### Dr Jonathan Bevan

Senior Lecturer

### Biography

### Biography

- 1995-1999: Undergraduate at St. Anne's College, Oxford
- 1999-2003: Graduate student of Prof. J. Ball, Oxford
- 2002-2005: Stipendiary Lecturer in Mathematics, Balliol College, Oxford
- 2003-2005: EPSRC Postdoctoral Research Fellow, Oxford
- 2005-2006: MULTIMAT Postdoctoral Research Fellow, Max Planck Insititute for Mathematics in the Sciences, Leipzig
- 2005-2010: RCUK Academic Fellow in Mathematics, Surrey
- 2010-2016 Lecturer in Mathematics, Surrey
- 2016- Senior Lecturer in Mathematics, Surrey

### Research interests

- Calculus of variations
- Analysis and properties of polyconvex functions
- Regularity theory in nonlinear elasticity

### My publications

### Publications

Bevan J On double-covering stationary points of a constrained Dirichlet energy, Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis 31 (2) pp. 391-411

The double-covering map udc:R2 R2 is given byudc(x)=1/?2|x|(x22-x 122x1x2) in cartesian coordinates. This paper examines the conjecture that udc is the global minimizer of the Dirichlet energy I(u)=+B|?u|2dx among all W 1,2 mappings u of the unit ball B R2 satisfying (i) u=udc on ?B, and (ii) det?u=1 almost everywhere. Let the class of such admissible maps be A. The chief innovation is to express I(u) in terms of an auxiliary functional G(u-udc), using which we show that udc is a stationary point of I in A, and that udc is a global minimizer of the Dirichlet energy among members of A whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about udc in A using ODE techniques, we also show that udc is a local minimizer among variations whose tangent È to A at udc obeys G(È o)>0, where È o is the odd part of È. In addition, a Lagrange multiplier corresponding to the constraint det?u=1 is identified by an analysis which exploits the well-known Fefferman-Stein duality. © 2013 Elsevier Masson SAS. All rights reserved.

Bevan Jonathan J., Zeppieri Caterina Ida (2016) A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation, Calculus of Variations and Partial Differential Equations 55 (42) Springer Berlin Heidelberg

In this note we formulate a sufficient condition for the quasiconvexity at $x \mapsto \l x$ of certain functionals $I(u)$ which model the stored-energy of elastic materials subject to a deformation $u$. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to M\"{u}ller and Spector, on admissible deformations. Deformations obey the condition $u(x)= \lambda x$ whenever $x$ belongs to the boundary of the domain initially occupied by the material. In terms of the parameters of the models, our analysis provides an explicit $\lambda_0>0$ such that for every $\lambda\in (0,\lambda_0]$ it holds that $I(u) \geq I(u_{\lambda})$ for all admissible $u$, where $u_{\lambda}$ is the linear map $x \mapsto \lambda x$ applied across the entire domain. This is the quasiconvexity condition referred to above.

Bevan JJ (2011) Austenite as a Local Minimizer in a Model of Material Microstructure with a Surface Energy Term, SIAM Journal on Mathematical Analysis 43 (2) pp. 1041-1073 Society for Industrial and Applied Mathematics

Bevan JJ (2014) A remark on a stability criterion for the radial cavitating map in CrossMark nonlinear elasticity, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 425 (2) pp. 954-982 ACADEMIC PRESS INC ELSEVIER SCIENCE

Bevan JJ (2014) Explicit examples of Lipschitz, one-homogeneous solutions of log -singular planar elliptic systems, Nonlinear Analysis, Theory, Methods and Applications 125 pp. 659-680

© 2015 Elsevier Ltd. All rights reserved.We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u(R,¸)=Rg(¸), where (R,¸) are plane polar coordinates and g takes values in B W(x,u(x))dx, where

^{Rm}, m2. The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals I(u)=^{DF}W(x,F) behaves like 1|x| as |x|0 and W satisfies an ellipticity condition. Such solutions cannot exist when |x|^{DF}W(x,F)0 as |x|0, so the condition is optimal. The associated analysis exploits the well-known Fefferman-Stein duality (Fefferman and Stein, 1972). We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.Bevan JJ (2010) On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions, P ROY SOC EDINB A 140 pp. 449-475 ROYAL SOC EDINBURGH

We extend a result from Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist; or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to W-1,W-2 one-homogeneous solutions, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method.

