mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

fock space representations of affine lie algebras and integral representations in the wess zumino witten models fock space representations of affine lie algebras are studied explicit forms of correction terms adding to the currentsf i z are determined it is proved that the sugawara energy momentum tensor on the fock spaces is quadratic in free bosons furthermore screening operators are constructed this implies the existence of generalized hypergeometric integrals satisfying the knizhnik zamolodchikov equation

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

on invariant analysis of some time fractional nonlinear systems of partial differential equations i an investigation of lie point symmetries for systems of time fractional partial differential equations including ito system coupled burgers equations coupled korteweg de vries equations hirota satsuma coupled kdv equations and coupled nonlinear hirota equations has been done using the obtained symmetries each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving erdelyi kober fractional differential operator depending on a parameter

fractional noether s theorem in the riesz caputo sense abstract we prove a noether s theorem for fractional variational problems with riesz caputo derivatives both lagrangian and hamiltonian formulations are obtained illustrative examples in the fractional context of the calculus of variations and optimal control are given

on the gradient set of lipschitz maps we prove that the essential range of the gradient of planar lipschitz maps has a connected rank one convex hull as a corollary in combination with the results in 7 we obtain a complete characterization of incompatible sets of gradients for planar maps in terms of rank one convexity 2008 walter de gruyter berlin new york

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

monotone hopf harmonics we introduce the concept of monotone hopf harmonics in 2d as an alternative to harmonic homeomorphisms much of the foregoing is motivated by the principle of non interpenetration of matter in the mathematical theory of nonlinear elasticity ne the question we are concerned with is whether or not a dirichlet energy minimal mapping between jordan domains with a prescribed boundary homeomorphism remains injective in the domain the classical theorem of rado kneser choquet asserts that this is the case when the target domain is convex an alternative way to deal with arbitrary target domains is to minimize the dirichlet energy subject to only homeomorphisms and their limits this leads to the so called hopf laplace equation among its solutions some rather surreal are continuous monotone mappings of sobolev class mathscr w 1 2 text loc called monotone hopf harmonics it is at the heart of the present paper to show that such solutions are correct generalizations of harmonic homeomorphisms and in particular are legitimate deformations of hyperelastic materials in the modern theory of ne we make this clear by means of several examples

noether symmetries and conserved quantities for fractional forced birkhoffian systems abstract in this paper noether symmetries and conserved quantities for fractional forced birkhoffian systems are studied firstly a new fractional pfaff birkhoff variational principle with riemann liouville derivatives and generalized force is established and from which the fractional forced birkhoff equations are derived secondly the definitions and the criteria of the invariance of a functional under an e parameter group of infinitesimal transformations are given finally the fractional noether theorems for forced birkhoffian systems are obtained and an illustrative example is presented

new counterexamples to the cell formula in nonconvex homogenization in this article we show that for the homogenization of multiple integrals the quasiconvexification of the cell formula is different from the asymptotic formula in general to this aim we construct three examples in three different settings the homogenization of a discrete model the homogenization of a composite material and the homogenization of a homogeneous material on a perforated domain

on the n dimensional dirichlet problem for isometric maps we exhibit explicit lipschitz maps from r n to r n which have almost everywhere orthogonal gradient and are equal to zero on the boundary of a cube we solve the problem by induction on the dimension n

fractional hamiltonian analysis of higher order derivatives systems the fractional hamiltonian analysis of 1 1 dimensional field theory is investigated and the fractional ostrogradski s formulation is obtained the fractional path integral of both simple harmonic oscillator with an acceleration squares part and a damped oscillator are analyzed the classical results are obtained when fractional derivatives are replaced with the integer order derivatives

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

strong unique continuation for the lame system with lipschitz coefficients in this paper we prove the strong unique continuation property for a lame system with lipschitz coefficients in the plane the proof relies on reducing the lame system to a first order elliptic system and suitable carleman estimates with polynomial weights

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

the failure of rank one connections this article is concerned with interface problems for lipschitz mappings f n n and f n n in the half spaces which agree on the common boundary n 1 n n these naturally occur in mathematical models for material microstructures and crystals the task is to determine the relationship between the sets of values of the differentials df and df for some time it has been thought that the polyconvex hulls df pc and df pc satisfy hadamard s jump condition or are at least rank one connected our examples here refute this idea the estimates of the jacobians we obtain in the course of solving the so called monge ampere inequalities seem also to be of independent interest as an application we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar cauchy riemann equations for which the unique continuation property fails

realization of the space of conformal blocks in lie algebra modules abstract using integrable irreducible representations of generalized twisted affine lie algebra modules we give a realization of the space of conformal blocks of conformal field theory on a stable algebraic curve many basic properties of the conformal blocks such as finite dimensionality of the space invariance of the conformal blocks under suitable formal neighborhood changes and the property of propagation of vacua are discussed finally a relative local 1 form around a fixed point of the order two automorphism of the curve is given

fractional actionlike variational problems fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems in this note we address multidimensional fractional actionlike problems of the calculus of variations

the fractional supertrace identity and its application to the super ablowitz kaup newell segur hierarchy in this paper a fractional supertrace identity on superalgebras is presented based on the fractional supertrace identity we can construct fractional super hamiltonian structures of the generalized zero curvature equations associated with lie superalgebras as an application we get the fractional super ablowitz kaup newell segur akns hierarchy and its super hamiltonian structure by using of fractional supertrace identity this method can be used to get more fractional super hierarchies

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

generalized natural boundary conditions for fractional variational problems in terms of the caputo derivative this paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a lagrangian depending on the free end points the fractional derivatives are defined in the sense of caputo

fractional variational problems from extended exponentially fractional integral in this letter we study the fractional variational problems from extended exponentially fractional integral both the lagrangian and hamiltonian formulations are explored and discussed in some details some interesting consequences are revealed