Maximum principles and symmetry results in sub-Riemannian settings.

*(English)*Zbl 1232.35026
Farina, Alberto (ed.) et al., Symmetry for elliptic PDEs. 30 years after a conjecture of De Giorgi and related problems. Selected papers presented at the INdAM school, Rome, Italy, May 25–29, 2009. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4804-3/pbk). 17-33 (2010).

The author presents some weak and strong maximum principles for linear second order partial differential equations with nonnegative characteristic form on bounded and unbounded domains.

The author considers the linear second-order PDE of the form \[ \mathcal{L}:=\sum_{i,j=1}^Na_{ij}(x)\frac{\partial ^2u}{ \partial x_i\partial x_j}+\sum_{j=1}^Nb_j(x) \frac{ \partial u}{\partial x_j}+c(x), \tag{1} \] where the coefficients \(a_{ij},b_j\) and \(c\) are real functions defined on a domain \(D\subset \mathbb{R}^N\), and the matrix \(A(x):=(a_{ij}(x))_{i,j=1,\dots ,N}\) is symmetric and nonnegative definite at any point \(x\in D\).

The paper is structured in 5 sections.

In Section 1 the author shows some weak maximum principle including the classical one due to Picone. In Section 2, some sufficient conditions are given for the operator \( \mathcal{L}\) to satisfy the following.

Strong maximum principle: Let \(\Omega \subset D\) be a connected open set and let \(u\in C^2(\Omega ,\mathbb{R})\) be such that \( \mathcal{L}u\geq 0\), \(u\leq 0\text{ in }\Omega\). Assume that there exists \(x_0\in \Omega \) such that \(u(x_0) =0.\) Then \(u\equiv 0\) in \(\Omega\). The conditions stated follow from a Hopf-type lemma together with the Nagumo-Bony’s nonsmooth analysis result. The operator \(\mathcal{L}\) in (1) will be assumed to satisfy the maximum principle on small domains. Moreover, continuity of \(a_{ij}\) and local boundedness of \(b_j\) and \(c\) are assumed.

In Section 3, the author deals with maximum principles for sub-Laplacians on unbounded domains. Section 4 is dedicated to some monotonicity and one-dimensional symmetry results for the following semilinear equation in \(\mathbb{R}^N:\) \[ \mathcal{L}u+f(u)=0, \] where \(\mathcal{L}\) is now a sub-Laplacian in \(\mathbb{R}^N\), \(f\) is of class \(C^1\) in an open interval containing \([0,1]\), \(f(0)=f(1)=0,\) and there exists \(\delta \in (0,1)\) such that \(f\) is decreasing on \([1-\delta,1]\) and on \([-1,-1+\delta].\) In Section 5 the author gives a negative answer to a one-dimensional symmetry problem for sub-Laplacians on the Heisenberg group. The final section deals with the Modica-Mortola equation for sub-Laplacians.

For the entire collection see [Zbl 1200.35001].

The author considers the linear second-order PDE of the form \[ \mathcal{L}:=\sum_{i,j=1}^Na_{ij}(x)\frac{\partial ^2u}{ \partial x_i\partial x_j}+\sum_{j=1}^Nb_j(x) \frac{ \partial u}{\partial x_j}+c(x), \tag{1} \] where the coefficients \(a_{ij},b_j\) and \(c\) are real functions defined on a domain \(D\subset \mathbb{R}^N\), and the matrix \(A(x):=(a_{ij}(x))_{i,j=1,\dots ,N}\) is symmetric and nonnegative definite at any point \(x\in D\).

The paper is structured in 5 sections.

In Section 1 the author shows some weak maximum principle including the classical one due to Picone. In Section 2, some sufficient conditions are given for the operator \( \mathcal{L}\) to satisfy the following.

Strong maximum principle: Let \(\Omega \subset D\) be a connected open set and let \(u\in C^2(\Omega ,\mathbb{R})\) be such that \( \mathcal{L}u\geq 0\), \(u\leq 0\text{ in }\Omega\). Assume that there exists \(x_0\in \Omega \) such that \(u(x_0) =0.\) Then \(u\equiv 0\) in \(\Omega\). The conditions stated follow from a Hopf-type lemma together with the Nagumo-Bony’s nonsmooth analysis result. The operator \(\mathcal{L}\) in (1) will be assumed to satisfy the maximum principle on small domains. Moreover, continuity of \(a_{ij}\) and local boundedness of \(b_j\) and \(c\) are assumed.

In Section 3, the author deals with maximum principles for sub-Laplacians on unbounded domains. Section 4 is dedicated to some monotonicity and one-dimensional symmetry results for the following semilinear equation in \(\mathbb{R}^N:\) \[ \mathcal{L}u+f(u)=0, \] where \(\mathcal{L}\) is now a sub-Laplacian in \(\mathbb{R}^N\), \(f\) is of class \(C^1\) in an open interval containing \([0,1]\), \(f(0)=f(1)=0,\) and there exists \(\delta \in (0,1)\) such that \(f\) is decreasing on \([1-\delta,1]\) and on \([-1,-1+\delta].\) In Section 5 the author gives a negative answer to a one-dimensional symmetry problem for sub-Laplacians on the Heisenberg group. The final section deals with the Modica-Mortola equation for sub-Laplacians.

For the entire collection see [Zbl 1200.35001].

Reviewer: Cristian Chifu (Cluj-Napoca)