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A symmetry result for cooperative elliptic systems with singularities. (English) Zbl 1447.35148

The authors investigate symmetry results for the solutions of an elliptic system of equations possessing a cooperative structure in a domain that may have “holes” or “cuts” or, more in general, “small vacancies” along which the solutions can become singular. More precisely, they consider a convex open set \(\Omega \subseteq \mathbb{R}^n\) of class \(C^\infty\) which is bounded and symmetric with respect to the hyperplane \(\{x_1=0\}\) and a closed set \(\Gamma \subseteq \Omega \cap \{x_1=0\}\) which consists of a point if \(n=2\) or verifying \(\mathrm{Cap}_2(\Gamma)=0\) if \(n\geq 3\). Then they analyse a second-order cooperative elliptic system \[ \left\{ \begin{array}{ll} -\Delta u_i =f_i(u_1,\dots,u_m) & \mathrm{in}\ \Omega \setminus \Gamma\, ,\\ u_i>0 & \mathrm{in}\ \Omega \setminus \Gamma\, ,\\ u_i \equiv 0 & \mathrm{on}\ \partial \Omega\, , \end{array} \right. \] where \(f_1,\dots, f_m \in \mathrm{Lip}(\mathbb{R}^m)\). Under suitable assumptions, they show that the solutions \(u_1, \dots u_m\) are symmetric with respect to the hyperplane \(\{x_1=0\}\) and increasing in the \(x_1\)-direction in \(\Omega \cap \{x_1<0\}\) and that for every \(i \in {1,\dots,m}\) one has \[ \frac{\partial u_i}{\partial x_1}(x) \] for every \(x \in \Omega \cap \{x_1<0\}\).

MSC:

35J57 Boundary value problems for second-order elliptic systems
35B06 Symmetries, invariants, etc. in context of PDEs
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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