# Seminario de Ecuaciones Diferenciales

#### 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 $W_{0}^{1,2}(\Omega) \times W_{0}^{1,2}(\Omega)$, $$\label{j} J(v,\psi ) = \frac12 \int_{\Omega} \,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega} v\,E(x) \nabla\psi - \int_{\Omega} f(x)\,v\,.$$ where $A(x)$, $M(x)$ are symmetric measurable matrices such that $$\label{al} \begin{cases} A(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |A(x)| \leq \beta\,, \\ M(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |M(x)| \leq \beta\,, \end{cases}$$ for almost every $x$ in $\Omega$, for every $\xi$ in $\mathbb{R}^{N}$, with $0 < \alpha \leq \beta$, and $$\label{fm} f\in L^{m}(\Omega)\,,\ m\geq 2_{*} =\frac{2N}{N+2},$$ $$\label{e} E\in(L^{N}(\Omega))^N.$$ 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 $$\label{i} I(v,\psi ) = \frac12\int_{\Omega}\,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega}|v|^r\psi - \int_{\Omega} f(x)\,v\,.$$
Fecha: 4 de junio de 2018, 13:00 - 14:00
Lugar: Sala de Conferencias, Facultad de Ciencias