sharp convex lorentz sobolev inequalities new sharp lorentz sobolev inequalities are obtained by convexifying level sets in lorentz integrals via the lp minkowski problem new lp isocapacitary and isoperimetric inequalities are proved for lipschitz star bodies it is shown that the sharp convex lorentz sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

combined state adding and state deleting approaches to type iii multi step rationally extended potentials applications to ladder operators and superintegrability type iii multi step rationally extended harmonic oscillator and radial harmonic oscillator potentials characterized by a set of k integers m1 m2 mk such that m1 m2 mk with mi even resp odd for i odd resp even are considered the state adding and state deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators the eigenstates of the hamiltonians are shown to separate into mk 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebras these ladder operators are then used to build a higher order integral of motion for seven new infinite families of superintegrable two dimensional systems separable in cartesian coordinates the finite dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one dimensional hamiltonian eigenstates and provide an algebr

scaling theory for systems with instantaneous gelation partial results we study a system of kinetic equations describing irreversible aggregation the so called smoluchowski equations for a peculiar case in which even starting from initial conditions for which only a monomer is present at arbitrarily small times an infinite cluster appears and causes the total mass of the system to decrease this phenomenon known as instantaneous gelation is known to arise in cases where the reaction rates between large clusters increase strongly as the cluster size goes to infinity we resolve two puzzles linked to this behaviour first from general results it is well known that a power law behaviour must form instantly at large masses for arbitrarily small times we show that the small time behaviour can be analysed for these systems but this must be done in an essentially different manner than in the regular case the correct treatment provides a mechanism for the instantaneous appearance of a power law tail at large masses in the aggregate size distribution we additionally show that instantaneous gelation is connected to the presence of an essential singularity in the solution and also show how from the numerical study of the formal power series describing the solution the nature of the singularity can be guessed and the time dependence of the mass scale for the onset of the power law behaviour conjectured

the two dimensional lp minkowski problem and nonlinear equations with negative exponents abstract in this paper we present various existence results for nonlinear differential equations related to the l p minkowski problem in the plane and the one dimensional conformal curvature problem in particular the supercritical case that is p 2 for the l p minkowski problem with sign changing data is addressed a new geometric sharp inequality is also obtained

optimal boundary regularity for nonlinear singular elliptic equations abstract in this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary by a careful construction of super and sub solutions we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument

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

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

lp john ellipsoids it is shown that the classical john ellipsoid the petty ellipsoid and a recently discovered dual of the legendre ellipsoid are all special cases p infty p 1 and p 2 of a family of l p ellipsoids which can be associated with a fixed convex body this insight allows for a unified view of alternative approaches to and extensions of some basic results in convex geometry

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

a generalized rotationally symmetric case of the centroaffine minkowski problem abstract in this paper the centroaffine minkowski problem a critical case of the l p minkowski problem in the n 1 dimensional euclidean space is studied by its variational structure and the method of blow up analyses we obtain two sufficient conditions for the existence of solutions for a generalized rotationally symmetric case of the problem

nonuniqueness of solutions to the l p minkowski problem abstract in this paper we show by example that there is no uniform estimate for the l p minkowski problem as a result we obtain the nonuniqueness of solutions to the problem

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

the planar orlicz minkowski problem in the l1 sense abstract in this paper we solve l p minkowski problem for l 1 data and all p 0 and orlicz minkowski problem with two nonlinear terms in l 1 sense a byproduct is the blaschke santalo inequality which was previously established for only constant data and now is shown to hold for l 1 data