Uniqueness theorems for variational problems by the method of transformation groups (Q1889977)

The book is devoted to the question of uniqueness of critical points of integral functionals of the type \(I(u)= \int_\Omega L(x, u,Vu)\,dx\), where \(\Omega\subset\mathbb{R}^n\) is a bounded domain and \(u\) belongs to an appropriate Sobolev space \(V\) (mostly \(W^1_\infty(\Omega)\)). As a sufficient condition for uniqueness is considered the existence of a one-parameter group \(G= \{g_\varepsilon\}\) of transformations of \(V\). As a classical example we point out to the case of a star-shaped domain with \(0\in\Omega\) and the integrand \[ L(x,u,\nabla u)= {1\over 2}\nabla u^2-{1\over p+1}|u|^{p+1},\tag{1} \] where the corresponding transformations are \[ q_\varepsilon u\to\widetilde u,\quad\widetilde u(\widetilde x)= e^{{n-2\over 2}\varepsilon} u(e^\varepsilon x)\text{ for }\widetilde x\in e^{-\varepsilon}\Omega,\;\varepsilon> 0. \] The book gives a systematic approach to the questions of the existence and properties of such groups on Riemannian manifolds and applications to the integrands of the type (1), including the case of the \(p\)-Laplacian. The last chapter of the book discusses some vectorial problems in Euclidean spaces, e.g., the Emden-Fowler system and some cases of nonlinear elasticity.