. 2.

Theorem 1.8

(Strong Maximum Principle) Let $\Omega \subseteq \mathbb {R}^N$ be an open and connected set. For any non-negative supersolution $u\in \text {LSC}(\Omega )$ of (1.7) we have $u\equiv 0$ in $\Omega$ or $u>0$ in $\Omega$.

Remark 1.9

Assume that F satisfies condition (F-2). Suppose $u\in \text {LSC}(\Omega )$ is a supersolution of (1.7) bounded from below in $\Omega$. Let $c=\inf _\Omega u$. We make the following observations.

(i):

If $c\le 0$, H satisfies (H-d) and $s\mapsto H(s,t)$ is non-decreasing in $\mathbb {R}$ for each $t\in \mathbb {R}^+_0$, then as an immediate consequence of the strong maximum principle we conclude that either $u\equiv c$ or $u>c$ in $\Omega$. Indeed this is true, as in this case $u-c$ is a non-negative supersolution of (1.7), and therefore Theorem 1.8 applies. Simple examples show the above conclusion may not hold if $c>0$.

(ii):

The statement in (i) above may not be true if the monotonicity assumption of $s\mapsto H(s,t)$ is replaced by condition (H-i). An example is given in the appendix.

Next, we present a strong comparison principle associated with $\mathcal {F}[u]=H(u,|Du|)$, where F and H satisfy suitable conditions. We state these conditions below.

(F-2)$^*$::

Given a constant $M>0$, there is a constant $\kappa _M:=\kappa (M)$ such that

$$\begin{aligned}F(x,q,Y)-F(x,p,X)\ge \varpi (|p|)\mathcal {M}_{\lambda ,\Lambda }^-(Y-X)-\kappa _M |q-p|\end{aligned}$$

for all $(x,q,Y),(x,p,X)\in \Omega \times \mathbb {R}^N\times \mathcal {S}^{m\times m}$ with $|x|,|p|,|q|,\Vert X\Vert ,\Vert Y\Vert \le M$, and for some $\varpi :\mathbb {R}^+_0\rightarrow \mathbb {R}^+_0$ that satisfies conditions $\varvec{\mathcal {W}(i)}$ and $\varvec{\mathcal {W}(ii)}$.

We assume that $H:\mathbb {R}\times \mathbb {R}^+_0\rightarrow \mathbb {R}$ satisfies the following conditions.

(H-L)::

Given a constant $M>0$ there is a constant $\mu _M=\mu (M)$ such that

$$\begin{aligned}|H(s_1,t_1)-H(s_2,t_2)|\le \mu _M(|s_1-s_2|+|t_1-t_2|)\end{aligned}$$

for all $(s_j,t_j)\in \mathbb {R}^2$ with $|s_j|,|t_j|\le M,\, j=1,2.$

(H-M)::

$s\mapsto H(s,t)$ is non-decreasing on $\mathbb {R}$ for each $t\in \mathbb {R}^+_0$.

The method of proof of the the strong maximum principle, Theorem 1.8, can be adapted to obtain the following strong comparison principle for the equation

$$\begin{aligned} \mathcal {F}[u]=H(u, |D_{\mathcal {X}}u|). \end{aligned}$$

(1.11)

Theorem 1.10

(Strong Comparison Principle) Let $\Omega \subset \mathbb {R}^n$ be an open connected set and suppose that F satisfies (F-2)$^*$. Assume that $H:\mathbb {R}\times \mathbb {R}_0^+\rightarrow \mathbb {R}$ satisfies conditions (H-L) and (H-M). Let $u\in C^2(\Omega )$ be a subsolution and $w\in \text {LSC}(\Omega )$ be a supersolution of (1.11) in $\Omega$, i.e.,

$$\begin{aligned}\mathcal {F}[u]\ge H(u,|D_{\mathcal {X}}u|),\;\;\;\text {and}\;\;\;\mathcal {F}[w]\le H(w,|D_{\mathcal {X}}w|).\end{aligned}$$

If $\varpi (0)=0$, we also suppose that $|D_\mathcal {X}u|>0$ in $\Omega$. If $u\le w$ in $\Omega$, then exactly one of the following is true in $\Omega$.

$$\begin{aligned}u\equiv w\;\;\;\;\text {or}\;\;\;\;u<w.\end{aligned}$$

Proofs of the main results appear in Section 4.

2 Preliminary results

In this section, we gather some well-known results and include a weak comparison principle for solutions of equations of the kind as in (1.7).

We begin with the following two theorems and their corollaries. These will be useful in the proof of the strong maximum/comparison principle. We refer to [7, 8] for proofs of these results.

Theorem A

Suppose that $\Omega \subseteq \mathbb {R}^N$ is an open set. Let $E\subseteq \Omega$ be a relatively closed subset and X be a smooth vector field in $\Omega$. If X is tangent to E, then any integral curve of X that intersects E is entirely contained in E.

The second theorem we need is stated as follows.

Theorem B

Let $\Omega \subseteq \mathbb {R}^N$ be an open set and $E\subseteq \Omega$ be a relatively closed subset. If $Y_1,\cdots , Y_k$ are smooth vector fields in $\Omega$ each of which is tangent to E, then the Lie algebra generated by these vector fields is also tangent to E.

An important consequence of Theorem A and Theorem B is the following observation.

Corollary A

Let $\Omega \subseteq \mathbb {R}^N$ be an open and connected set, and let $E\subseteq \Omega$ be a relatively closed subset, and $X_1,\cdots , X_m$ be smooth vector fields in $\Omega$ that satisfy Hörmander’s finite rank condition (1.4). If each vector field $X_j$ is tangent to E, then $E=\emptyset$ or $E=\Omega$.

For our work in this paper the following simple weak comparison principle suffices.

Lemma 2.1

(Weak Comparison Principle) Let $\Omega \subseteq \mathbb {R}^N$ be a bounded open set and $G: \mathbb {R}\times \mathbb {R}_0^+\rightarrow \mathbb {R}$ be such that $s\mapsto G(s,t)$ is a non-decreasing function for each $t\ge 0$. Suppose that $v\in C^2(\Omega )\cap C(\overline{\Omega })$ and $w\in \text {LSC}(\overline{\Omega })$ are such that

$$\begin{aligned} \mathcal {F}[v] \ge G(v,|D_{\mathcal {X}}v|)\;\;\text {in}\;\Omega \;\;\;\text {and}\;\;\;\mathcal {F}[w]\le G(w,|D_{\mathcal {X}}w|)\;\;\text {in}\;\Omega . \end{aligned}$$

(2.1)

Assume that at least one of the inequalities is strict. If $v\le w$ on $\partial \Omega$, then $v< w$ in $\Omega$.

