on an inequality by n trudinger and j moser and related elliptic equations it has been shown by trudinger and moser that for normalized functions u of the sobolev space w 1 n whereis a bounded domain in r n one has r exp njuj n n 1 dx cn where n is an explicit constant depending only on n and cn is a constant depending only on n and carleson and chang proved that there exists a corresponding extremal function in the case that is the unit ball in r n in this paper we give a new proof a generalization and a new interpretation of this result in particular we give an explicit sequence that is maximizing for the above integral among all normalized concentrating sequences as an application the existence of a nontrivial solution for a related elliptic equation with trudinger moser growth is proved c 2002 john wiley sons inc

steady flow of non newtonian fluids monotonicity methods in generalized orlicz spaces abstract our purpose is to show the existence of weak solutions to steady flow of non newtonian incompressible fluids with nonstandard growth conditions of the cauchy stress tensor we are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening since l p framework is not sufficient to capture the described model we describe the growth conditions with the help of a general x dependent convex function and formulate our problem in generalized orlicz spaces

poincare meets korn via maxwell extending korn s first inequality to incompatible tensor fields for a bounded domain r3 r3 with lipschitz boundary and some relatively open lipschitz subset t t of we prove the existence of some c 0c 0 such that equation 0 1 c t l2 r3 3 symt l2 r3 3 curlt l2 r3 3 holds for all tensor fields in h curl h curl i e for all square integrable tensor fields t r3 3t r3 3 with square integrable generalized rotation curlt r3 3 having vanishing restricted tangential trace on t t if t t 0 1 still holds at least for simply connected and for all tensor fields t h curl t h curl which are l2 l2 perpendicular to so 3 so 3 i e to all skew symmetric constant tensors here both operations curl and tangential trace are to be understood row wise for compatible tensor fields t vt v 0 1 reduces to a non standard variant of the well known korn s first inequality in r3r3 namely c v l2 r3 3 sym v l2 r3 3 for all vector fields v h1 r3 v h1 r3 for which vn vn n 1 3n 1 3 are normal at t t on the other hand identifying vector fields v h1 r3 v h1 r3 having the proper boundary conditions with skew symmetric tensor fields t 0 1 turns to poincare s inequality since 2c v l2 r3 c t l2 r3 3 curlt l2 r3 3 2 v l2 r3 therefore 0 1 may be viewed as a natural common generalization of korn s first and poincare s inequality from another point of view 0 1 states that one can omit compatibility of the tensor field t at the expense of measuring the deviation from compatibility through curl t decisive tools for this unexpected estimate are the classical korn s first inequality helmholtz decompositions for mixed boundary conditions and the maxwell estimate

on rediscovered methods for solving equations abstract in the recent papers of gerlach siam rev 36 1994 272 276 and ford and pennline siam rev 38 1996 658 659 a class of iterative methods for solving a single equation f x 0 with arbitrary rate of convergence has been presented in this paper we show that this class is equivalent to five other classes of iterative methods derived earlier in various ways and expressed in different forms the proofs of equivalence of all considered iteration formulas are given

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

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

sobolev inequalities for the symmetric gradient in arbitrary domains abstract a form of sobolev inequalities for the symmetric gradient of vector valued functions is proposed which allows for arbitrary ground domains in r n in the relevant inequalities boundary regularity of domains is replaced with information on boundary traces of trial functions the inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of sobolev functions in regular domains furthermore they involve constants independent of the geometry of the domain and hence yield novel results yet for smooth domains our approach relies upon a pointwise estimate for the functions in question via a riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace

new simultaneous root finding methods with accelerated convergence for analytic functions a new iterative method of the fourth order for the simultaneous determination of zeros of a class of analytic functions is proposed further improvements of the basic method are attained by using newton s and halley s corrections giving the orders of convergence five and six respectively the improved convergence is achieved with negligible number of additional calculations which significantly increases the computational efficiency of the accelerated methods numerical examples demonstrate a good convergence properties fitting very well theoretical results

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

optimal sobolev trace embeddings optimal target spaces are exhibited in arbitrary order sobolev type embeddings for traces of n dimensional functions on lower dimensional subspaces sobolev spaces built upon any rearrangement invariant norm are allowed a key step in our approach consists in showing that any trace embedding can be reduced to a one dimensional inequality for a hardy type operator depending only on n and on the dimension of the relevant subspace this can be regarded as an analogue for trace embeddings of a well known symmetrization principle for first order sobolev embeddings for compactly supported functions the stability of the optimal target space under iterations of sobolev trace embeddings is also established and is part of the proof of our reduction principle as a consequence we derive new trace embeddings with improved optimal target spaces for classical sobolev lorentz sobolev and orlicz sobolev spaces

