A Resolution of the Poisson Problem for Elastic Plates

Lio, Francesca Da; Palmurella, Francesco; Rivière, Tristan

doi:10.1007/s00205-020-01499-2

Cited by 12 publications

(9 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Prescribing additional conditions and showing existence has also been very successful, i.e. the isoperimetric ratio in [23] and [36], the area in [25], boundary conditions in [6,32,35] and [9]. Finally, the Willmore conjecture was shown to be true in [28].…”

Section: Introductionmentioning

confidence: 99%

Lower semicontinuity for the Helfrich problem

Eichmann

2020

Ann Glob Anal Geom

View full text Add to dashboard Cite

We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of $${{\mathbb {R}}}^3$$ R 3 .

show abstract

Section: Introductionmentioning

confidence: 99%

Lower semicontinuity for the Helfrich problem

Eichmann

2020

Ann Glob Anal Geom

View full text Add to dashboard Cite

show abstract

“…For the Helfrich functional an existence result for branched immersions in R 3 was found in [19]. This is somehow related to [11], where an area constraint was imposed in order to minimise the Willmore functional.…”

Section: Introductionmentioning

confidence: 90%

“…There are several applications of the Helfrich energy in e. g. biology in modelling red blood cells or lipid bilayers (see e. g. [8,27,34,36]) and in modelling thin elastic plates (see e. g. [24]). As noted in [11,34], the Willmore functional was already considered in the early 19th century (see [24,37]) and has been periodically reintroduced according to the availability of more powerful mathematical tools e.g. in the early 1920s (see [45]) and most successfully by Willmore (see [46]) in the 1960s.…”

Section: Introductionmentioning

confidence: 99%

Boundary value problems for a special Helfrich functional for surfaces of revolution: existence and asymptotic behaviour

2021

View full text Add to dashboard Cite

The central object of this article is (a special version of) the Helfrich functional which is the sum of the Willmore functional and the area functional times a weight factor $$\varepsilon \ge 0$$ ε ≥ 0 . We collect several results concerning the existence of solutions to a Dirichlet boundary value problem for Helfrich surfaces of revolution and cover some specific regimes of boundary conditions and weight factors $$\varepsilon \ge 0$$ ε ≥ 0 . These results are obtained with the help of different techniques like an energy method, gluing techniques and the use of the implicit function theorem close to Helfrich cylinders. In particular, concerning the regime of boundary values, where a catenoid exists as a global minimiser of the area functional, existence of minimisers of the Helfrich functional is established for all weight factors $$\varepsilon \ge 0$$ ε ≥ 0 . For the singular limit of weight factors $$ \varepsilon \nearrow \infty $$ ε ↗ ∞ they converge uniformly to the catenoid which minimises the surface area in the class of surfaces of revolution.

show abstract

“…In the case where the curve and the tangent plane are prescribed along the boundary, there are substantial existence and regularity results for minimizers by Schätzle [15] and Da Lio, Palmurella and Rivière [4]. It is clearly of interest to develop an analogous theory for the free boundary problems in the case of curved supporting surfaces or curves.…”

Section: Introductionmentioning

confidence: 99%