Event Details
Día: 6 de Noviembre de 2020
Hora: 11:30 - 12:30
Lugar: Videoconferencia Sala EUROPA de la UGR,
Acceso Sala Virtual Europa
Contraseña de la reunión: 237948
Ponente: Alberto Roncoroni (Universidad de Granada, España)
Título: Classification of solutions to the critical p-Laplace equations Resumen: We consider the following critical p-Laplace equation: (1)$$\begin{equation}(1)\quad \Delta_p u+u^{p^{\ast}-1}=0 \quad \text{ in \(\mathbb{R}^n\)} \, , \end{equation}$$ with \(n \geq 2\) and \(1 < p < n\). Equation (1) has been largely studied in the PDE’s and geometric analysis’ communities, since extremals of Sobolev inequality solve (1) and, for p = 2, the equation is related to the Yamabe’s problem. In particular it has been recently shown, exploiting the moving planes method, that positive solutions to (1) such that $$u\in L^{p^\ast}(\mathbb{R}^n)$$ and $$\nabla u\in L^p(\mathbb{R}^n)$$ can be completely classified. In the talk we will consider the anisotropic critical p-Laplace equation in convex cones of \(\mathbb{R}^n\). Since the moving plane method strongly relies on the symmetries of the equation and of the domain, in the talk a different approach to this problem will be presented. In particular this approach gives a complete classification of the solutions in an anisotropic setting. More precisely, we characterize solutions to the critical p-Laplace equation induced by a smooth norm inside any convex cone of \(\mathbb{R}^n\). This is a joint work with G. Ciraolo and A. Figalli.
Título: Classification of solutions to the critical p-Laplace equations Resumen: We consider the following critical p-Laplace equation: (1)$$\begin{equation}(1)\quad \Delta_p u+u^{p^{\ast}-1}=0 \quad \text{ in \(\mathbb{R}^n\)} \, , \end{equation}$$ with \(n \geq 2\) and \(1 < p < n\). Equation (1) has been largely studied in the PDE’s and geometric analysis’ communities, since extremals of Sobolev inequality solve (1) and, for p = 2, the equation is related to the Yamabe’s problem. In particular it has been recently shown, exploiting the moving planes method, that positive solutions to (1) such that $$u\in L^{p^\ast}(\mathbb{R}^n)$$ and $$\nabla u\in L^p(\mathbb{R}^n)$$ can be completely classified. In the talk we will consider the anisotropic critical p-Laplace equation in convex cones of \(\mathbb{R}^n\). Since the moving plane method strongly relies on the symmetries of the equation and of the domain, in the talk a different approach to this problem will be presented. In particular this approach gives a complete classification of the solutions in an anisotropic setting. More precisely, we characterize solutions to the critical p-Laplace equation induced by a smooth norm inside any convex cone of \(\mathbb{R}^n\). This is a joint work with G. Ciraolo and A. Figalli.
