New regularity and uniqueness results in the multidimensional Calculus of Variations

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New regularity and uniqueness results in the

multidimensional Calculus of Variations

Marcel Dengler

Thesis submitted to the University of Surrey

for the degree of Doctor of Philosophy

Department of Mathematics

University of Surrey

Guildford GU2 7XH, United Kingdom

E-mail address: m.dengler@surrey.ac.uk

arXiv:2210.14356v1 [math.AP] 25 Oct 2022

Scientiﬁc abstract

In the ﬁrst part of the Thesis we develop a Regularity Theory for a polyconvex functional in com-

pressible elasticity. In particular, we consider energy minimizers/stationary points of the functional

Ipuq “ ˆ

Ω

2|∇u|2`ρpdet ∇uqdx, (0.1)

where ΩĂR2is open and bounded, uPW1,2pΩ,R2qand ρ:RÑR`

0smooth and convex with

ρpsq “ 0for all sď0and ρbecomes aﬃne when sexceeds some value s0ą0.Additionally, we may

impose boundary conditions.

The ﬁrst general result we will establish is that every stationary point needs to be locally H¨

older-

continuous. Secondly, we prove that if the growth of ρis ‘small’ s.t. the integrand is still uniformly

convex, then all stationary points have to be in W2,2

loc .Next, a higher-order regularity result is shown.

We show that all stationary points that are additionally of class W2,2

loc and whose Jacobian is H¨

older-

continous are of class C8

loc.In particular, these results show that all stationary points have to be

smooth for ρ1‘small’ enough.

The theory described above works for fairly general domains and boundary conditions. We specify

those by restricting to the unit ball, and we consider M-covering maps, which take the unit sphere to

itself, covering the image Mtimes in the process, on the boundary. Under these circumstance, we

construct radial symmetric M-covering stationary points to the functional, as given in (0.1), which

are at least of class C1.In certain situations, depending mainly on the behaviour of ρand the sta-

tionary point itself, we are even able to guarantee maximal smoothness.

In the second part, we will concentrate on uniqueness questions in various situations of ﬁnite elasticity.

Starting in incompressible elasticity, the central point is to show that for problems with uniformly

convex integrands with ”small” pressure, a unique global minimizer can be guaranteed. We make

use of this statement by considering various examples and applications. One such application is the

construction of a counterexample to regularity. Indeed, on the unit ball and for smooth boundary

conditions we give a non-autonomous uniformly convex functional fpx, ξqdepending smoothly on ξ

however discontinously on x, where the unique global minimizer is Lipschitz but no better. Then

we generalise the main result in various ways, for instance, we show that if the pressure is too large

to guarantee uniqueness in the full class of admissible maps, one can still guarantee uniqueness up

to the ﬁrst Fourier-modes. Lastly, we discuss analogous statements for polyconvex integrands in

compressible elasticity.

Keywords and AMS Classiﬁcation Codes: Calculus of Variations, Polyconvexity, Regularity, Unique-

ness

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NewregularityanduniquenessresultsinthemultidimensionalCalculusofVariationsMarcelDenglerThesissubmittedtotheUniversityofSurreyforthedegreeofDoctorofPhilosophyDepartmentofMathematicsUniversityofSurreyGuildfordGU27XH,UnitedKingdomCopyright©2021byMarcelDengler.Allrightsreserved.E-mailaddress:m.dengler@s...

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New regularity and uniqueness results in the multidimensional Calculus of Variations

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