Dr Jonathan Bevan

+44 (0)1483 682620
26 AA 04

Academic and research departments

School of Mathematics and Physics.



Research interests


Marcel Dengler, Jonathan J. Bevan (2023)A uniqueness criterion and a counterexample to regularity in an incompressible variational problem, In: Nonlinear Differential Equations and Applications Springer

In this paper we consider the problem of minimizing functionals of the form E(u) = B f (x, ∇u) dx in a suitably prepared class of incompressible, planar maps u : B → R 2. Here, B is the unit disk and f (x, ξ) is quadratic and convex in ξ. It is shown that if u is a stationary point of E in a sense that is made clear in the paper, then u is a unique global minimizer of E(u) provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional f (x, ξ), depending smoothly on ξ but discontinuously on x, whose unique global minimizer is the so-called N −covering map, which is Lipschitz but not C 1 .

J. J. Bevan, J. H. B. Deane (2023)A Study of Certain Sharp Poincare Constants as Set Functions of Their Domain, In: Applied mathematics & optimization88(2)55 Springer Nature

For bounded, convex sets Omega subset of R-d, the sharp Poincare constant C(Omega), which appears in || f - (f) over bar (Omega)||(L infinity(Omega))

Jonathan Deane, Jonathan Bevan (2018)A hydrostatic model of the Wirtz pump, In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences474(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.

JJ Bevan (2010)On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions, In: P ROY SOC EDINB A140pp. 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.

Jonathan J. Bevan, Caterina Ida Zeppieri (2016)A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation, In: Calculus of Variations and Partial Differential Equations55(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.

Jonathan Bevan (2013)On double-covering stationary points of a constrained Dirichlet energy, In: G Francfort (eds.), 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.

Jonathan Bevan, Jonathan Deane (2018)A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity, In: SIAM Journal on Mathematical Analysis50(3)pp. 2566-2587 Society for Industrial and Applied Mathematics

In this paper we give an explicit sufficient condition for the affine map uλ(x):=λx to be the global energy minimizer of a general class of elastic stored-energy functionals I(u)=∫ΩW(∇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(⋅) at λ1, 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 λ1(∇u) of a competing deformation u, that is necessary for the inequality I(u)

J Bevan, X Yan (2008)Minimizers with topological singularities in two dimensional elasticity, In: ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS14(1)pp. 192-209 EDP SCIENCES S A
JJ Bevan (2015)A remark on a stability criterion for the radial cavitating map in nonlinear elasticity, In: Journal of Mathematical Analysis and Applications425(2)pp. 954-982 Elsevier

We study the integral functional I(w) := ∫|adj∇w(w|w|)|qdx on suitable maps w:B⊂R → R and where 2q∈(2, 3). The inequality I(w)≥I(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 x in B which produces a cavity about the origin), or vanish at exactly one point x in B, in which case w is a diffeomorphism in a neighbourhood of x. We show that I({dot operator}) behaves like a polyconvex functional and associate with it another functional, K({dot operator}), satisfying I(w)≥I(i)+q(K(w)-K(i)). We give conditions under which K(w)=K(i), and from these infer I(w)≥I(i). It is also shown that (i) K is strictly decreasing along paths of admissible functions that move x away from the origin and (ii) K(w) exhibits some quite pathological behaviour when w is sufficiently close to i.

JJ Bevan (2011)Austenite as a Local Minimizer in a Model of Material Microstructure with a Surface Energy Term, In: SIAM Journal on Mathematical Analysis43(2)pp. 1041-1073 Society for Industrial and Applied Mathematics
JJ Bevan (2017)A condition for the Holder regularity of strong local minimizers of a nonlinear elastic energy in two dimensions, In: Archive for Rational Mechanics and Analysis Springer Verlag

We prove the local Holder continuity of strong local minimizers of the stored energy functional ____[E(u)=____int_____Omega ____lambda|____nabla u|^{2}+h(____det ____nabla 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 $s____to 0+$, linearly as $s ____to +____infty$, and satisfies $h(s)=+____infty$ if $s ____leq 0$. 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 strong local minimizer has positive twist a.e. on a ball then a variational inequality holds and a Caccioppoli inequality can be derived from it. The claimed Holder continuity then follows by adapting some well-known elliptic regularity theory.

J Bevan (2006)On convex representatives of polyconvex functions, In: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS136pp. 23-51 ROYAL SOC EDINBURGH
J Bevan, P Pedregal (2006)A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals, In: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS136pp. 701-708 ROYAL SOC EDINBURGH
J Bevan (2005)Singular minimizers of strictly polyconvex functionals in R-2X2, In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS23(3)pp. 347-372 SPRINGER
J Bevan (2003)An example of a C-1,C-1 polyconvex function with no differentiable convex representative, In: COMPTES RENDUS MATHEMATIQUE336(1)pp. 11-14 EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER
Jonathan J. Bevan, Jonathan H. B. Deane (2019)A rigorous mathematical treatment of the shape of a dissipative rope fountain, In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences475(2231)pp. 1-15 Royal Society

This is a theoretical and numerical study of a model of a rope fountain subject to a drag force that depends linearly on the rope velocity. A precise, analytical description of the long-term shape adopted by the rope is given, and various consequences are derived from it. Using parameters that naturally appear in the model, we distinguish between cases wherein energy is conserved by means of a constant tension far from the rope source (the ‘free’ case) and where energy conservation is a consequence of a non-constant tension (the ‘braked’ case). The model is used, among other things, to generate rope fountain shapes based on approximate experimental estimates of the parameter values and a careful numerical treatment.