Proof

We suppose that $\mathcal {F}[v] > G(v,|D_{\mathcal {X}}v|)$. Assume, contrary to the conclusion of the theorem, that $v\ge w$ at some point in $\Omega$. Then $v-w$ attains its non-negative maximum on $\overline{\Omega }$ at some point $x_0\in \Omega$. In other words, $w-v$ has a minimum at $x_0\in \Omega$. Since w is a viscosity solution of $\mathcal {F}[w]\le G(w,|D_{\mathcal {X}}w|)$ in $\Omega$, and $v\in C^2(\Omega )$ we have

$$\begin{aligned} \mathcal {F}[v](x_0)\le G(w(x_0),|D_\mathcal {X}v(x_0)|). \end{aligned}$$

(2.2)

Since $v(x_0)\ge w(x_0)$ and G(s, t) is non-decreasing in the variable s we conclude from (2.2) the following.

$$\begin{aligned} \mathcal {F}[v](x_0)\le G(v(x_0),|D_\mathcal {X}v(x_0)|). \end{aligned}$$

(2.3)

However, this contradicts the hypothesis that $\mathcal {F}[v]>G(v,|D_\mathcal {X}v|)$ in $\Omega$. We may argue similarly if $\mathcal {F}[w] < G(w,|D_{\mathcal {X}}w|)$. $\square$

Remark 2.2

Suppose the requirement of strict inequality does not hold. If G(s, t) is strictly increasing in s for each $t\ge 0$, then we conclude that $v\le w$ in $\Omega$.

3 Solutions of two-point boundary value problems

In this section, we present the construction of an auxiliary function that will prove useful in showing the strong maximum principle in Theorem 1.8.

Let $\varpi :\mathbb {R}_0^+\rightarrow \mathbb {R}^+_0$ be a non-decreasing function that satisfies conditions $\varvec{\mathcal {W}(i)}$ and $\varvec{\mathcal {W}(ii)}$. Let

$$\begin{aligned} \omega (t)=\int _0^t \varpi (s)ds,\;\;t\ge 0. \end{aligned}$$

(3.1)

In this section we fix a continuous function $H: \mathbb {R} \times \mathbb {R}_0^+\rightarrow \mathbb {R}^+_0$ that satisfies (H-i) and (H-d) for all $a\ge a_0$, and some constant $a_0\ge 1$. The constant $a_0$ will remain fixed through out this section.

Given constants $r_1>0$ and $\varepsilon >0$, we consider the following two-point boundary value problem for a second-order ODE for all $a\ge a_0$.

Our goal is to find a $C^2$ function $\varphi$ defined on $[0,r_1)$ that satisfies (BVP).

We begin with the following remark.

Remark 3.1

If $g:\mathbb {R}_0^+\rightarrow \mathbb {R}_0^+$ is a non-decreasing function such that

$$\begin{aligned} \int _0^1\frac{dt}{g(t)}< \infty ,\;\;\;\text {then}\;\;\;\lim _{t\rightarrow 0^+}\frac{t}{g(t)}=0. \end{aligned}$$

The claim follows from the observation that for any $t>0$

$$\begin{aligned}\frac{t}{g(t)}\le 2\int _{t/2}^t\frac{ds}{g(s)}\le 2\int _0^t\frac{ds}{g(s)}.\end{aligned}$$

$\square$

Remark 3.1 leads to the following lemma.

Lemma 3.2

Let $g:\mathbb {R}_0^+\rightarrow \mathbb {R}_0^+$ be a continuous function. Suppose that, for some $k>0$, the function $t\mapsto kt+g(t)$ is non-decreasing. Then

$$\begin{aligned}\int _0^1\frac{dt}{g(t)}=\infty \quad \text {if and only if} \quad \int _0^1\frac{dt}{g(t)+kt}=\infty .\end{aligned}$$

Proof

First we note that the backward implication is obvious. To prove the forward implication, we assume

$$\begin{aligned} \int _0^1\frac{dt}{ g(t)+kt}<\infty . \end{aligned}$$

(3.2)

Since $t\mapsto g(t)+kt$ is non-decreasing, by Remark 3.1 we have

$$\begin{aligned}\lim _{t\rightarrow 0^+}\frac{t}{ g(t)+kt}=0,\end{aligned}$$

implying that for some $\delta >0$,

$$\begin{aligned}\frac{t}{ g(t)+kt}\le \frac{1}{k+1},\;\;\;\text {for any }0<t<\delta .\end{aligned}$$

Consequently, we have

$$\begin{aligned} \frac{t}{ g(t)}\le 1,\;\;\;\;\;0<t<\delta . \end{aligned}$$

(3.3)

Therefore from (3.2) and (3.3) we conclude that

$$\begin{aligned} \int _0^\delta \frac{dt}{g(t)} =\int _0^\delta \left( 1+\frac{kt}{g(t)}\right) \frac{dt}{g(t)+kt}\le (k+1)\int _0^1\frac{dt}{g(t)+kt}<\infty . \end{aligned}$$

This proves the lemma.$\square$

We now commence our study of (BVP). Let $\omega$ be as in (3.1) and we recall that H satisfies (H-i) and (H-d). Let us set

$$\begin{aligned} h(t)=H(t,t). \end{aligned}$$

(3.4)

Note that according to (H-d) we have

$$\begin{aligned}\int _0^1\frac{dt}{h(a\omega ^{-1}(t))}=\infty ,\;\;\;\text {for all }a\ge a_0.\end{aligned}$$

Furthermore, by condition (H-i) and Remark 1.2 (i), we see that for each $a\ge a_0$ the map $t\mapsto 2at+h(a\omega ^{-1}(t))$ is non-decreasing. Therefore, by Lemma 3.2 we have

$$\begin{aligned} \int _0^1\frac{dt}{2at+h(a\omega ^{-1}(t))}=\infty ,\;\;\text {for}\;a\ge a_0. \end{aligned}$$

(3.5)

We are now ready to show that Problem (BVP) admits a positive solution.

Proposition 3.3

Given $0<r_1\le 1,\,\varepsilon >0$ and $a\ge a_0$, there exists a positive convex solution $\varphi \in C^2([0,r_1])$ of Problem (BVP) such that $\varphi >0$ for $0<r\le r_1$ and $\varphi '>0$ for $0\le r\le r_1$.

Proof

Let $a\ge a_0$, $0<r_1\le 1$ and $\varepsilon >0$ be given. Fix

$$\begin{aligned} 0<\gamma _1<\omega (\varepsilon /r_1). \end{aligned}$$

(3.6)

Let h(t) be as in (3.4) above. We recall the conclusion in (3.5). Hence, we take $\gamma _0>0$ such that

$$\begin{aligned} \int _{\gamma _0}^{\gamma _1}\frac{dt}{2at+h(a\omega ^{-1}(t))}=r_1. \end{aligned}$$

(3.7)

Set

$$\begin{aligned}\Psi (r)=\int _{\gamma _0}^r\frac{dt}{2at+h(a\omega ^{-1}(t))},\;\;\;\gamma _0\le r\le \gamma _1. \end{aligned}$$

Clearly, $\Psi$ is a $C^1$ function such that $\Psi '(r)>0$ for $r\in [\gamma _0,\gamma _1]$ and $\Psi (\gamma _0)=0,\;\Psi (\gamma _1)=r_1$.

