Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions.Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u=(u1,…,uN) : EL[u,X]=⎧⎩⎨⎪⎪Δu=div(P(x)cof∇u)det∇u=1u≡φinX,inX,on∂X, where X is a finite, open, symmetric N -annulus (with N≥2 ), P=P(x) is an unknown hydrostatic pressure field and φ is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N=3 , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N=2 , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions N≥4 and discuss a number of closely related issues

Solution of the Kirchhoff-Plateau problem

The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments

Recent developments in empirical IO: dynamic demand and dynamic games

Empirically studying dynamic competition in oligopoly markets requires dealing with large states spaces and tackling difficult computational problems, while handling heterogeneity and multiple equilibria. In this paper, we discuss some of the ways recent work in Industrial Organization has dealt with these challenges. We illustrate problems and proposed solutions using as examples recent work on dynamic demand for differentiated products and on dynamic games of oligopoly competition. Our discussion of dynamic demand focuses on models for storable and durable goods and surveys how researchers have used the "inclusive value" to deal with dimensionality problems and reduce the computational burden. We clarify the assumptions needed for this approach to work, the implications for the treatment of heterogeneity and the different ways it has been used. In our discussion of the econometrics of dynamics games of oligopoly competition, we deal with challenges related to estimation and counterfactual experiments in models with multiple equilibria. We also examine methods for the estimation of models with persistent unobserved heterogeneity in product characteristics, firms’ costs, or local market profitability. Finally, we discuss different approaches to deal with large state spaces in dynamic games.Industrial Organization; Oligopoly competition; Dynamic demand; Dynamic games; Estimation; Counterfactual experiments; Multiple equilibria; Inclusive values; Unobserved heterogeneity.

Recent Developments in Empirical IO: Dynamic Demand and Dynamic Games

Empirically studying dynamic competition in oligopoly markets requires dealing with large states spaces and tackling difficult computational problems, while handling heterogeneity and multiple equilibria. In this paper, we discuss some of the ways recent work in Industrial Organization has dealt with these challenges. We illustrate problems and proposed solutions using as examples recent work on dynamic demand for differentiated products and on dynamic games of oligopoly competition. Our discussion of dynamic demand focuses on models for storable and durable goods and surveys how researchers have used the \Industrial Organization; Oligopoly competition; Dynamic demand; Dynamic games; Estimation; Counterfactual experiments; Multiple equilibria; Inclusive values; Unobserved heterogeneity.

On the uniqueness and monotonicity of energy minimisers in the homotopy classes of incompressible mappings and related problems

The goal of this paper is to prove the existence and uniqueness of the so-called energy minimisers in homotopy classes for the variational energy integral F[u; X] = Z X F(|x| 2 , |u| 2 )|∇u| 2 /2 dx, with F ≥ c > 0 of class C 2 and satisfying suitable conditions and u lying in the Sobolev space of weakly differentiable incompressible mappings of a finite open symmetric plane annulus X onto itself, specifically, lying in A(X) = {u ∈ W 1,2 (X, R 2 ) : det ∇u = 1 a.e. in X, and u ≡ x on ∂X}. It is well known that the space A(X) admits a countably infinite homotopy class decomposition A(X) = S Ak (with k ∈ Z). We prove that the energy integral F has a unique minimiser in each of these homotopy classes Ak. Furthermore we show that each minimiser is a homeomorphic, monotone, radially symmetric twist mapping of class C 3 (X, X) or as smooth as F allows thereafter whilst also being a local minimiser of F over A(X) with respect to the L 1 -metric. To our best knowledge this is the first uniqueness result for minimisers in homotopy classes in the context of incompressible mappings

Equilibrium Multiplicity: A Systematic Approach using Homotopies, with an Application to Chicago

Discrete choice models with social interactions or spillovers may exhibit multiple equilibria. This paper provides a systematic approach to enumerating them for a quantitative spatial model with discrete locations, social interactions, and elastic housing supply. The approach relies on two homotopies. A homotopy is a smooth function that transforms the solutions of a simpler city where solutions are known, to a city with heterogeneous locations and finite supply elasticity. The first homotopy is that, in the set of cities with perfectly elastic floor surface supply, an economy with heterogeneous locations is homotopic to an economy with homogeneous locations, whose solutions can be comprehensively enumerated. Such an economy is epsilon close to an economy whose equilibria are the zeros of a system of polynomials. This is a well-studied area of mathematics where the enumeration of equilibria can be guaranteed. The second homotopy is that a city with perfectly elastic housing supply is homotopic to a city with an arbitrary supply elasticity. In a small number of cases, the path may bifurcate and a single path yields two or more equilibria. By running the method on thousands of cities, we obtain a large number of equilibria. Each equilibrium has different population distributions. We provide a method that is computationally feasible for economies with a large number of locations choices, with an empirical application to the City of Chicago. There exist multiple ``counterfactual Chicagos'' consistent with the estimated parameters. Population distribution, prices, and welfare are not uniquely pinned down by amenities. The paper's method can be applied to models in trade and IO. Further applications of algebraic geometry are suggested

On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.Comment: An extended abstract will appear in SODA 201

Non-Uniqueness of Minimizers for Strictly Polyconvex Functionals

In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557-611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals, $$\mathcal {F}(u)=\int_\Omega f(\nabla u(x)) {\rm d}x\quad{\rm and}\quad u\vert_{\partial\Omega}=u_0,$$ where Ω is homeomorphic to a ball. We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surface