We show that the energy–momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion into the orthogonal group and in particular these solutions can be nowhere $C^1$. We prove that these solutions are not stationary points if the Lagrangian $W$ is $C^1$ and strictly rank-one convex. In view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for solution of the energy–momentum equation in dimension 2, we give a sufficient condition for the non-existence of a partial $C^1$ -regularity result even under the condition that the mappings satisfy a positive Jacobian determinant condition. Finally, we consider a number of well-known functionals studied in non-linear elasticity and geometric function theory and show that these do not satisfy this obstruction to partial regularity.

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

Optimal growth theory as it stands today does not work. Using strictly concave utility functions systematically inflicts on the economy distortions that are either historically unobserved or unacceptable by society. Moreover, we show that the traditional approach is incompatible with competitive equilibrium: Any economy initially in such equilibrium will always veer away into unwanted trajectories if its investment is planned using a concave utility function. We then propose a rule for the optimal savings-investment rate based on competitive equilibrium that simultaneously generates three intertemporal optima for society. The rule always leads to reasonable time paths for all central economic variables, even under very different hypotheses about the future evolution of population and technical progress.

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy 𝓗 subject to the constraint 𝓘 = 2µ, where 𝓘 is the wave momentum and 0 < µ ≪ 1. Since 𝓗 and 𝓘 are both conserved quantities, a standard argument asserts the stability of the set Dµ of minimizers: solutions starting near Dµ remain close to Dµ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as µ ↓ 0.

In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of the calculus of variations are described. Necessary optimality conditions in the main problem of the calculus of variations and in the problem of Bolza in the nonsmooth case are derived. Examples comparing presented results with other approaches to nonsmooth problems of the calculus of variations are given.

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.

The Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

Searching for the optimal partitioning of a domain leads to the use of the adjoint methodin topological asymptotic expansions to know the influence of a domain perturbation on acost function. Our approach works by restricting to local subproblems containing theperturbation and outperforms the adjoint method by providing approximations of higherorder. It is a universal tool, easily adapted to different kinds of real problems and doesnot need the fundamental solution of the problem; furthermore our approach allows toconsider finite perturbations and not infinitesimal ones. This paper provides theoreticaljustifications in the linear case and presents some applications with topologicalperturbations, continuous perturbations and mesh perturbations. This proposed approach canalso be used to update the solution of singularly perturbed problems.

In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. Theproof is based on a reduction argument and the Goh condition for minimality of singularcurves. The Goh condition is deduced from a reformulation and a calculus of the end-pointmapping which boils down to the graded structures of Carnot groups.

In this paper, an analysis of the large-amplitude dynamic-plastic behavior of the circular plates with a rigid perfectly plastic material is presented. The plate is subjected to a short-time high-intensity impulsive load uniformly distributed over the surface. Modeling is complemented by using specific convex yield criteria. Corresponding to boundary conditions of the plate, it can be deformed through more than one mechanism, so, the mathematical formulation is based on the principle of calculus of variations in which the transverse displacement fields are assumed as a combination of appropriate paths. Based on the upper bound approach, the different terms of kinetic and consumed plastic energies likewise the applied impulse energy derived to produce an energy functional with unknown coefficients which is minimized through the displacement path. Finally, calculating the constants maximum residual deflection and strain distribution are obtained. Results of present model show satisfactory correlation with the empirical data for the different levels of the pulsed loads.

Recall that a smooth Riemannian metric on a simply connected domain canbe realized as the pull-back metric of an orientation preserving deformation ifand only if the associated Riemann curvature tensor vanishes identically.When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem byintroducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the properscaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W2,2 isometric immersion of a given 2d metricinto $\mathbb R^3$.

We prove that the critical points of the 3d nonlinear elasticity functionalon shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0to the critical points of the vonKármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)].This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ. 33 (2008) 1018–1032], derived for the case of plates when $S\subset\mathbb{R}^2$.The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.

Newton's problem of the body of minimal aerodynamic resistance is traditionallystated in the class of convex axially symmetric bodies withfixed length and width. We state and solve the minimal resistanceproblem in the wider class of axially symmetric but generallynonconvex bodies. The infimum in this problem is not attained. Weconstruct a sequence of bodies minimizing the resistance. Thissequence approximates a convex body with smooth front surface, whilethe surface of approximating bodies becomes more and morecomplicated. The shape of the resulting convex body and the value ofminimal resistance are compared with the corresponding results forNewton's problem and for the problem in the intermediate class ofaxisymmetric bodies satisfying the single impact assumption[Comte and Lachand-Robert, J. Anal. Math.83 (2001) 313–335]. In particular, the minimal resistance in our class issmaller than in Newton's problem; the ratio goes to 1/2 as(length)/(width of the body) → 0, and to 1/4 as(length)/(width) → +∞.

An a priori Campanato type regularity condition is established for a class of W1X local minimisers $\overline{u}$ of the general variational integral $\int_{\Omega} F(\nabla u(x))\,{\rm d}x$ where $\Omega \subset \mathbb{R}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition $F(\xi)\leq c(1+|\xi|^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space $\mathcal{L}^{p,\mu} (\Omega,\mathbb{R}^{N\times n})$ , µ < n, Campanato space $\mathcal{L}^{p,n}(\Omega,\mathbb{R}^{N\times n})$ and the space of bounded mean oscillation $ {\rm BMO}\Omega,\mathbb{R}^{N\times n})$ . The admissible maps $u\colon \Omega \to \mathbb{R}^N$ are of Sobolev class W1,p , satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W1BMO local minimisers is extended from Lipschitz maps to this admissible class.

We prove partial regularity with optimal Hölder exponent ofvector-valued minimizers u of the quasiconvex variational integral $\intF( x,u,Du) \,{\rm d}x$ under polynomial growth. We employ the indirectmethod of the bilinear form.

The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations.

In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that $f_o$ is a homeomorphism of the circle with $f_{o}^{-1} \in {\cal W}^{1/2, 2}$. Then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non-vanishing Jacobian determinant. The condition on $f_o$ is sharp.

Classically the mapping $f_o$ is assumed to be quasisymmetric. Then there is an extremal quasiconformal mapping with boundary values $f_o$, but it is not always unique and it is seldom smooth. Indeed, even when $f_o$ is quasisymmetric, the ${\cal L}^1$-minimiser for the distortion function will almost never be quasiconformal.

We further find that there are many new and unexpected phenomena concerning existence, uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex. These seem to differ markedly from phenomena observed when studying multi-well type functionals.