Let $\psi :[0,r_1]\rightarrow [\gamma _0,\gamma _1]$ be the inverse of $\Psi$ so that

$$\begin{aligned} \int _{\gamma _0}^{\psi (r)}\frac{dt}{2at+h(a\omega ^{-1}(t))}=r,\;\;\;\;\;0\le r\le r_1. \end{aligned}$$

(3.8)

We note that $\psi \in C^1([0,r_1])$ is increasing and from (3.7) and the identity (3.8) we find

$$\begin{aligned} \psi '(r)=2a\psi (r)+h(a\omega ^{-1}(\psi (r)); \quad \psi (0)=\gamma _0,\, \quad \psi (r_1) =\gamma _1. \end{aligned}$$

(3.9)

Let us now set

$$\begin{aligned}\varphi (r):=\int _0^r \omega ^{-1}(\psi (s))\,ds,\;\;\;\;0\le r\le r_1.\end{aligned}$$

Then $\varphi '=\omega ^{-1}(\psi )>0$ on $[0,r_1]$ and $\varphi \in C^2([0,r_1])$. Note that $\varphi '(0)=\omega ^{-1}(\gamma _0)>0.$ Moreover, for $0\le r\le r_1$ we have

$$\begin{aligned} \varphi (r)&=\int _0^r\omega ^{-1}(\psi (s))\,ds\le r\omega ^{-1}(\psi (r)) \nonumber \\&\le r_1\omega ^{-1}(\psi (r_1)) =r_1\omega ^{-1}(\gamma _1)<\varepsilon . \end{aligned}$$

(3.10)

Here we have used (3.6) and (3.9). Thus,

$$\begin{aligned}\varphi (0)=0\;\;\;\text {and}\;\;\varphi (r_1)\le \varepsilon .\end{aligned}$$

Recalling $0<r_1\le 1$, we should also note from (3.10) that

$$\begin{aligned} \varphi (r)\le \omega ^{-1}(\psi (r)),\;\;\;\;\;\;r\in [0,r_1]. \end{aligned}$$

(3.11)

On differentiating the equation $\omega (\varphi ')=\psi$ on $(0,r_1)$ and using Equation (3.9), we find

$$\begin{aligned} \varpi (\varphi ')\varphi ''&=\psi '= 2a\psi +h(a\omega ^{-1}(\psi ))=a\omega (\varphi ')+a\psi +H(a\omega ^{-1}(\psi ),a\varphi ')\\&=a\omega (\varphi ')+ a\omega (\omega ^{-1}(\psi ))+H(a\omega ^{-1}(\psi ),a\varphi '). \end{aligned}$$

Next, using (3.11) and the monotonicity assumption stipulated in (H-i), we obtain that

$$\begin{aligned} \varpi (\varphi ')\varphi ''\ge a\omega (\varphi ')+ a\omega (\varphi )+H(a\varphi ,a\varphi '). \end{aligned}$$

(3.12)

This completes the proof of the proposition.$\square$

4 Proofs of the Main Results

In this section, we present proofs of the main results stated in Section 1, under the respective assumptions.

We begin with some calculations that will be used in the proofs of Theorems 1.6 and Theorem 1.7.

Let $\mathcal {X}=\{ X_1,\; X_2,\cdots , X_m\}$ be a set of smooth vector fields. Given $z\in \mathbb {R}^N$ we let

$$\begin{aligned} \sigma _i(x)=\langle X_i(x), x-z\rangle ,\;\;\;\forall \;1\le i\le m. \end{aligned}$$

(4.1)

For $\alpha >0$ and $r>0$ we let

$$\begin{aligned} \vartheta (x)=e^{-\alpha |x-z|^2}\;\;\;\text {and}\;\;\;\;w(x)=\vartheta (x)-e^{-\alpha r^2}. \end{aligned}$$

(4.2)

Given a non-negative, increasing and $C^2$ convex function $\varphi$, we set for $k>0$

$$\begin{aligned}v(x):=\varphi (k w(x)).\end{aligned}$$

We now compute, using the notations in (4.1) and (4.2) to obtain the following for $1\le i,j\le m$.

$$\begin{aligned}X_iw(x)=-2\alpha \vartheta (x)\sigma _i(x),\;\;\;\; X_jX_i w(x)=4\alpha ^2\vartheta (x)\sigma _j(x)\sigma _i(x)-2\alpha \vartheta (x)X_j\sigma _i(x).\end{aligned}$$

Since

$$\begin{aligned}X_iv=k\varphi '(kw)X_iw,\;\;\text {and}\;\;X_jX_iv=k^2\varphi ''(kw)X_jwX_iw+k\varphi '(kw)X_jX_iw,\end{aligned}$$

we have

$$\begin{aligned} X_jX_iv&=4\alpha ^2k^2\vartheta ^2(x)\varphi ''(kw)\sigma _j(x)\sigma _i(x)+4k\alpha ^2 \vartheta (x)\varphi '(kw) \sigma _j(x)\sigma _i(x)\\&\quad {}-2k\alpha \vartheta (x)\varphi '(kw)X_j\sigma _i(x),\;\;\;\;\forall \;1\le i,j \le m. \end{aligned}$$

Let $A\in \mathcal {A}_{\lambda ,\Lambda }$. Since $\lambda I_m\le A\le \Lambda I_m$, we estimate $\mathcal {L}_A((D^2_{\mathcal {X}}v)^*)$ as follows. If $\Omega \subseteq \mathbb {R}^N$ is a bounded open set and the vector fields $X_1,\cdots , X_m$ belong to $C^2(\Omega )$, then in any open set $\mathcal {O}\Subset \Omega$ the following estimate holds.

$$\begin{aligned} \mathcal {L}_A((D^2_{\mathcal {X}}v)^*)&\ge 4\lambda \alpha ^2k^2\vartheta ^2(x)\varphi ''(kw)|\sigma (x)|^2 \nonumber \\&+2k\alpha \vartheta (x)\varphi '(kw)\left[ 2\alpha \lambda |\sigma (x)|^2-N^2\Lambda \max _{1\le i,j\le m} \max _{\overline{\mathcal {O}}}|X_j\sigma _i|\right] . \end{aligned}$$

(4.3)

We recall a few items before presenting the proof of Theorem 1.6. The functions $\varpi$, $\omega$ and the quantities $\lambda$ and $\Lambda$ are as in Condition (F-2). We recall also the solution $\varphi$ of

The function $\varphi$ can be found for any $a\ge a_0$, where $a_0\ge 1$, and $0<r_1<1$. See Condition (H-i) and Proposition 3.3.

Proof of Theorem 1.6

We recall that we assume the conditions (F-1), (F-2) on F and the conditions (H-i) and (H-d) on H.

