Nonvariational systems

A typical nonvariational elliptic system has the form

\begin{displaymath} (NV) \;\; \left \{ \begin{array}{ll} -\Delta u= f(x; u,v), &... ... v}{\partial n}=0 &\, x\in \partial \Omega \end{array} \right. \end{displaymath}

where $\Omega\subset {\mathbb{R}}^N (N\ge 1)$ is an open bounded domain, $f(x; u,v), g(x; u,v) \in \mathcal{C}^{1}(\overline{\Omega} \times \mathbb{R}^2; \mathbb{R})$ in the variables $(u,v) \in \mathbb{R}^2$. Here, we further assume that there exists no function $G(x;u,v)$ with its gradient $\nabla G =(f, \pm g)$ or $\nabla G =(g, f)$. Under this assumption, it is easy to see that problem (NV) is nonvariational.





Created by Xianjin Chen

Oct-17-2006