Friedemann Brock

A symmetry problem in the calculus of variations

AbstractLet Ω be a ball in ℝN, centered at zero, and letu be a minimizer of the nonconvex functional

A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative

R(v) = \int_\Omega {\tfrac{1}{{1 + |\nabla v(x)|^2 }}dx}

Rearrangements and Applications to Symmetry Problems in PDE

over one of the classesCM := {w ∈Wloc1,∞(∖) ∣ 0 ≤w(x) ≤M inΩ,w concave} orEM := {w ∈Wloc1,2 (Ω) ∣ 0 ≤w(x) ∖M in≤,Δw∖ 0 inL′(∖)}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newtons rotational “body of minimal resistance“ through symmetry breaking.

On isoperimetric inequalities with respect to infinite measures

This work presents new results and applications for the continuous Steiner symmetrization. There are proved some functional inequalities, e.g. for Dirichlet-type integrals and convolutions and also continuity properties in Sobolev spacesW1,p. Further it is shown that the local minimizers of some variational problems and the nonnegative solutions of some semilinear elliptic problems in symmetric domains satisfy a weak, ‘local’ kind of symmetry.

A relative isoperimetric inequality

We develop a new method to prove symmetry results for overdetermined boundary value problems. This method is based on the use of continuous Steiner symmetrization together with derivative with respect to the domain. It allows to consider nonlinear equations in divergence form with dependence inr=|x| in the nonlinearity. By using the notion of “local symmetry” introduced by the first author, we prove that the domain is necessarily a ball. We also give an example where the solution of the overdetermined problem is not radially symmetric.

Symmetry and asymmetry of minimizers of a class of noncoercive functionals

We prove some new symmetry results for positive solutions of elliptic problems in ℝn and on the sphere. The proofs are based on the moving plane method, rearrangement arguments and stereographic projection.

An approach to symmetrization via polarization

Abstract This survey is meant as a general introduction to the theory of rearrangements on ℝ N . We give proofs of all the basic integral inequalities for symmetrization, and a number of applications to symmetry problems in PDE. A particular role plays the method of two-point rearrangement.

An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem

We study isoperimetric problems with respect to infinite measures on