As in the statement of the theorem, we set $E:=\{x\in \Omega \;:\; u(x)=0\}$. Since u is non-negative in $\Omega$ and $u\in \text {LSC}(\Omega )$, E is closed relative to $\Omega$. Assume that

$$\begin{aligned}E\ne \emptyset \;\;\;\text {and}\;\;\;E\ne \Omega .\end{aligned}$$

Since $\Omega$ is connected, we note that $E^*\not =\emptyset$ (see [7, p.297]).

For the sake of contradiction, suppose there is $y\in E^*$, a vector ${\varvec{v}}$ with ${\varvec{v}}\perp E$ at y such that

$$\begin{aligned} \langle X_\ell (y),{\varvec{v}}\rangle \not =0\;\;\;\text { for some }1\le \ell \le m. \end{aligned}$$

(4.4)

Let

$$\begin{aligned} z=y+{\varvec{v}}\;\;\;\text {and}\;\;\;r=|{\varvec{v}}| \end{aligned}$$

(4.5)

so that $\overline{B(z,r)}\subseteq (\Omega \setminus E)\cup \{y\}$. As a consequence of this, we notice that

$$\begin{aligned}u>0\;\;\;\text {in}\;\;\;B(z,r).\end{aligned}$$

With z as in (4.5) and recalling the notation (4.1), we consider the vector field

$$\begin{aligned}\sigma (x):=(\sigma _1(x),\cdots ,\sigma _m(x)),\;\;\;\;\;\;\forall \; x\in \Omega .\end{aligned}$$

By (4.4), we see that $|\sigma (y)|>0$. Since $y\in \Omega \cap \partial E$, we take $\delta _0>0$ such that $B(y,\delta _0)\Subset \Omega$ and

$$\begin{aligned} |\sigma (x)|>0,\;\;\;\forall \;x\in B(y,\delta _0). \end{aligned}$$

Let $M_1,M_2$ and $M_3$ be positive constants such that

$$\begin{aligned} M_1\le |\sigma (x)|\le M_2,\;\;\;\text {and}\;\;\; \max _{i,j} |X_j \sigma _i(x)|\le M_3,\;\;\;\forall \;x\in B(y,\delta _0). \end{aligned}$$

(4.6)

Next, we proceed to construct an auxiliary function. Choose constants k and $\alpha$ as follows. First we fix $\alpha >0$ so that

$$\begin{aligned} \alpha >\frac{ M_3N^2\Lambda +M_2 }{\lambda M_1^2}. \end{aligned}$$

(4.7)

Following this choice of $\alpha$, we fix $k>0$ such that

$$\begin{aligned} 4\alpha ^2 k^2 e^{-2\alpha r^2}\ge \frac{1}{M_1^2}\max \{1,\lambda ^{-1}\}, \end{aligned}$$

(4.8)

where r is as in (4.5).

We fix $0<\varrho <r$ close enough to r so that

$$\begin{aligned} r_1:=k\left( e^{-\alpha \varrho ^2}- e^{-\alpha r^2}\right) \le 1. \end{aligned}$$

(4.9)

Hereafter, we take $0<\delta <\min \{\delta _0,r-\varrho \}$ so that $B(y,\delta )\cap B(z,r)\subseteq B(z,r)\setminus \overline{B(z,\varrho )}.$

Now, we fix a such that

$$\begin{aligned} a\ge \max \{a_0,2k\alpha M_2\}. \end{aligned}$$

(4.10)

Here $a_0\ge 1$ is the positive constant that appears in conditions (H-i) and (H-d).

Next we compare u with an auxiliary function, to be defined below, in the open set

$$\begin{aligned} \mathcal {O}:=B(y,\delta )\,\cap \, B(z,r). \end{aligned}$$

(4.11)

See Fig. 1 above.

Since $\overline{B(z,r)}\subseteq (\Omega \setminus E)\cup \{y\}$, there is an $\varepsilon >0$ such that

$$\begin{aligned} u(x)\ge \varepsilon \;\;\;\text {on}\;\;\; \partial B(y,\delta )\cap B(z,r). \end{aligned}$$

(4.12)

Corresponding to the constants $a, r_1$ (see (4.9) and (4.10)) and $\varepsilon ,$ let $\varphi \in C^2([0,r_1])$ be the solution of (BVP) given in Proposition 3.3. Since $\varphi$ is increasing we find that

$$\begin{aligned} \varphi \left( kw(x)\right) \le \varphi (r_1)\le \varepsilon \;\;\;\;\text {for}\;\,\varrho \le |x-z|\le r, \end{aligned}$$

(4.13)

where k and w are is as in (4.8) and (4.2) respectively.

Fig. 1

The shaded part represents $\mathcal {O}=B(y,\delta )\cap B (z,r)$.

With the constants k and $\alpha$ determined according to (4.7) and (4.8) above, we introduce the following function.

$$\begin{aligned} v(x):=\varphi (k w(x)),\;\;\;\;\;x\in \overline{\mathcal {O}}. \end{aligned}$$

(4.14)

On observing that $v=0$ on $\partial B(z,r)$ and taking both (4.12) and (4.13) into account, we find that

$$\begin{aligned} v\le u \;\;\;\text {on}\;\;\partial \mathcal {O}. \end{aligned}$$

(4.15)

Note that in $\mathcal {O}$, applying (4.6) and (4.8), we have

$$\begin{aligned} |D_{\mathcal {X}}v|&=k\varphi '(kw)|D_{\mathcal {X}}w|=2\alpha k\vartheta |\sigma |\varphi '(kw)\end{aligned}$$

(4.16)

$$\begin{aligned}&\ge (2k\alpha e^{-\alpha r^2}M_1)\varphi '(kw) \nonumber \\&\ge \varphi '(kw)>0. \end{aligned}$$

(4.17)

Next, we use the estimates in (4.3) and (4.6) to obtain the following in $\mathcal {O}$.

$$\begin{aligned}\mathcal {L}_A((D^2_{\mathcal {X}}v)^*)\ge 4\lambda \alpha ^2k^2\vartheta ^2(x)\varphi ''(kw)|\sigma (x)|^2 +2k\alpha \vartheta (x)\varphi '(kw)\left[ 2\alpha \lambda |\sigma (x)|^2- M_3N^2\Lambda \right] .\end{aligned}$$

Consequently, the inequality $\omega (t)\le t\varpi (t)$, together with (4.16) shows that the following holds.

$$\begin{aligned}&\varpi (|D_\mathcal {X}v|)\mathcal {L}_A((D^2_{\mathcal {X}}v)^*)-\omega (|D_\mathcal {X}v|)\ge \varpi (|D_\mathcal {X}v|)\left[ \mathcal {L}_A((D^2_{\mathcal {X}}v)^*)-|D_\mathcal {X}v|\right] \nonumber \\&\ge \varpi (|D_\mathcal {X}v|)\left[ 4\lambda \alpha ^2k^2\vartheta ^2\varphi ''(kw)|\sigma |^2\right. \nonumber \\&{}\left. +2k\alpha \vartheta \varphi '(kw)\left( 2\alpha \lambda |\sigma |^2-M_3N^2\Lambda -M_2\right) \right] \;\,\text {in}\;\mathcal {O}. \end{aligned}$$

