Minimal Surfaces II

Abstract

Library ofCongress Cataloging-in-Publication Data Minimal surfaceslUlrich Dierkes ... [etaI.] v. cm.-(Grundlehren der mathematischen Wissenschaften; 295-296) Includes bibliographical references and indexes. Contents: I. Boundary value problems-2. Boundary regularity.

Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems

“…In the mathematical context, triply periodic minimal surfaces (TPMS) have been investigated extensively [21,22]. In order to locally minimize their surface area, minimal surfaces have vanishing mean curvature H = (1/R 1 + 1/R 2 )/2 at every point on the surface, where R 1 and R 2 are the two principal radii of curvature.…”

“…Exact representations are known for four cubic TPMS [21,22,60,61]. Consider the composite mapping of the surface into the complex plane, which consists of two parts; first, each point of the surface is mapped to a point on unit sphere which is defined by the normal vector on the surface, then the unit sphere is mapped into the complex plane by stereographic projection.…”

Plateau’s problem for infinite contours

“…The Douglas-Radò existence theorem [ 2 ] guarantees the existence of disc type minimal surfaces spanning Γ R ∪ β R ; more precisely, there exist mappings…”

“…Analogous to the classical Plateau problem [2,Chpt. 4.3] it seems natural to pose the problem of the existence of minimal surfaces spanning general unbounded curves.…”

The discrete Douglas Problem: theory and numerics

“…Although in this work we study the specific problem of approximating annulus-like minimal surfaces, we will often refer to it (with some abuse of notation) as the Douglas case, or Douglas problem. Comprehensive references for the classical theory of minimal surfaces are the books by Dierkes, Hildebrandt, Küster and Wohlrab [1], [2], and by J. C. C. Nitsche [9]; more specifically, the Douglas problem is considered in the works by J. Jost [8], [7], and the references given there.…”