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

a new approach to counterexamples to l 1 estimates korn s inequality geometric rigidity and regularity for gradients of separately convex functions the derivation of counterexamples to l1 estimates can be reduced to a geometric decomposition procedure along rank one lines in matrix space we illustrate this concept in two concrete applications firstly we recover a celebrated and rather complex counterexample by ornstein proving the failure of korn s inequality and of the corresponding geometrically nonlinear rigidity result in l1 secondly we construct a function f 2 which is separately convex but whose gradient is not in bvloc in the sense that the mixed derivative 2f x1 x2 is not a bounded measure

on the simultaneous determination of the zeros of an analytic function inside a simple smooth closed contour in the complex plane abstract a family of higher order iterative methods for the simultaneous determination of all simple or multiple zeros of an analytic function inside a simple smooth closed contour is obtained using earlier results of the author with the help of circular arithmetic the interval variant of this family is proposed many parallel iterative methods of the literature are special cases of this family

fundamental solution for the q laplacian and sharp moser trudinger inequality in carnot groups abstract for a general carnot group g with homogeneous dimension q we prove the existence of a fundamental solution of the q laplacian u q and a constant a q 0 such that exp a q u q is a homogeneous norm on g this implies a representation formula for smooth functions on g which is used to prove the sharp carnot group version of the celebrated moser trudinger inequality

korn type inequalities in orlicz spaces abstract we exhibit balance conditions between a young function a and a young function b for a korn type inequality to hold between the l b norm of the gradient of vector valued functions and the l a norm of its symmetric part in particular we extend a standard form of the korn inequality in l p with 1 p and an orlicz version involving a young function a satisfying both the 2 and the 2 condition

a class of simultaneous methods for the zeros of analytic functions abstract a family of iterative methods for simultaneously approximating simple zeros of analytic functions inside a simple smooth closed contour in the complex plane is presented the order of convergence of the considered methods in m 2 m 1 2 where m is the order of the highest derivative of analytic function appearing in the iterative formula a special attention is paid to the total step and single step methods with newton s and halley s corrections because of their high computational efficiency numerical examples are also included

sobolev inequalities in arbitrary domains abstract a theory of sobolev inequalities in arbitrary open sets in r n is established boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order the relevant sobolev inequalities exhibit the same critical exponents as in the classical framework moreover they involve constants independent of the geometry of the domain and hence yield genuinely new results even in the case when just smooth domains are considered our approach relies upon new representation formulas for sobolev functions and on ensuing pointwise estimates which hold in any open set

two level systems space curve formalism berry s phase and gauss bonnet theorem the authors study the projective space representation of a general two level system introduced by feynman et al 1957 this allows a mapping onto a space curve geometric phases berry s phase and its complementary phase are calculated the relation with the gauss bonnet theorem is discussed and an inequality for the berry phase is derived

solenoidal lipschitz truncation and applications in fluid mechanics abstract we extend the lipschitz truncation method to the setting of solenoidal functions in particular we approximate a solenoidal sobolev function by a solenoidal lipschitz function which differs from the original function only on a small set our main application is the existence of weak solutions to the two dimensional prandtl eyring fluid model which has almost linear growth in this situation a correction via bogovski operators does not work furthermore we extend the concept of almost a harmonicity to the fluid context in the pressure free formulation

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

orlicz sobolev boundary trace embeddings we prove a sharp trace embedding theorem for orlicz sobolev spaces into orlicz spaces on the boundary an optimal trace embedding into more general rearrangement invariant spaces on the boundary is also established

parallel transport along a space curve and related phases the authors investigate geometrical properties of a space curve and of its spherical images they show that a space curve is characterised by two phase like quantities and comment on the relation of these quantities to the berry phase

on schroder s families of root finding methods schroder s methods of the first and second kind for solving a nonlinear equation f x 0 originally derived in 1870 are of great importance in the theory and practice of iteration processes they were rediscovered several times and expressed in different forms during the last 130 years it was proved in the paper of petkovic and herceg 1999 7 that even seven families of iteration methods for solving nonlinear equations are mutually equivalent in this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in schroder s generalized method of the second kind presented in 1870 in the continuation we consider smale s open problem from 1994 about possible link between schroder s methods of the first and second kind and state the link in a simple way