(4.18)

Upon using the choices of $\alpha$ and k (see (4.7 and 4.8)), the inequality in (4.18) leads to

$$\begin{aligned} \varpi (|D_\mathcal {X}v|)\mathcal {L}_A((D^2_{\mathcal {X}}v)^*)-\omega (|D_\mathcal {X}v|)\ge \varpi (\varphi '(kw))\varphi ''(kw)\;\,\text {in}\;\mathcal {O}. \end{aligned}$$

(4.19)

Taking the infimum in (4.19) over all $A\in \mathcal {A}_{\lambda ,\Lambda }$ and recalling (BVP), we conclude that the following holds in $\mathcal {O}$.

$$\begin{aligned}&\varpi (|D_\mathcal {X}v|)\mathcal {M}^-_{\lambda ,\Lambda }((D_{\mathcal {X}}^2v)^*)-\omega (|D_\mathcal {X}v|)\nonumber \\&\ge a\omega (\varphi '(kw))+a\omega (\varphi (kw))+H(a\varphi (kw),a\varphi '(kw)). \end{aligned}$$

(4.20)

Recalling (4.6), the choice of a in (4.10), and that $0<\vartheta \le 1$ we get the following from (4.16) in $\mathcal {O}$:

$$\begin{aligned} a\varphi '(kw(x))\ge 2\alpha k\vartheta |\sigma |\varphi '(kw)=|D_{\mathcal {X}}v|. \end{aligned}$$

(4.21)

Using (4.21) in (4.20), condition (H-i) i.e., that $t\mapsto a\omega (t)+H(s,at)$ is non-decreasing, we obtain from (F-2)

$$\begin{aligned} \mathcal {F}[v]&\ge \varpi (|D_\mathcal {X}v|)\mathcal {M}^-_{\lambda ,\Lambda }((D^2v)^*)-\omega (|D_\mathcal {X}v|)\nonumber \\&\ge a\omega (a^{-1}|D_\mathcal {X}v|)+a\omega (v)+H(av,|D_\mathcal {X}v|)\,\;\text {in}\;\mathcal {O}. \end{aligned}$$

(4.22)

Since $|D_\mathcal {X}v|>0$ in $\mathcal {O}$ by (4.17), we get the following strict inequality from (4.22).

$$\begin{aligned} \mathcal {F}[v] > a\omega (v)+H(av,|D_{\mathcal {X}}v|)\;\;\;\text {in}\;\mathcal {O}.\end{aligned}$$

(4.23)

Moreover, on using condition (H-i) ($a\ge a_0\ge 1$), we have

$$\begin{aligned} \mathcal {F}[u]&\le H(u,|D_\mathcal {X}u|)\le a\omega (a^{-1}u)+H(u,|D_\mathcal {X}u|)\nonumber \\&\le a\omega (u)+H(au,|D_\mathcal {X}u|)\;\;\text {in}\;\Omega . \end{aligned}$$

(4.24)

Observe the boundary inequality in (4.15), in conjunction with (2.3) and (4.24). By the weak comparison principle, Lemma 2.1 applied to $G(s,t)=a\omega (s)+H(as,t)$, we conclude that

$$\begin{aligned} v\le u\;\;\; \text {in}\;\; \overline{\mathcal {O}}. \end{aligned}$$

(4.25)

We now wish to extend the inequality (4.25) to a suitable open set containing y. Towards this end, we first extend $\varphi$ to an interval containing $[0,r_1]$. Since $\varphi '(0)>0$ we can choose $0<r_0\le r_1$, sufficiently small, such that

$$\begin{aligned} -t\varphi '(0)\ge t^2 \varphi ''(0), \;\;\;\;\forall \; t\in [-r_0,0]. \end{aligned}$$

(4.26)

Now let

$$\begin{aligned}\Theta (x):=\Phi (kw(x)),\;\;\;\;\varrho \le |x-z|\le R,\end{aligned}$$

where

$$\begin{aligned}\Phi (t):=\left\{ \begin{array}{cl} \varphi (t) &{} \text {if}\;t\in [0,r_1]\\ t\varphi '(0)+\displaystyle {\frac{1}{2}}\varphi ''(0) t^2 &{} \text {if}\; t\in [-r_0,0] \end{array} \right. \end{aligned}$$

and $R>r$ is a suitable constant that depends on $\alpha , k$ and $r_0$. It is clear that $\Phi \in C^2([-r_0,r_1])$ is an extension of $\varphi$. As a consequence of (4.26) we notice that $\Theta (x)\le 0$ (as $w\le 0$) in $B(y,\delta )\setminus B(z, r)$, where $\delta >0$ is chosen small enough such that $B(y,\delta )\subseteq B(z,R)$.

Recalling from (4.25) that $u(x)\ge v(x)=\Theta (x)$ in $\overline{\mathcal {O}}$, we note that

$$\begin{aligned}u\ge \Theta \;\;\;\text {in }B(y,\delta )\;\;\;\;\text {and}\;\;\;\;\Theta (y)=u(y)=0.\end{aligned}$$

Moreover, $D_{\mathcal {X}}\Theta (y)=D_{\mathcal {X}}v(y)$ and $(D^2_{\mathcal {X}} \Theta )^*(y)=(D^2_{\mathcal {X}}v)^*(y).$ Since $\Theta \in C^2(B(y,\delta ))$, $u-\Theta$ has a local minimum at y and u is viscosity supersolution of (1.7), we see that

$$\begin{aligned} \mathcal {F}[v](y)=\mathcal {F}[\Theta ](y)\le H(u(y), |D_{\mathcal {X}} \Theta (y)|)=H(v(y), |D_{\mathcal {X}} v(y)|). \end{aligned}$$

(4.27)

To obtain the desired contradiction let us first observe the following inequality from (4.17) and (4.22).

$$\begin{aligned} \mathcal {F}[v](x)-H(av(x),|D_\mathcal {X}v(x)|)&\ge a\omega (a^{-1}|D_\mathcal {X}v(x)|)\nonumber \\&\ge a\omega (a^{-1} \varphi ' (kw(x) ),\;\;\;x\in \mathcal {O}. \end{aligned}$$

(4.28)

On letting $\mathcal {O}\ni x\rightarrow y$ in (4.28), we conclude (recall that $u(y)=v(y)=0$)

$$\begin{aligned} \mathcal {F}[v](y)-{ H(av(y),|D_\mathcal {X}v(y)|) } \ge a\omega (a^{-1} \varphi '(0) )>0, \end{aligned}$$

(4.29)

which is in contradiction with the inequality in (4.27) and the theorem follows. $\square$

We next proceed to prove Theorem 1.7. The argument uses the auxiliary function v defined in (4.14) together with the differential inequality it satisfies.

