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

conductor and capacitary inequalities for functions on topological spaces and their applications to sobolev type imbeddings in 1972 the author proved the so called conductor and capacitary inequalities for the dirichlet type integrals of a function on a euclidean domain both were used to derive necessary and sufficient

trees and branches in banach spaces an infinite dimensional notion of asymptotic structure is considered this notion is developed in terms of trees and branches on banach spaces every countably infinite countably branching tree t of a certain type on a space x is presumed to have a branch with some property it is shown that then x can be embedded into a space with an fdd e i so that all normalized sequences in x which are almost a skipped blocking of e i have that property as an application of our work we prove that if x is a separable reflexive banach space and for some 1 0 there exists a subspace of x having finite codimension which c 2 e embeds into the l p sum of finite dimensional spaces

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

minimisers and kellogg s theorem we extend the celebrated theorem of kellogg for conformal mappings to the minimizers of dirichlet energy namely we prove that a diffeomorphic minimizer of dirichlet energy of sobolev mappings between doubly connected domains d and omega having mathscr c n alpha boundary is mathscr c n alpha up to the boundary provided mathrm mod d geqslant mathrm mod omega if mathrm mod d mathrm mod omega and n 1 we obtain that the diffeomorphic minimizer has mathscr c 1 alpha extension up to the boundary for alpha alpha 2 alpha it is crucial that every diffeomorphic minimizer of dirichlet energy has a very special hopf differential and this fact is used to prove that every diffeomorphic minimizer of dirichlet energy can be locally lifted to a certain minimal surface near an arbitrary point inside and at the boundary this is a complementary result of an existence results proved by iwaniec et al invent math 186 3 667 707 2011

the sharp quantitative sobolev inequality for functions of bounded variation abstract the classical sobolev embedding theorem of the space of functions of bounded variation bv r n into l n r n is proved in a sharp quantitative form

operators on lp 2 p which factor through xp abstract let t be a bounded linear operator on l p 2 p we show that t factors through x p if and only if there is a change of density of l p so that the induced operator t satisfies an upper l p l 2 tree estimate an operator which factors through x p but fails the upper l p l 2 tree estimate is constructed it is also shown that if t is a closed z p strictly singular operator on l p 2 p then t factors through x p however there exists a z p strictly singular operator which does not factor through x p

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

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

asymmetric affine lp sobolev inequalities abstract a new sharp affine l p sobolev inequality for functions on r n is established this inequality strengthens and implies the previously known affine l p sobolev inequality which in turn is stronger than the classical l p sobolev inequality

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

tartar s conjecture and localization of the quasiconvex hull in mathbb r 2 times 2 we give a concrete and surprisingly simple characterization of compact sets k subset mathbb r 2 times 2 for which families of approximate solutions to the inclusion problem du k are compact in particular our condition is algebraic and can be tested algorithmically we also prove that the quasiconvex hull of compact sets of 2 2 matrices can be localized this is false for compact sets in higher dimensions in general

dual orlicz brunn minkowski theory dual orlicz lϕ affine and geominimal surface areas abstract this paper aims to develop the basic theory for the dual orlicz l affine and geominimal surface areas of star bodies basic properties of these new affine invariants are established moreover related orlicz affine isoperimetric cyclic santalo type and alexandrov fenchel type inequalities are obtained

commutators on lq l1 abstract let t be a bounded linear operator on x l q l 1 with 1 q t is said to be x strictly singular if the restriction of t on any subspace of x that is isomorphic to x is not an isomorphism it is shown that the unique proper maximal ideal in l x is the set of all x strictly singular operators with some more efforts we prove that t is a commutator in l x if and only if for all non zero c the operator t i is not x strictly singular

a characterization of subspaces and quotients of reflexive banach spaces with unconditional bases we prove that the dual or any quotient of a separable reflexive banach space with the unconditional tree property has the unconditional tree property this is used to prove that a separable reflexive banach space with the unconditional tree property embeds into a reflexive banach space with an unconditional basis this solves several long standing open problems in particular it yields that a quotient of a reflexive banach space with an unconditional finite dimensional decomposition embeds into a reflexive banach space with an unconditional basis

the sharp sobolev and isoperimetric inequalities split twice this paper shows that each of the sharp endpoint sobolev inequality and the isoperimetric inequality can be split into two sharp and stronger inequalities through either the 1 variational capacity or the 1 integral affine surface area furthermore some related sharp analytic and geometric inequalities are also explored

small subspaces of l_p we prove that if x is a subspace of lp 2 p 1 then either x embeds isomorphically into p 2 or x contains a subspace y which is isomorphic to p 2 we also give an intrinsic characterization of when x embeds into p 2 in terms of weakly null trees in x or equivalently in terms of the innite asymptotic game played in x this solves problems concerning small subspaces of lp originating in the 1970 s the techniques used were developed over several decades the most recent being that of weakly null trees developed in the 2000 s

