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.

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).