Proof of Theorem 1.7

Our assumptions (F-1), and (F-2) on F and the conditions (H-i) and (H-d) on H remain in place. Note that u is a non-negative supersolution of (1.7). Using the assumptions (ii) and (iii) and with the same notations as in the proof of Theorem 1.6 we construct the auxiliary function v as in the proof Theorem 1.6. Using the assumption (i) we proceed in exactly the same manner to obtain $0<v\le u$ in $\mathcal {O}$, where $\mathcal {O}:=B(z,r)\cap B(y,\delta )$ with $\delta >0$ defined as in the proof of Theorem 1.6. See Figure 1, where the dotted line now represents a portion of the boundary $\partial \Omega$ that contains y. Consequently, on recalling that $v(y)=0$, we have

$$\begin{aligned} \liminf _{t\rightarrow 0^+}\frac{u(y+t{\varvec{\eta }})-u(y)}{t}&\ge \liminf _{t\rightarrow 0^+}\frac{v(y+t{\varvec{\eta }})-v(y)}{t}\\&=\langle Dv(y),{\varvec{\eta }}\rangle \\&= 2k\alpha e^{-\alpha r^2}\varphi '(0)\langle z-y,{\varvec{\eta }}\rangle >0. \end{aligned}$$

Finally, if $s\mapsto H(s,t)$ is non-decreasing and $u(y)\le 0$, then $u-u(y)$ is a non-negative supersolution of (1.7) that vanishes at y. Therefore, by what we already proved above the conclusion follows. $\square$

A proof of the strong maximum principle for non-negative viscosity supersolutions of equation (1.7) now follows.

Proof of Theorem 1.8

We prove the theorem by contradiction. Suppose $u\in \text {LSC}(\Omega )$ is a non-negative viscosity supersolution of (1.7) and let us consider the relatively closed set $E:=\{x\in \Omega : u(x)=0\}$. If the conclusion of the theorem is not true, then $E\ne \emptyset$ and $E\ne \Omega$. By Theorem 1.6 we see that $X_j$ is tangent to E for all $j=1,\cdots ,m$. By Theorem B, the Lie algebra generated by $\{X_1,\cdots ,X_m\}$ is tangent to E. But then, by Corollary A, we conclude that $E=\emptyset$ or $E=\Omega$, which is a contradiction. $\square$

We now present the proof of the strong comparison principle, Theorem 1.10, for equations of the type $\mathcal {F}[u]=H(u, |D_{\mathcal {X}}u|)$ where F and H satisfy (F-2)$^*$ and (H-L), respectively.

Proof of Theorem 1.10

Let us consider the relatively closed subset $E:=\{x\in \Omega : u(x)=w(x)\}$ of $\Omega$. Note that $u(x)<w(x)$ on $\Omega \setminus E$. Suppose, contrary to the conclusion we desire to show, that $\emptyset \ne E$ and $E\ne \Omega$. According to Corollary A, there is $y\in E^*$ and an index $\ell$ in $\{1,\cdots ,m\}$ such that $\langle X_\ell (y)\,,\,{\varvec{\nu }}\rangle \ne 0$ for some $\varvec{\nu }\perp E$ at y. Let $z:=y+{\varvec{\nu }}$ and $r:=|{\varvec{\nu }}|=|y-z|.$ Consider the smooth vector field $\sigma (x):=(\sigma _1(x),\cdots ,\sigma _m(x))$, where

$$\begin{aligned}\sigma _i(x):=\langle X_i(x),x-z\rangle ,\;\;\;\;i=1,\cdots ,m.\end{aligned}$$

By assumption $\sigma _\ell (y)\ne 0$, and hence there is a small $\delta >0$ such that

$$\begin{aligned}|\sigma (x)|>0,\;\;\;\;x\in B(y,\delta ).\end{aligned}$$

As in the proof of Theorem 1.6 we take positive constants $M_1,M_2$ and $M_3$ such that

$$\begin{aligned}M_1\le |\sigma (x)|\le M_2\;\;\;\;\text {and}\;\;\;|X_k\sigma _i(x)|\le M_3\;\;\,\text {in}\;B(y,\delta )\;\;\text {for all}\;\;1\le i,k\le m.\end{aligned}$$

For $\rho >0$, to be determined, we set

$$\begin{aligned}\xi _\rho (x)=\frac{1}{\rho ^2}(\vartheta (x)-\vartheta (y)),\;\;\;\text {where}\;\;\vartheta (x):=e^{-\rho |x-z|^2}.\end{aligned}$$

Direct computation leads to, for $1\le i,j\le m$,

$$\begin{aligned} X_i\xi _\rho (x)=-\frac{2}{\rho }\vartheta (x)\sigma _i(x),\;\;\; X_jX_i\xi _\rho (x)=4\vartheta (x)\sigma _j(x)\sigma _i(x)-\frac{2}{\rho }\vartheta (x)X_j\sigma _i(x).\end{aligned}$$

Therefore, the following holds in $\mathcal {O}:=B(y,\delta )\cap B(z,r)$, (see Figure 1).

$$\begin{aligned}&\mathcal {L}_A((D^2_{\mathcal {X}}\xi _\rho )^*)\ge 4\lambda \vartheta (x)|\sigma (x)|^2-\frac{2}{\rho } \, \vartheta (x)\Lambda N^2M_3 \nonumber \\&\ge \vartheta (x)\left( 4\lambda M_1^2-\frac{2}{\rho }\Lambda N^2M_3\right) , \end{aligned}$$

(4.30)

and as a consequence, we get the following in $\mathcal {O}$, for a sufficiently large $\rho$.

$$\begin{aligned} \mathcal {M}_{\lambda ,\Lambda }^-((D^2_{\mathcal {X}}\xi _\rho )^*)\ge \vartheta (x)\left( 4\lambda M_1^2-\frac{2}{\rho }\Lambda N^2M_3\right) . \end{aligned}$$

(4.31)

Using Condition (F-2)$^*$ and inequality (4.31), we obtain the following in $\mathcal {O}.$

$$\begin{aligned} \mathcal {F}[u+\xi _\rho ]-\mathcal {F}[u]&=F(x,D_\mathcal {X}(u+\xi _\rho ),(D^2_{\mathcal {X}}(u+\xi _\rho ))^*)-F(x,D_\mathcal {X}u,(D_{\mathcal {X}}^2u)^*)\nonumber \\&\ge \varpi (|D_\mathcal {X}u|)\mathcal {M}_{\lambda ,\Lambda }^-((D^2_\mathcal {X}\xi _\rho )^*)-\kappa |D_\mathcal {X}\xi _\rho |\nonumber \\&\ge 2\vartheta (x)\left[ \varpi (|D_\mathcal {X}u|)\left( 2\lambda M_1^2-\frac{1}{\rho }\Lambda N^2 M_3\right) -\frac{\kappa M_2}{\rho }\right] . \end{aligned}$$

