A general variational identity. (English) Zbl 0625.35027

Pucci, Patrizia; Serrin, James

Indiana Univ. Math. J. 35, 681-703 (1986).

The authors consider extremals of the general variational problem, \(\delta \int_{\Omega}F(x,u,Du)du=0\) which are \(C^ 2\) and satisfy the Euler-Lagrange equation, \(div(F_ p(x,u,Du))=F_ u(x,u,Du),\quad x\in \Omega.\) Under certain assumptions on F, the Euler-Lagrange equation has no nontrivial solution \(C^ 2(\Omega)\cap C^ 1(\Omega)\) which vanishes on \(\partial \Omega\). Similar results are established for higher-order extremals and for vector-valued extremals, which gives some new results for certain elliptic systems.

Reviewer: S.M.Lenhart

MSC:

35J50	Variational methods for elliptic systems
49J20	Existence theories for optimal control problems involving partial differential equations
35A15	Variational methods applied to PDEs
35J60	Nonlinear elliptic equations

Keywords:

extremals; Euler-Lagrange equation; higher-order extremals; vector-valued extremals