| Non-Lipschitz minimizers for smooth strongly convex functionals by Xiadong Yan, Research Fellow, Department of Mathematics, New York University, New York, New York Abstract We consider variational integral of the form I(u)=\int_{W}f(\nabla u) dx. Here W is asmooth bounded domain in Rn, u:W\rightarrow Rm. f is a smooth strongly convex function defined onm \times n matrix space. We shall show that for n \geq 3, m \geq 5, there exists f smooth, strongly convex with bounded second derivative, while the minimizer u forthe corresponding I(u) is not smooth (i.e. I(u) \leq I(u+f) for any compactlysupported f). In fact, when n \geq 5, minimizer u is not even bounded. Similarexamples can also be constructed for n=4, m=3. |
