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In the spectral theory it does make a difference, whether we consider differential operators on bounded or unbounded domains. In order to treat eigenvalue problems on the whole Euclidean space, we construct Sobolev spaces over ℝn, which are weighted by the Gaussian normal distribution. By the methods presented in Chapters 2, 8, and 10 of the treatise F. Sauvigny: Partial Differential Equations 1 and 2, Springer Universitext (2012), we can prove an analogue of the Sobolev embedding theoremand a Rellich selection theoremfor the Sobolev spaces W01,p(ℝn,γ)weighted by γ- with vanishing values towards infinity. We achieve these specific results for our entire Sobolev spacesW1,p(ℝn,γ), since we concentrate on the Gaussian normal distribution γas our weight function. Even our notion of the weighted partial derivativedepends on this weight function. Within the so-called Gauß–Rellich spaceW01,2(ℝn,γ)we shall investigate the discrete spectrum of weighted elliptic operators over ℝnby spectral methods. There we rely on the treatise F. Sauvigny: Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren, Springer Spektrum (2019). By reflection methods, we solve eigenvalue problems for elliptic differential operators on the sectorial domain ℝ+nunder vanishing and mixed boundary conditions.

AbstractWe derive an energy bound, estimating a weighted Dirichlet integral of two solutions for the nonparametric equation with prescribed mean curvature in ndimensions in terms of the L1-norm for the difference of their values on the boundary. Furthermore, a similar estimate is established for solutions of the equation divFp(·,∇u)=nH(·,u), where F(x,p)denotes an elliptic Lagrangian with linear growth in p. These results are used to remove singularities of solutions to these equations.

Assuming stability and integral conditions we show that a sequence of immersed surfaces of constant mean curvatureH converges to an immersedH-surface. The latter theorem depends on an oscillation estimate forH-surfaces based on an isoperimetric inequality. These compactness results are utilized to prove that certain Jordan curvesG only bound finitely many stable and unstable, immersed, smallH-surfaces.

Abstract: The aim of this paper is to give a new proof of the gradient estimate for graphs of prescribed mean curvatureH=H(x,y,z). Similarly as in [2] where the caseH=H(x,y) is studied, we introduce conformal parameters for the surface. Then we employ the differential equation for the unit normal of the surface derived in [3] Satz 1. By this method, which is contained in [4] Satz 4, we prove the following

Abstract: We give an adequate parametric description of surfaces of minimal surface type, satisfying the weighted relation ϱ1κ1+ϱ2κ2 with the positive factors ϱ j for their principal curvatures κ j , by the introduction of weighted conformal parameters. We then establish apriori estimates of the principal curvatures for certain classes of surfaces. These estimates imply new theorems of Bernstein type.

The minimal surfaces spanning a polygon in Rp (p?2) correspond to the critical points of an analytic function T in finitely many variables, namely Shiffman's function. We shall prove that the Morse index of the minimal surface coincides with the Morse index of T at the corresponding critical point. Alternatively expressed, the Schwarz operator of the minimal surface and the Hessian of T have the same number of negative eigenvalues. Finally we control the degeneration of the critical points.

Abstract. In this paper we continue our investigations of the boundary behaviour of minimal surfaces X which are stationary in a configuration $ \langle {\mit\Gamma} ,{\cal S} \rangle $. Here $ \cal S $ is the boundary of a wedge $ {\cal W}_{\alpha } $ with the edge $ \cal G $ and the opening angle $ \pi + \alpha \pi $, and $ {\mit\Gamma} $ is a Jordan arc contained in $ \overline {\cal W}_{\alpha } $ whose endpoints P₁ and P₂ lie on distinct faces of $ \cal S $. First we construct a minimal surface of this type whose free boundary on $ \cal S $ attaches to the edge $ \cal G $ in a full interval. Secondly we derive a condition which forces minimal surfaces stationary in $ \langle {\mit\Gamma} , {\cal S} \rangle $ to exhibit this edge-creeping behaviour. Apparently this phenomenon was discovered by Y. W. Chen when treating an exterior free boundary value problem for minimal surfaces. Later S. Hildebrandt and J. C. C. Nitsche proved existence and optimal regularity of solutions to a partially free boundary value problem where edge creeping occurs. They derived a uniqueness theorem proving that, under suitable assumptions on $ {\mit\Gamma} $ and for $ \alpha = 1 $, the solutions can be viewed as graphs over two-dimensional slit domains.

Abstract.: Continuing our papers [4]–[6] we study the global behaviour of minimal surfaces in a wedge which meet the faces of the wedge perpendicularly and are transversal to the edge. We derive a Hlder estimate for the Gauss maps of such surfaces which immediately implies a Bernstein–type result: Each complete minimal sector in the wedge of the type described before is necessarily a plane sector. The method uses in an essential way ideas developed in [2], [3], [5], and [9].

In this paper we investigate the boundary behaviour of minimal surfaces in a wedge which are either of class ℐ (Γ, ℱ) or of class ℰ(Γ, ℱ) Here Γ denotes a Jordan curve in ℝ3whose endpoints lie on a support surface ℱ which is assumed to be the boundary of a wedge. Then, roughly speaking, ℐ (Γ, ℱ) denotes the set of disk-type minimal surfaces which are stationary in 〈 Γ, ℐ 〉 and whose free traces on ℱ intersect the edge ℒ of the wedge “trans versally”, while ℰ(Γ, ℱ) denotes the class of corresponding surfaces whose free traces attach to ℒ in full intervals. We derive asymptotic expansions of Xwand Nwwhere X(w), w≠ B, is a minimal surface of class ℐ(Γ, ℱ) or ℰ(Γ, ℱ), and N(w)denotes the Gauss map corresponding to X.