TY - JOUR

T1 - Fine Level Set Structure of Flat Isometric Immersions

AU - Hornung, Peter

PY - 2011/3

Y1 - 2011/3

N2 - A result by Pogorelov asserts that C-1 isometric immersions u of a bounded domain S subset of R-2 into R-3 whose normal takes values in a set of zero area enjoy the following regularity property: the gradient f := del u is 'developable' in the sense that the nondegenerate level sets of f consist of straight line segments intersecting the boundary of S at both endpoints. Motivated by applications in nonlinear elasticity, we study the level set structure of such f when S is an arbitrary bounded Lipschitz domain. We show that f can be approximated by uniformly bounded maps with a simplified level set structure. We also show that the domain S can be decomposed (up to a controlled remainder) into finitely many subdomains, each of which admits a global line of curvature parametrization.

AB - A result by Pogorelov asserts that C-1 isometric immersions u of a bounded domain S subset of R-2 into R-3 whose normal takes values in a set of zero area enjoy the following regularity property: the gradient f := del u is 'developable' in the sense that the nondegenerate level sets of f consist of straight line segments intersecting the boundary of S at both endpoints. Motivated by applications in nonlinear elasticity, we study the level set structure of such f when S is an arbitrary bounded Lipschitz domain. We show that f can be approximated by uniformly bounded maps with a simplified level set structure. We also show that the domain S can be decomposed (up to a controlled remainder) into finitely many subdomains, each of which admits a global line of curvature parametrization.

UR - http://www.scopus.com/inward/record.url?scp=79951809317&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s00205-010-0375-x

U2 - 10.1007/s00205-010-0375-x

DO - 10.1007/s00205-010-0375-x

M3 - Article

SN - 0003-9527

VL - 199

SP - 943

EP - 1014

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 3

ER -