(4.32)

Next, we use condition (H-L) to estimate

$$\begin{aligned} |H(u,|D_{\mathcal {X}}u|)-H(u+\xi _\rho ,|D_{\mathcal {X}}(u+\xi _\rho )|)|&\le \mu (|\xi _\rho |+|D_\mathcal {X}\xi _\rho |)\\&\le \mu \left( \frac{1}{\rho ^2}+\frac{2M_2}{\rho }\right) . \end{aligned}$$

Consequently the following is true in $\mathcal {O}$.

$$\begin{aligned} H(u, |D_{\mathcal {X}} u| )\ge H(u+\xi _\rho ,|D_{\mathcal {X}}(u+\xi _\rho )|)- \mu \left( \frac{1}{\rho ^2}+\frac{2M_2}{\rho }\right) . \end{aligned}$$

(4.33)

Let $\eta :=\min _{\overline{\mathcal {O}}}\left( |D_\mathcal {X}u|\right)$. Since $\varpi$ is non-decreasing, we have $\varpi \left( |D_\mathcal {X}u|\right) \ge \varpi (\eta )$ in $\mathcal {O}$. By our hypothesis on u we note that if $\eta =0$, then $\varpi (\eta )=\varpi (0)>0$. Using this, (4.32) and (4.33), and recalling that $\mathcal {F}[u]\ge H(u, |D_{\mathcal {X}}u|)$, we find the following in $\mathcal {O}$.

$$\begin{aligned}&\mathcal {F}[u+\xi _\rho ] \ge \mathcal {F}[u]+ 2\vartheta (x)\left[ \varpi (|D_\mathcal {X}u|)\left( 2\lambda M_1^2-\frac{1}{\rho }\Lambda N^2 M_3\right) -\frac{\kappa M_2}{\rho }\right] \\&\ge H(u+\xi _\rho ,|D_{\mathcal {X}}(u+\xi _\rho )|)+2\vartheta (y)\left[ \varpi (\eta )\left( 2\lambda M_1^2-\frac{1}{\rho } \Lambda N^2 M_3\right) -\frac{\kappa M_2}{\rho }\right] \\&{}-\mu \left( \frac{1}{\rho ^2}+\frac{2M_2}{\rho }\right) . \end{aligned}$$

On taking $\rho >0$ large enough we obtain the strict inequality

$$\begin{aligned} \mathcal {F}[u+\xi _\rho ]>H(u+\xi _\rho ,|D_{\mathcal {X}}(u+\xi _\rho )|)\;\;\;\text {in}\;\mathcal {O}. \end{aligned}$$

(4.34)

Clearly, the strict inequality in (4.34) extends to $\overline{\mathcal {O}}$. Since $\xi _\rho \equiv 0$ on $\partial B(z,r)$, and $u\le w$ in $\Omega$ we see that $u+\xi _\rho \le w$ on $B(y,\delta )\cap \partial B(z,r)$. On $\partial B(y,\delta )\cap B(z,r)$ we have $u(x)<w(x)$. By choosing $\rho$ sufficiently large we can guarantee that $u+\xi _\rho <w$ on $\partial B(y,\delta )\cap B(z,r)$. Thus $u+\xi _\rho \le w$ on $\partial \mathcal {O}$.

As a consequence of (4.34), condition (H-M) and the fact that w is a supersolution of (1.7), we invoke the weak comparison principle, Lemma 2.1 to obtain

$$\begin{aligned} u+\xi _\rho \le w\;\;\;\text {in}\;\,\mathcal {O}=B(z,r)\cap B(y,\delta ). \end{aligned}$$

(4.35)

Since, by assumption, $u\le w$ in $\Omega$ and $\xi _\rho \le 0$ on $\Omega \setminus B(z,r)$ we conclude that $u+\xi _\rho \le w$ in $\Omega \setminus B(y,\delta )$. This, together with (4.35), then shows that $u+\xi _\rho \le w$ in $B(y,\delta )$. Note that $(w-(u+\xi _\rho ))(y)=0$, and therefore $w-(u+\xi _\rho )$ has a local minimum at y. Since $u+\xi _\rho \in C^2(\Omega )$ and w is a supersolution, by definition, we find

$$\begin{aligned}\mathcal {F}[u+\xi _\rho ](y)\le H(w(y),|D_\mathcal {X}(u+\xi _\rho )(y)|)=H((u+\xi _\rho )(y),|D_\mathcal {X}(u+\xi _\rho )(y)|).\end{aligned}$$

Obviously this conclusion is contrary to (4.34). This completes the proof of the theorem.$\square$

Appendix

Let us begin by providing a proof of Remark 1.2 (ii). Let us first note that for any $a>0,s>0$ we have

$$\begin{aligned} a\,\omega \left( \frac{s}{a}\right) =a\int _0^{s/a}\varpi (\tau )\,d\tau =\int _0^s\varpi \left( \frac{\tau }{a}\right) \,d\tau . \end{aligned}$$

(5.1)

For a fixed s, we take the limit in (5.1) as $a\rightarrow \infty$ to get

$$\begin{aligned} \lim _{a\rightarrow \infty }a\,\omega \left( \frac{s}{a}\right) =s\,\varpi (0). \end{aligned}$$

(5.2)

Given $t\ge 0$, assume that the function $s\mapsto a\omega (s)+ H(as,t)$ is non-decreasing in s for all $a\ge a_0\ge 1$. Let $\alpha \le \beta$. Then, by assumption, we have

$$\begin{aligned}a\,\omega \left( \frac{\alpha }{a}\right) +H\left( a\left( \frac{\alpha }{a}\right) \,,\,t\right) \le a\,\omega \left( \frac{\beta }{a}\right) +H\left( a\left( \frac{\beta }{a}\right) \,,\,t\right) \end{aligned}$$

Sending $a\rightarrow \infty$, and using the limit in (5.2) we conclude that

$$\begin{aligned}H(\beta ,t)-H(\alpha ,t )\ge (\alpha -\beta )\varpi (0).\end{aligned}$$

If $\varpi (0)=0$, then $H(\beta , t)\ge H(\alpha ,t).$ $\square$

In the remainder of this appendix we provide some examples to illustrate the conditions on F and H that were used to obtain the strong maximum and comparison principles.

