Dr Jonathan Bevan

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.

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

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.

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))

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)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 .

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.

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.