Bevan JJ, Zeppieri CI (2015) A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation, CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 55 (2) ARTN 42 SPRINGER HEIDELBERG

Bevan JJ (2017) A condition for the Holder regularity of strong local minimizers of a nonlinear elastic energy in two dimensions, Archive for Rational Mechanics and Analysis Springer Verlag

We prove the local H¨older continuity of strong local minimizers of

the stored energy functional E(u) =

Z

©

»|?u|2 + h(det?u) dx subject to a condition of ?positive twist?. The latter turns out to be equivalent

to requiring that u maps circles to suitably star-shaped sets. The convex function

h(s) grows logarithmically as s 0+, linearly as s +?, and satisfies

h(s) = +? if s d 0. These properties encode a constitutive condition which

ensures that material does not interpenetrate during a deformation and is one

of the principal obstacles to proving the regularity of local or global minimizers.

The main innovation is to prove that if a local minimizer has positive twist

a.e. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli

inequality can be derived from it. The claimed H¨older continuity then follows

by adapting some well-known elliptic regularity theory. We also demonstrate

the regularizing effect that the term

R

© h(det?u) dx can have by analysing

the regularity of local minimizers in the class of ?shear maps?. In this setting

a more easily verifiable condition than that of positive twist is imposed, with

the result that local minimizers are H¨older continuous.

the stored energy functional E(u) =

Z

©

»|?u|2 + h(det?u) dx subject to a condition of ?positive twist?. The latter turns out to be equivalent

to requiring that u maps circles to suitably star-shaped sets. The convex function

h(s) grows logarithmically as s 0+, linearly as s +?, and satisfies

h(s) = +? if s d 0. These properties encode a constitutive condition which

ensures that material does not interpenetrate during a deformation and is one

of the principal obstacles to proving the regularity of local or global minimizers.

The main innovation is to prove that if a local minimizer has positive twist

a.e. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli

inequality can be derived from it. The claimed H¨older continuity then follows

by adapting some well-known elliptic regularity theory. We also demonstrate

the regularizing effect that the term

R

© h(det?u) dx can have by analysing

the regularity of local minimizers in the class of ?shear maps?. In this setting

a more easily verifiable condition than that of positive twist is imposed, with

the result that local minimizers are H¨older continuous.

Bevan JJ (2011) Extending the Knops-Stuart-Taheri technique to C^{1} weak local minimizers in nonlinear elasticity, Proceedings of the American Mathematical Society 139 (5) pp. 1667-1679 American Mathematical Society

Bevan J, Yan X (2008) Minimizers with topological singularities in two dimensional elasticity, ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 14 (1) pp. 192-209 EDP SCIENCES S A

Bevan J (2005) Singular minimizers of strictly polyconvex functionals in R-2X2, CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 23 (3) pp. 347-372 SPRINGER

Bevan JJ (2011) A remark on the structure of the busemann representative of a polyconvex function, Journal of Convex Analysis 18 (1) pp. 203-208

Under mild conditions on a polyconvex function W : R

R, its largest convex representative, known as the Busemann representative, may be written as the supremum over all affine functions ¦ : R

R satisfying ¦(¾ det ¾) d W(¾) for all 2 × 2 matrices ¾. In this paper, we construct an example of a polyconvex W : R

R whose Busemann representative is, on an open set, strictly larger than the supremum of all affine functions ¦ as above and which also satisfy ¦(¾

, det ¾\

) = W(¾

) for at least one 2×2 matrix

. © Heldermann Verlag.

R, its largest convex representative, known as the Busemann representative, may be written as the supremum over all affine functions ¦ : R

R satisfying ¦(¾ det ¾) d W(¾) for all 2 × 2 matrices ¾. In this paper, we construct an example of a polyconvex W : R

R whose Busemann representative is, on an open set, strictly larger than the supremum of all affine functions ¦ as above and which also satisfy ¦(¾

, det ¾\

) = W(¾

) for at least one 2×2 matrix

. © Heldermann Verlag.

Bevan Jonathan (2013) On double-covering stationary points of a constrained Dirichlet energy, Annales de l'Institut Henri Poincare / Analyse non lineaire Elsevier

This paper examines the conjecture that $\udc$ is the global minimizer of the Dirichlet energy $I(\bu) = \int_{B}|\nabla \bu|^{2}\,d\bx$ among all $W^{1,2}$ mappings $\bu$ of the unit ball $B \subset \mathbb{R}^{2}$ satisfying (i) $\bu =\udc$ on $\partial B$, and (ii) $\det \nabla \bu = 1$ almost everywhere.