For $(x,p,Y)\in \Omega \times \mathbb {R}^N\times S^{m\times m},$ we set

$$\begin{aligned}F(x,p, Y):= b_{ij}(x, p ) Y_{ij} \end{aligned}$$

where $b:= (b_{ij} )\in S^{m\times m}$ satisfies

$$\begin{aligned} \lambda \varpi (|p|) I_m \le b(x,p) \le \Lambda \varpi (|p|) I_m,\;\;\;\;(x,p)\in \Omega \times \mathbb {R}^N. \end{aligned}$$

(5.3)

We take $0<\lambda \le \Lambda <\infty$ and the function $\varpi :\mathbb {R}^+_0\rightarrow \mathbb {R}^+_0$ is a non-decreasing function such that $\varpi (t)>0$ for $t>0$. It follows from (5.3) that F satisfies conditions (F-1) and (F-2). Thus, in this case a supersolution u of (1.7) would satisfy

$$\begin{aligned}\text {trace}\left( b(x,D_\mathcal {X}u) (D^2_{\mathcal {X}} u)^* \right)&= \frac{1}{2}b_{ij}(x, D_{\mathcal {X}} u(x)) \left[ X_i (X_j u)+X_j (X_i u) \right] \\&\le H(u, |D_{\mathcal {X}} u|)\;\;\text{ in }\;\;\Omega , \end{aligned}$$

where H satisfies (H-i) and (H-d). A specific example can be given by taking $b(x,p)=|p|^\gamma I_m$ and $H(s,t)=\max \{0,s\}^\alpha +t^\beta$, where $\alpha ,\beta$ and $\gamma$ are non-negative numbers. This leads to

$$\begin{aligned} |D_{\mathcal {X}} u|^\gamma \sum _{i=1}^m X_i^2 u\le u^{\alpha }+|D_{\mathcal {X}}u|^{\beta }\;\;\text{ in } \Omega , \end{aligned}$$

(5.4)

where $\Omega \subseteq \mathbb {R}^N$ is a connected open subset. If $\min (\alpha , \beta )\ge \gamma +1$ and u is a non-negative viscosity solution of (5.4), then Theorem 1.8 shows that either $u\equiv 0$ in $\Omega$ or $u>0$ in $\Omega$

We mention here that if $b_{ij}=b_{ij}(x)$ with $\lambda I_m\le b\le \Lambda I_m$, and $c_i(x)$ and d(x) are continuous and bounded, the elliptic inequality

$$\begin{aligned} b_{ij}\left[ X_i(X_j u)+X_j(X_iu ) \right] +c_i (X_iu)+d u \le 0 \end{aligned}$$

(5.5)

is also included in our work. To see this, define $H(u, |D_{\mathcal {X}} u|)=M( u+|D_{\mathcal {X}} u|)$, where $M\ge \max \{ \max |d|,\; \max _{1\le i\le m}|c_i | \}$. Thus, (5.5) may be written as

$$\begin{aligned}F(x, ( D^2_{\mathcal {X}} u)^*)\le H(u, |D_{\mathcal {X}} u|) \;\;\text{ in } \Omega ,\end{aligned}$$

where F satisfies (F-1) and (F-2) and H satisfies (H-i) and (H-d).

Let h be a $C^1$ function on $\mathbb {R}_0^+$, and set

$$\begin{aligned}\varpi (t):=h(t^2)\;\;\text {for}\;t\ge 0.\end{aligned}$$

Then

$$\begin{aligned} \varpi (|q|)-\varpi (|p|)&=\int _0^1 \frac{d}{dt}h(|p+t(q-p)|^2)\,dt\\&=2(q-p)\cdot \int _0^1h'(|p+t(q-p)|^2)(p+t(q-p))\,dt\\&:=\ell (p,q). \end{aligned}$$

Let $\mathcal {K}\subseteq \mathbb {R}^N\times \mathbb {S}^{m\times m}$ be a given compact set. Then for all $(p, X),(q,Y)\in \mathcal {K}$ we have

$$\begin{aligned} \varpi (|q|)\mathcal {M}_{\lambda ,\Lambda }^-(Y)-\varpi (|p|)\mathcal {M}_{\lambda ,\Lambda }^-(X)&=\left( \varpi (|p|)+\ell (p,q)\right) \mathcal {M}_{\lambda ,\Lambda }^-(Y)-\varpi (|p|)\mathcal {M}_{\lambda ,\Lambda }^-(X)\\&=\varpi (|p|)\left\{ \mathcal {M}_{\lambda ,\Lambda }^-(Y)-\mathcal {M}_{\lambda ,\Lambda }^-(X)\right\} +\ell (p,q)\mathcal {M}_{\lambda ,\Lambda }^-(Y)\\&\ge \varpi (|p|)\mathcal {M}_{\lambda ,\Lambda }^-(Y-X)-\kappa |q-p|, \end{aligned}$$

where $\kappa$ depends on $\mathcal {K}$. See [9] for the last inequality used above.

Therefore $F(x,p,X):=\varpi (|p|)\mathcal {M}_{\lambda ,\Lambda }^-(X)$ satisfies condition (F-2)$^*$. Obviously, F satisfies (F-1) and (F-2).

Next we look at some examples with the goal of showing the following.

(i):

If H satisfies (H-d) and (H-i) but $s\mapsto H(s,t)$ is not monotonic non-decreasing in s for each $t\ge 0$, then Theorem 1.8 may not hold for supersolutions u with $\inf _\Omega u<0$. See Remark 1.9 (ii).

(ii):

The strong comparison principle, Theorem 1.10, may fail if the assumption $|Du|>0$ is not met when $\varpi (0)=0$.

For (i), we consider $F(x,p,X)=\text {trace}(X)$ and $H(s,t)=\sin ^2 s+Nt^2$. Then H satisfies both (H-i) and (H-d), but $s\mapsto H(s,t)$ is not monotonic. Let

$$\begin{aligned}u(x)=\frac{1}{8N}|x|^2-\frac{\pi }{2}.\end{aligned}$$

We note that u is a sign-changing function in the ball $B:=B\left( o,2\sqrt{5\pi N/3}\right)$ such that

$$\begin{aligned}\frac{1}{4}\le \sin ^2\left( \frac{1}{8N}|x|^2-\frac{\pi }{2}\right) +\frac{1}{16N}|x|^2,\;\;\;\;\;\text {for}\;x\in B.\end{aligned}$$

Since $F(x,Du,D^2u)=\Delta u=1/4$, we see that $F(x,Du,D^2u)\le H(u,|Du|)$ holds in the ball B.

Let us now look at (ii). Let $F(x,p,X)=|p|^2\,\text {trace}(X)$ and $H(s,t)= 4N\,|t|$, where N is the dimension of $\mathbb {R}^N$. As noted earlier, F satisfies (F-2)$^*$ with $\varpi (t)=t^2$. Clearly F satisfies (F-1). Note that $u\equiv 0$ is a subsolution and $w(x)=|x|^2$ is a supersolution of

$$\begin{aligned} F(x,Dw,D^2w)=H(w,|Dw|) \;\;\text {in}\;B(o,1)\subseteq \mathbb {R}^N. \end{aligned}$$

(5.6)

This demonstrates the necessity of the condition $|Du|>0$ when $\varpi (0)=0$ in the strong comparison principle, Theorem (1.10).