Band 7, Heft 2

Abstract. We obtain everywhere 𝒞 α -regularity for vector solutions to a class of nonlinear elliptic systems whose principal part is the Euler operator to a variational integral ∫F(u,∇u)dx${\int F(u,\nabla u)\, dx}$ with quadratic growth in ∇u${\nabla u}$ and which satisfies a generalized splitting condition that cover the case F(u,∇u):=∑iQi,$ {F(u,\nabla u):= \sum _i Q_i},\vspace*{-2.27621pt} $ where Qi:=∑αβAiαβ(u,∇u)∇uα·∇uβ${Q_i := \sum _{\alpha \beta } A^{\alpha \beta }_i(u,\nabla u) \nabla u^{\alpha } \cdot \nabla u^{\beta }}$, or the case F(u,∇u):=∏i(1+Qi)θi.$ {F(u,\nabla u) := \prod _i (1+Q_i)^{\theta _i}}.\vspace*{-2.13394pt} $ A crucial assumption is the one-sided condition Fu(u,η)·u≥-K$ F_u (u,\eta ) \cdot u \ge -K\vspace*{-0.62596pt} $ and related generalizations. In the elliptic case we obtain existence of 𝒞 α -solutions. If the leading operator is not necessarily elliptic but coercive, possible minima are everywhere Hölder continuous and the same holds also for Noether solutions, i.e., extremals which are also stationary with respect to inner variations. In particular if Aαβ(u,∇u)=Aαβ(u)${A^{\alpha \beta }(u,\nabla u)=A^{\alpha \beta } (u)}$, our result generalizes a result of Giaquinta and Giusti. The technique of our proof (using weighted norms and inhomogeneous hole-filling method) does not rely on L∞${L^{\infty }}$-a priori estimates for the solution.

Abstract. In this paper we investigate the Dirichlet problem for the minimal surface equation on certain nonconvex domains of the plane. In our first result, we give, by an independent proof, a numerically explicit version of Williams' existence theorem. Our main result concerns the Dirichlet problem on exterior domains. It was shown by Krust (1989) and Kuwert (1993) that between two different solutions with the same normal at infinity there is a continuum of solutions foliating the space in between. We investigate the space of solutions further and show that, unless it is empty, it contains a maximal and a minimal solution if the boundary data is rectifiable. In the case of sufficiently smooth data we parametrize the set of solutions in terms of the extremal inclinations which the normal of the graph of a solution reaches at the boundary. We show that all theoretically possible values are realized including the horizontal position of the normal for the minimal and maximal solutions. We moreover give an example where the maximal and the minimal solution coincide so that there is exactly one with given normal at infinity. This answers a natural question which has not been touched in the previous papers.

Abstract. The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak*-compactness theorem for integral currents in dual spaces of separable Banach spaces. Our theorems generalize results of Ambrosio–Kirchheim, Lang, the author, and recent results of Ambrosio–Schmidt.

Abstract. We study an anisotropic version of the outer Minkowski content of a closed set in ℝn${\mathbb {R}}^n$. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.