Bevan J (2006) On convex representatives of polyconvex functions, PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 136 pp. 23-51 ROYAL SOC EDINBURGH

Bevan J, Pedregal P (2006) A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals, PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 136 pp. 701-708 ROYAL SOC EDINBURGH

Bevan J (2003) An example of a C-1,C-1 polyconvex function with no differentiable convex representative, COMPTES RENDUS MATHEMATIQUE 336 (1) pp. 11-14 EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER

Bevan JJ (2015) A remark on a stability criterion for the radial cavitating map in nonlinear elasticity, Journal of Mathematical Analysis and Applications 425 (2) pp. 954-982

© 2015 Elsevier Inc.We study the integral functional I(w) := +B|adj?w(w|w|3)|qdx on suitable maps w:BR3 R3 and where 2q?(2, 3). The inequality I(w)eI(i), which we establish on a subclass of the admissible maps, was first proposed in [13] as one of two possible necessary conditions for the stability, i.e. local minimality, of the radial cavitating map in nonlinear elasticity. Here, i is the identity map. Admissible maps w either do not vanish (and in this case possess a single discontinuity x0 in B which produces a cavity about the origin), or vanish at exactly one point x0 in B, in which case w is a diffeomorphism in a neighbourhood of x0. We show that I({dot operator}) behaves like a polyconvex functional and associate with it another functional, K({dot operator}), satisfying I(w)eI(i)+q(K(w)-K(i)). We give conditions under which K(w)=K(i), and from these infer I(w)eI(i). It is also shown that (i) K is strictly decreasing along paths of admissible functions that move x0 away from the origin and (ii) K(w) exhibits some quite pathological behaviour when w is sufficiently close to i.

Bevan Jonathan, Kabisch Sandra (2019) Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians, Proceedings of the Royal Society of Edinburgh Section A: Mathematics Cambridge University Press

In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine in particular the relationship between the positivity of the Jacobian det?u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit \emph{explicit} twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer u Ã :©R 2 in a model, two-dimensional case. The shear map minimizer has the properties that (i) det?u Ã is strictly positive on one part of the domain © , (ii) det?u Ã =0 necessarily holds on the rest of © , and (iii) properties (i) and (ii) combine to ensure that ?u Ã is not continuous on the whole domain.

Bevan Jonathan, Deane Jonathan (2018) A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity, SIAM Journal on Mathematical Analysis 50 (3) pp. 2566-2587 Society for Industrial and Applied Mathematics

In this paper we give an explicit sufficient condition for the affine map u»(x):=»x to be the global energy minimizer of a general class of elastic stored-energy functionals I(u)=+©W(?u)dx in three space dimensions, where W is a polyconvex function of 3×3 matrices. The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible. In the language of the calculus of variations, the condition ensures the quasiconvexity of I(Å) at »1, where 1 is the 3×3 identity matrix. Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of 3×3 matrices), on the previous work Bevan & Zeppieri, 2015, and on a careful numerical treatment to make the calculation of certain constants tractable. We also derive a new condition, which seems to depend heavily on the smallest singular value »1(?u) of a competing deformation u, that is necessary for the inequality I(u)

Deane J, Bevan J (2018) A hydrostatic model of the Wirtz pump, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474 (2211) The Royal Society

The Wirtz pump is not only an excellent example of alternative technology,

using as it does the kinetic energy of a stream to raise a proportion of its water,

but its mathematical modelling also poses several intriguing problems.

We give some history of the Wirtz pump and describe its operation. Taking

a novel dynamical systems approach, we then derive a discrete mathematical

model in the form of a mapping that describes its hydrostatic behaviour. Our

model enables us to explain several aspects of the behaviour of the pump

as well as to design one that gives approximately maximal, and maximally

constant, output pressure.

using as it does the kinetic energy of a stream to raise a proportion of its water,

but its mathematical modelling also poses several intriguing problems.

We give some history of the Wirtz pump and describe its operation. Taking

a novel dynamical systems approach, we then derive a discrete mathematical

model in the form of a mapping that describes its hydrostatic behaviour. Our

model enables us to explain several aspects of the behaviour of the pump

as well as to design one that gives approximately maximal, and maximally

constant, output pressure.