Sesión Ecuaciones Diferenciales y Probabilidad

Diciembre 14, 17:50 ~ 18:10

Soluciones periódicas de ecuaciones de Euler-Lagrange en el marco de espacios de Orlicz-Sobolev anisotrópicos

ACINAS, Sonia Ester

\newtheorem{thm}{Theorem} \newcommand{\wphi}{W^{1}\lphi} \newcommand{\lphi}{L^{\Phi}} Let $\Phi:\mathbb{R}^d\to [0,+\infty)$ be a differentiable, convex function such that $\Phi(0)=0$, $\Phi(y)>0$ if $y\neq 0$, $\Phi(-y)=\Phi(y)$, and $\lim\limits_{|y|\to\infty}\frac{\Phi(y)}{|y|}=+\infty.$ We denote by $\Phi^*$ the complementary (in the sense of convex analysis) function of $\Phi$. For $T>0$, we assume that $F:[0,T]\times\mathbb{R}^d\to\mathbb{R}^d$ with $F$ and $\nabla_x F$ Carathéodory functions satisfying \begin{equation*}\label{eq:phi-lagrange} |F(t,x)| + |\nabla_x F(t,x)| \leq a(x)b(t), \mbox{ for a.e. } t\in [0,T], \end{equation*} where $a\in C\left(\mathbb{R}^d,[0,+\infty)\right)$ and $0\leq b\in L^1([0,T],\mathbb{R})$. Our goal is to obtain existence of solutions for the following problem: \begin{equation}\label{eq:ProbPhiLapla} \left\{% \begin{array}{ll} \frac{d}{dt} \nabla \Phi(u'(t))= \nabla_{x}F(t,u(t)), \quad \hbox{for a.e. } t \in (0,T),\\ u(0)-u(T)=u'(0)-u'(T)=0, \end{array}% \right. \tag{${P_{\Phi}}$} \end{equation} minimizing the functional \begin{equation*} I(u):=\int_{0}^T \Phi(u'(t))+ F(t,u(t))\ dt, \end{equation*} on the set $H:=\{u\in \wphi | u(0)=u(T)\}$, where $\wphi$ is the anisotropic Orlicz-Sobolev space associated to $\Phi$. We say that $\Phi \in \Delta_2$, if there exists a constant $C>0$ such that \begin{equation*} \Phi(2x)\leq C \Phi(x)+1,\quad x\in\mathbb{R}^d. \end{equation*} We write $\Phi_0 \prec \Phi$ if for every $k>0$ there exists $C=C(k)>0$ such that \begin{equation*}\label{eq:orden} \Phi_1(x)\leq C+\Phi_2(kx),\quad x\in\mathbb{R}^d.\end{equation*} The following is our main theorem which contains, as a particular case, known results of existence of periodic solutions of $p$-laplacian and $(p,q)$-laplacian systems. \begin{thm}\label{coercitividad-r} tLet $\Phi^{\star} \in \Delta_2$. t\begin{enumerate} t\item If there exist $\Phi_0$ with $\Phi_0 \prec \Phi$ tand a function $d \in L^1([0,T],\mathbb{R})$, with $d\geq 1$, such that \begin{equation*}\label{eq:propiedad-coercividad-phi0} \Phi^{\star}(d^{-1}(t)\nabla_x F)\leq \Phi_0(x)+1,\;\;\mbox{and } \lim_{|x|\to\infty}\frac{\int_{0}^{T}F(t,x)\ dt}{\Phi_0(2x)}=+\infty, \end{equation*} then $I$ attains a minimum on $H$. \item If $u$ is a minimum and $d(u',L^{\infty}([0,T],\mathbb{R}^d))<1$, then $u$ is solution of $(P_\Phi)$. \end{enumerate} \end{thm} Additionally, we discuss conditions under which $d(u',L^{\infty}([0,T],\mathbb{R}^d))<1$ is satisfied.

Autores: ACINAS, Sonia Ester / Mazzone, Fernando .