Geometrische Analysis, insbesondere im Hinblick auf ihre Anwendungen in Biologie und Kontinuumsmechanik

• Applicant: Professor Dr. Peter Hornung
• Subject Area: Mathematics
• Term: from 2011 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 200238036

• Final Report Year: 2020

Final Report Abstract

We have addressed applied analysis problems, mainly arising in nonlinear elasticity, in which geometric effects play an important role. This includes the interplay between homogenisation and thin film asymptotics; the analysis of narrow and thin elastic ribbons; as well as a first step towards shape optimisation for nonlinearly elastic plates. More general results include a suitable notion of stationarity for intrinsically strained plate theories in non Euclidean elasticity, along with some surprising examples of stationary points. Moreover, we have sharpened some earlier regularity results about stationary points of Kirchhoff’s plate theory, and we have constructed examples proving the optimality of those results. We have also addressed questions of rigidity and its failure for isometric immersions and the related Monge-Ampère equation. Finally, we have continued our study of intrinsically biharmonic maps between manifolds.

Publications

• Continuation of infinitesimal bendings on developable surfaces and equilibrium equations for nonlinear bending theory of plates. Comm. Partial Differential Equations, 38(8):1368–1408, 2013
  Peter Hornung
  
• Derivation of a homogenized nonlinear plate theory from 3d elasticity. Calc. Var. Partial Differential Equations, 51(3-4):677–699, 2014
  Peter Hornung, Stefan Neukamm, and Igor Velčić
  
• Global structure of the singular set of energy minimising bendings. Nonlinearity, 28(11):3821–3844, 2015
  Anna Dall’Acqua and Peter Hornung
  
• A variational model for anisotropic and naturally twisted ribbons. SIAM J. Math. Anal., 48(6):3883–3906, 2016
  Lorenzo Freddi, Peter Hornung, Maria Giovanna Mora, and Roberto Paroni
  
• Existence of equivariant biharmonic maps. Int. Math. Res. Not. IMRN, (8):2397–2422, 2016
  Peter Hornung and Roger Moser
  
• Stationary points of nonlinear plate theories. J. Funct. Anal., 273(3):946–983, 2017
  Peter Hornung
  
• Regularity of intrinsically convex W 2,2 surfaces and a derivation of a homogenized bending theory of convex shells. J. Math. Pures Appl. (9), 115:1–23, 2018
  Peter Hornung and Igor Velčić
  
• Stochastic homogenization of the bending plate model. J. Math. Anal. Appl., 458(2):1236–1273, 2018
  Peter Hornung, Matthäus Pawelczyk, and Igor Velčić
  
• Convergence of equilibria for bending-torsion models of rods with inhomogeneities. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, pages 1–28, 2019
  Matthäus Pawelczyk
  
• Material optimization for nonlinearly elastic planar beams. ESAIM Control Optim. Calc. Var., 25:Art. 11, 19, 2019
  Peter Hornung, Martin Rumpf, and Stefan Simon