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.
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Notes
Du denotes the gradient of u relative to the standard canonical frame \(\{\frac{\partial }{\partial x_1},\cdots ,\frac{\partial }{\partial x_N}\}\) of \(\mathbb {R}^N\).
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Appendix
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
For a fixed s, we take the limit in (5.1) as \(a\rightarrow \infty \) to get
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
Sending \(a\rightarrow \infty \), and using the limit in (5.2) we conclude that
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
where \(b:= (b_{ij} )\in S^{m\times m}\) satisfies
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
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
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
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
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
Then
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
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
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
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
This demonstrates the necessity of the condition \(|Du|>0\) when \(\varpi (0)=0\) in the strong comparison principle, Theorem (1.10).
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Bhattacharya, T., Mohammed, A. On a strong maximum principle for fully nonlinear subelliptic equations with Hörmander condition. Calc. Var. 60, 9 (2021). https://doi.org/10.1007/s00526-020-01869-4
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DOI: https://doi.org/10.1007/s00526-020-01869-4