JJ Bevan (2011)Extending the Knops-Stuart-Taheri technique to C^{1} weak local minimizers in nonlinear elasticity, In: Proceedings of the American Mathematical Society139(5)pp. 1667-1679 American Mathematical Society
Jonathan Bevan, Sandra Kabisch (2019)Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians, In: 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 σ :Ω→R 2 in a model, two-dimensional case. The shear map minimizer has the properties that (i) det∇u σ is strictly positive on one part of the domain Ω , (ii) det∇u σ =0 necessarily holds on the rest of Ω , and (iii) properties (i) and (ii) combine to ensure that ∇u σ is not continuous on the whole domain.


For each natural number n and any bounded, convex domain Ω ⊂ R n we characterize the sharp constant C(n, Ω) in the Poincaré inequality ||f − ¯ f Ω || L ∞ (Ω;R) ≤ C(n, Ω)|||f || L ∞ (Ω;R). Here, ¯ f Ω denotes the mean value of f over Ω. In the case that Ω is a ball Br of radius r in R n , we calculate C(n, Br) = C(n)r explicitly in terms of n and a ratio of the volumes of the unit balls in R 2n−1 and R n. More generally, we prove that C(n, B r(Ω)) ≤ C(n, Ω) ≤ n n+1 diam(Ω), where B r(Ω) is a ball in R n with the same n−dimensional Lebesgue measure as Ω. Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.

J. J. Bevan, J. H. B. Deane (2023)A study of certain sharp Poincaré constants as set functions of their domain, In: Applied Mathematics & Optimization Springer

For bounded, convex sets Ω ⊂ R d , the sharp Poincaré constant C(Ω), which appears in ||f − f_Ω || L^∞(Ω) ≤ C(Ω)|||f || L^∞(Ω) , is given by C(Ω) = max ∂Ω ζ for a specific convex function ζ ([3, Theorem 1.1]). We study C(·) as a function on convex sets, in particular on polyhedra, and find that while a geometric characterization of C(Ω) for triangles is possible, for other polyhedra the problem of ordering ζ(V_i), where V_i are the vertices of Ω, can be formidable. In these cases, we develop estimates of C(Ω) from above and below in terms of more tractable quantities. We find, for example, that a good proxy for C(Q) when Q is a planar polygon with vertices V_i and centroid γ(Q) is the quantity D(Q) = maxi |V_i − γ(Q)|, with an error of up to ∼ 8%. A numerical study suggests that a similar statement holds for k−gons, this time with a maximal error across all k−gons of ∼ 13%. We explore the question of whether there is, for each Ω, at least one point M capable of ordering the ζ(V_i) according to the ordering of the |V_i − M |. For triangles, M always exists; for quadrilaterals, M seems always to exist; for 5−gons and beyond, they seem not to.

JONATHAN JAMES BEVAN, JONATHAN HUGH BERESFORD DEANE (2020)A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs, In: Nonlinear differential equations and applications : NoDEA286

We exhibit a family of convex functionals with infinitely many, equal-energy C 1 stationary points that (i) occur in pairs v± satisfying det v± = 1 on the unit ball B in R 2 and (ii) obey the boundary condition v± = id on ∂B. When the parameter upon which the family of functionals depends exceeds √ 2, the stationary points appear to 'buckle' near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps v±(x) and prove that they are proportional to (− 1//) ln |x| as x → 0 in B. The lowest-energy pairs v± are energy minimizers within the class of twist maps (see Taheri [30] or Sivaloganathan and Spector [22]), which, for each 0 ≤ r ≤ 1, take the circle {x ∈ B : |x| = r} to itself; a fortiori, all v± are stationary in the class of W 1,2 (B; R 2) maps w obeying w = id on ∂B and det w = 1 in B.

JONATHAN JAMES BEVAN, Jonathan H. B. Deane (2020)Energy minimizing N-covering maps in two dimensions, In: Calculus of variations and partial differential equations604 (2021) Springer Nature B.V

We show that the N-covering map, which in complex coordinates is given by uN(z):=z↦zN/N|z|N-1 and where N is a natural number, is a global minimizer of the Dirichlet energy D(v)=∫B|∇v(x)|2dx with respect to so-called inner and outer variations. An inner variation of uN is a map of the form uN∘φ, where φ belongs to the class A(B):={φ∈H1(B;R2):det∇φ=1a.e.,φ|∂B(x)=x} and B denotes the unit ball in R2, while an outer variation of uN is a map of the form ϕ∘uN, where ϕ belongs to the class A(B(0,1/N)). The novelty of our approach to inner variations is to write the Dirichlet energy of uN∘φ in terms of the functional I(ψ;N):=∫BN|ψR|2+1N|ψτ|2dy, where ψ is a suitably defined inverse of φ, and ψR and ψτ are, respectively, the radial and angular weak derivatives of ψ, and then to minimise I(ψ;N) by considering a series of auxiliary variational problems of isoperimetric type. This approach extends to include p-growth functionals (p>1) provided the class A(B) is suitably adapted. When 1

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u(R,θ)=Rg(θ), where (R,θ) are plane polar coordinates and g:R2→Rm, m≥2. The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals I(u)=∫BW(x,∇u(x))dx, where DFW(x,F) behaves like 1|x| as |x|→0 and W satisfies an ellipticity condition. Such solutions cannot exist when |x|DFW(x,F)→0 as |x|→0, so the condition is optimal. The associated analysis exploits the well-known Fefferman-Stein duality. We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.

JJ Bevan (2011)A remark on the structure of the busemann representative of a polyconvex function, In: Journal of Convex Analysis18(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 Φ : R →R satisfying Φ(ξ det ξ) ≤ W(ξ) for all 2 × 2 matrices ξ. 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 Φ as above and which also satisfy Φ(ξ , det ξ____ ) = W(ξ ) for at least one 2×2 matrix Ξ . © Heldermann Verlag.