Nonvariational systems

A typical nonvariational elliptic system has the form

(NV)\;\;\left\{\begin{array}[]{ll}-\Delta u=f(x;u,v),&\,x\in\Omega\\ -\Delta v=g(x;u,v),&\,x\in\Omega\\ u=v=0\;\mbox{or}\;\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=% 0&\,x\in\partial\Omega\end{array}\right.

where $\Omega\subset{\mathbb{R}}^{N}(N\geq 1)$ 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 $\nabla G=(f,\pm g)$ or $\nabla G=(g,f)$ . Under this assumption, it is easy to see that problem (NV) is nonvariational.

Title	Nonvariational systems
Canonical name	NonvariationalSystems1
Date of creation	2013-03-11 19:28:55
Last modified on	2013-03-11 19:28:55
Owner	linor (11198)
Last modified by	(0)
Numerical id	1
Author	linor (0)
Entry type	Definition