Detalles de Evento
Título: Existence and properties of saddle points of some integral functionals defined in $W_{0}^{1,2}(\Omega) \times W_{0}^{1,2}(\Omega)$
Conferenciante: Lucio Boccardo ("Sapienza" Università di Roma, Italia)
Resumen: Let \( \Omega \) be a bounded, open subset of \( \mathbb{R}^{N} \), with \( N > 2 \). Let us define, for \( (v,\psi) \) in \( W_{0}^{1,2}(\Omega) \times W_{0}^{1,2}(\Omega) \), \begin{equation} \label{j} J(v,\psi ) = \frac12 \int_{\Omega} \,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega} v\,E(x) \nabla\psi - \int_{\Omega} f(x)\,v\,. \end{equation} where \( A(x) \), \( M(x) \) are symmetric measurable matrices such that \begin{equation} \label{al} \begin{cases} A(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |A(x)| \leq \beta\,, \\ M(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |M(x)| \leq \beta\,, \end{cases} \end{equation} for almost every \( x \) in \( \Omega \), for every \( \xi \) in \( \mathbb{R}^{N} \), with \( 0 < \alpha \leq \beta \), and \begin{equation}\label{fm} f\in L^{m}(\Omega)\,,\ m\geq 2_{*} =\frac{2N}{N+2}, \end{equation} \begin{equation} \label{e} E\in(L^{N}(\Omega))^N. \end{equation} We study the existence of saddle points of the functional \( J \) defined above both in the regular case, i.e., if \( E \) belongs to \( (L^{N}(\Omega))^{N} \) and in the singular one, i.e., if \( E \) belongs to \( (L^{2}(\Omega))^{N} \). The second problem concerns the functional \begin{equation} \label{i} I(v,\psi ) = \frac12\int_{\Omega}\,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega}|v|^r\psi - \int_{\Omega} f(x)\,v\,. \end{equation}
Fecha: 4 de junio de 2018, 13:00 - 14:00
Lugar: Sala de Conferencias, Facultad de Ciencias
Conferenciante: Lucio Boccardo ("Sapienza" Università di Roma, Italia)
Resumen: Let \( \Omega \) be a bounded, open subset of \( \mathbb{R}^{N} \), with \( N > 2 \). Let us define, for \( (v,\psi) \) in \( W_{0}^{1,2}(\Omega) \times W_{0}^{1,2}(\Omega) \), \begin{equation} \label{j} J(v,\psi ) = \frac12 \int_{\Omega} \,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega} v\,E(x) \nabla\psi - \int_{\Omega} f(x)\,v\,. \end{equation} where \( A(x) \), \( M(x) \) are symmetric measurable matrices such that \begin{equation} \label{al} \begin{cases} A(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |A(x)| \leq \beta\,, \\ M(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |M(x)| \leq \beta\,, \end{cases} \end{equation} for almost every \( x \) in \( \Omega \), for every \( \xi \) in \( \mathbb{R}^{N} \), with \( 0 < \alpha \leq \beta \), and \begin{equation}\label{fm} f\in L^{m}(\Omega)\,,\ m\geq 2_{*} =\frac{2N}{N+2}, \end{equation} \begin{equation} \label{e} E\in(L^{N}(\Omega))^N. \end{equation} We study the existence of saddle points of the functional \( J \) defined above both in the regular case, i.e., if \( E \) belongs to \( (L^{N}(\Omega))^{N} \) and in the singular one, i.e., if \( E \) belongs to \( (L^{2}(\Omega))^{N} \). The second problem concerns the functional \begin{equation} \label{i} I(v,\psi ) = \frac12\int_{\Omega}\,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega}|v|^r\psi - \int_{\Omega} f(x)\,v\,. \end{equation}
Fecha: 4 de junio de 2018, 13:00 - 14:00
Lugar: Sala de Conferencias, Facultad de Ciencias
