×

Radially symmetric critical points of non-convex functionals. (English) Zbl 1153.49004

Summary: We investigate critical points of the functional
\[ E(u)= \int_{B_R(0)} W(\nabla u)+G(u)\,dx \]
over a ball in \(\mathbb R^n\). Here, \(W\) is radially symmetric but not convex. We embed the functional into a family of functionals
\[ E_{\varepsilon,\lambda}(u)= \int_{B_R(0)} \tfrac12 \varepsilon(\Delta u)^2+W(\lambda,\nabla u)+G(u)\,dx, \]
where \(E_{0,0}(u)= E(u)\). A global bifurcation analysis yields a branch of non-trivial critical points depending on \(\lambda\) and positive \(\varepsilon\), where we can set \(\lambda=0\). The geometric properties preserved on that branch, due to the maximum principle, prove compactness such that the singular limit as \(\varepsilon\searrow 0\) exists. Under natural conditions on \(W\) and \(G\) the critical point obtained in this way is a minimizer of the original functional. That plan can be carried out only under the restriction of radial symmetry, since the maximum principle applies only to special elliptic equations of fourth order. That restriction, however, is not essential since every minimizer of the functional is radially symmetric.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
93C70 Time-scale analysis and singular perturbations in control/observation systems
PDFBibTeX XMLCite
Full Text: DOI