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

asymmetric blaschke santalo functional inequalities abstract in this work we establish functional asymmetric versions of the celebrated blaschke santalo inequality as consequences of these inequalities we recover their geometric counter parts with equality cases as well as another inequality with strong probabilistic flavor that was firstly obtained by lutwak yang and zhang we present a brief study on an l p functional analogue to the center of mass that is necessary for our arguments and that might be of independent interest

the affine sobolev zhang inequality on bv rn abstract the affine sobolev zhang inequality is extended to b v r n the space of functions of bounded variation on r n and the equality cases are characterized as a consequence the petty projection inequality for sets of finite perimeter which implies the isoperimetric inequality for sets of finite perimeter is established

extremal mappings of finite distortion 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

the lattice of closed ideals in the banach algebra of operators on certain banach spaces very few banach spaces e are known for which the lattice of closed ideals in the banach algebra of all bounded linear operators on e is fully understood indeed up to now the only such banach spaces are up to isomorphism hilbert spaces and the sequence spaces c0 and lp for 1p we add a new member to this family by showing that there are exactly four closed ideals in for the banach space e l2n c0 that is e is the c0 direct sum of the finite dimensional hilbert spaces l21 l22 l2n

extremals of functionals with competing symmetries abstract we present a new method of producing optimizing sequences for highly symmetric functionals the sequences have good convergence properties built in we apply the method in different settings to give elementary proofs of some classical inequalities such as the hardy littlewood sobolev and the logarithmic sobolev inequality in their sharp form

deformations of annuli with smallest mean distortion we determine the extremal mappings with smallest mean distortion for mappings of annuli as a corollary we find that the nitsche harmonic maps are dirichlet energy minimizers among all homeomorphisms h mathbb a r r to mathbb a r r however outside the nitsche range of the modulus of the annuli within the class of homeomorphisms no such energy minimizers exist in this case we identify the bv limits of minimizers

the lp busemann petty centroid inequality abstract the ratio between the volume of the p centroid body of a convex body k in r n and the volume of k attains its minimum value if and only if k is an origin symmetric ellipsoid this result the lp busemann petty centroid inequality was recently proved by lutwak yang and zhang in this paper we show that all the intrinsic volumes of the p centroid body of k are convex functions of a time like parameter when k is moved by shifting all the chords parallel to a fixed direction the lp busemann petty centroid inequality is a consequence of this general fact

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

beltrami operators in the plane we determine optimal lp properties for the solutions of the general nonlinear elliptic system in the plane of the form fz h z fz h l c where h is a measurable function satisfying h z w1 h z w2 k w1 w2 and k is a constant k 1 we will also establish the precise invertibility and spectral properties in lp c for the operators i t i t and t where t is the beurling transform these operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions in particular we prove invertibility in lp c whenever 1 p 1 1 we also prove related results with applications to the regularity of weakly quasiconformal mappings

lipschitz regularity for inner variational equations we obtain lipschitz regularity results for a fairly general class of nonlinear first order pdes these equations arise from the inner variation of certain energy integrals even in the simplest model case of the dirichlet energy the inner stationary solutions need not be differ entiable everywhere the lipschitz continuity is the best possible but the proofs even in the dirichlet case turn out to relay on topological arguments the appeal to the inner stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity specifically neo hookian type problems

uniqueness of the maximal ideal of the banach algebra of bounded operators on c 0 ω1 let 1 be the smallest uncountable ordinal by a result of rudin bounded operators on the banach space c 0 1 have a natural representation as 0 1 0 1 matrices loy and willis observed that the set of operators whose final column is continuous when viewed as a scalar valued function on 0 1 defines a maximal ideal of codimension one in the banach algebra b c 0 1 of bounded operators on c 0 1 we give a coordinate free characterization of this ideal and deduce from it that b c 0 1 contains no other maximal ideals we then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range and finally we investigate the structure of the lattice of all closed ideals of b c 0 1

existence of energy minimal diffeomorphisms between doubly connected domains the paper establishes the existence of homeomorphisms between two planar domains that minimize the dirichlet energy

monotone sobolev mappings of planar domains and surfaces an approximation theorem of youngs duke math j 15 87 94 1948 asserts that a continuous map between compact oriented topological 2 manifolds surfaces is monotone if and only if it is a uniform limit of homeomorphisms analogous approximation of sobolev mappings is at the very heart of geometric function theory gft and nonlinear elasticity ne in both theories the mappings in question arise naturally as weak limits of energy minimizing sequences of homeomorphisms as a result of this the energy minimal mappings turn out to be monotone in the present paper we show that conversely monotone mappings in the sobolev space fancyscript w 1 p 1 p infty are none other than fancyscript w 1 p weak also strong limits of homeomorphisms in fact these are limits of diffeomorphisms by way of illustration we establish the existence of traction free energy minimal deformations for p harmonic type energy integrals

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

maximal smoothness of solutions to certain euler lagrange equations from nonlinear elasticity we investigate the maximal smoothness of stationary states for the multiple integral such variational problems are motivated by the study of nonlinear elasticity assuming certain structure conditions for and given a stationary state we derive an a priori l p estimate for for any p and where as a consequence we show that a c 1 stationary state necessarily satisfies det and is of class c 2 in nevertheless singular stationary states do exist we construct a nonsmooth c 1 solution for a particular in two dimensions such that det in and det vanishes at precisely one point in

a universal reflexive space for the class of uniformly convex banach spaces we show that there exists a separable reflexive banach space into which every separable uniformly convex banach space isomorphically embeds this solves problems raised by j bourgain b in 1980 and by w b johnson in 1977 jo we also give intrinsic characterizations of separable reflexive banach spaces which embed into a reflexive space with a block q hilbertian and or a block p besselian finite dimensional decomposition

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives