Bernd Kawohl

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

Remarks on a Finsler-Laplacian

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

ON SOME NONLOCAL VARIATIONAL PROBLEMS

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.

The Neumann eigenvalue problem for the

We consider a number of problems that are associated with the 1-Laplace operator Div (Du/|Du|), the formal limit of the p-Laplace operator for p → 1, by investigating the underlying variational problem. Since corresponding solutions typically belong to BV and not to , we have to study minimizers of functionals containing the total variation. In particular we look for constrained minimizers subject to a prescribed -norm which can be considered as an eigenvalue problem for the 1-Laplace operator. These variational problems are neither smooth nor convex. We discuss the meaning of Dirichlet boundary conditions and prove existence of minimizers. The lack of smoothness, both of the functional to be minimized and the side constraint, requires special care in the derivation of the associated Euler–Lagrange equation as necessary condition for minimizers. Here the degenerate expression Du/|Du| has to be replaced by a suitable vector field to give meaning to the highly singular 1-Laplace operator. For minimizers of a large class of problems containing the eigenvalue problem, we obtain the surprising and remarkable fact that in general infinitely many Euler–Lagrange equations have to be satisfied.

\infty

We investigate elementary properties of a Finsler-Laplacian operator Q that is associated to functionals containing (H(∇u)). Here H is convex and homogeneous of degree 1, and its polar H represents a Finsler metric on R. In particular we study the Dirichlet problem −Qu = 2n on a ball K = {x ∈ R : H(x) < 1}, a fundamental solution for Q, suitable maximum and comparison principles, and a mean value property for solutions of Qu = 0. 1 Preliminaries Throughout this paper let H : IR 7→ IR be a nonnegative convex function of class C2(IR\{0}) which is even and positively homogeneous of degree 1, so that H(tξ) = |t|H(ξ) for any t ∈ IR, ξ ∈ IR. (1.1) A typical example is H(ξ) = ( ∑ i |ξ|) for q ∈ [1,∞). We shall investigate Euler equations which involve functionals containing the expression ∫ Ω (H(∇u)) dx . (1.2) The differential equations contain operators of the form Qu := n ∑ i=1 ∂ ∂xi ((H(∇u))Hξi(∇u)) . (1.3) Note that these operators are not linear unless H is the Euclidean norm. In particular for H(ξ) = ( ∑ k |ξk|) the operator Q becomes Qu := n ∑

-Laplacian

We consider the

On the convexity of some free boundaries

p

Viscosity Solutions for Degenerate and Nonmonotone Elliptic Equations

--Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite