potential theory for the sum of the 1 laplacian and p laplacian abstract the variational problem of minimizing the functional u d u 1 p d u p a u on a domain r n under zero boundary values which appears for example in the theory of bingham fluids shows interesting phenomena such as the formation of a subset of with positive measure where the minimizer is constant a plateau the corresponding euler lagrange equation should be 1 u p u a but the 1 laplacian is not defined in points where d u 0 we show that it is nevertheless possible to define subsolutions and supersolutions for this equation we show that the comparison principle is valid and we begin developing a potential theory analogous to what is known about the p laplacian in addition we apply those notions to rotationally invariant solutions and we discuss properties of the plateau and a possible connection to the theory

canonical and differential realizations of osp 3 2 superalgebra the osp 3 2 superalgebra is realized first by means of two supervectorial operators canonically conjugated and then by differential operators acting on functions defined on e 3 2 as for the vectorial representation of osp 1 2 the eigenvalue problem yields to the superspherical harmonics

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

a simple differential operator realization of the super rotation algebra the super rotation algebra is realized as differential operators acting on functions defined on the e 1 2 superspace

regularity of quasi minimizers for non uniformly elliptic integrals abstract in this paper we consider a class of non uniformly elliptic integral functionals and we prove the local boundedness of the quasi minimizers our approach is based on a suitable adaptation of the celebrated de giorgi proof and it relies on an appropriate caccioppoli type inequality

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

rs compactness in l fuzzy topological spaces abstract a good definition of rs compactness in l fuzzy topological spaces is suggested for l a fuzzy lattice we define rs compactness on arbitrary l fuzzy sets and study some of its properties

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

minimizing non convex multiple integrals a density result we consider variational problems of the form where is a bounded open set in r n f r n r is a possibly non convex lower semicontinuous function with p growth at infinity for some 1 p u 0 is any function in w 1 p assuming that the convex envelope f of f is affine on each connected component of the set f f we prove the existence of solutions to p for every continuous function g such that i g has no strict local minima and ii every convergent sequence of extremum points of g eventually belongs to an interval where g is constant thus showing that the set of continuous functions g that yield existence to p is dense in the space of continuous functions on r

local regularity for solutions of anisotropic obstacle problems abstract this paper deals with anisotropic obstacle problems for the a harmonic equation div a x u x 0 a local regularity result is given under suitable assumptions

on the nash point equilibria in the calculus of variations abstract the existence and partial regularity of the nash point equilibria for a pair of multiple integrals j u v f x u v u v dx k u v g x u v u v dx are studied the conditions as well as the results are similar to those for local minima obtained by acerbi fusco and giaquinta giusti

the super rotation racah wigner calculus revisited it is shown that since the finite dimensional representations of the super rotation algebra are characterized by the superspin j and the parity of the representation space all features of the racah wigner calculus clebsch gordan coefficients recoupling coefficients as well as the wigner and racah symbols depend on both j and however it is noticed that the dependence on the parities of the wigner and racah symbols can be factorized out into phases so that one can define parity independent super s3 j and s6 j symbols the properties of these symbols are analyzed in particular it is shown that the s6 j symbols possess a symmetry similar to the regge symmetry satisfied by the rotation 6 j symbols analytical and numerical tables of the symbols are given for the lowest values of their arguments

functions and ai compactness on fuzzy topological spaces abstract the concept of i compactness is introduced and studied in the light of the concept of shading in a fuzzy setting we give some characterization of i compactness in terms of fuzzy regular semiopen fuzzy semiopen and fuzzy open sets we investigated the image and the inverse image of i compact spaces under some types of functions and we show that in a regular extremely disconnected fuzzy space the concepts compactness almost compactness near compactness s closedness and i compactness are equivalent also we improve and generalize in fuzzy setting some results on i compact spaces

supersymmetries of the superspace e 3 2 the superspace e 3 2 is the simplest superspace that generalizes the three dimensional ordinary space a super euclidean metric is defined on this superspace through its isomorphy with the vectorial representation superspace of the super rotation algebra osp 1 2 the supergroup uosp 3 2 is defined as the largest supersymmetry group acting in e 3 2 and preserving the scalar product the vectorial representation of the superrotation group uosp 1 2 acts also in e 3 2 the relations between these two supersymmetries are studied exhaustively

stability for the solutions of div u p 2 u f with varying p abstract elliptic partial differential equations with principal part div u p 2 u are applied in physics e g for the description of phenomena in glaceology the objective of our note is a natural question of stability for solutions as p varies

partial regularity of p x harmonic maps let g x and hij u be uniformly elliptic symmetric matrices and assume that hij u and p x 2 are sufficiently smooth we prove partial regularity of minimizers for the functional f u z

existence and boundedness of a minimiser for a constrained minimisation problem on rn with limiting exponent we use the concentration compactness principle to study the existence of a minimiser of the minimisation problem where u u 1 u n we also prove the boundedness of the minimiser of l 1 by using the reverse holder inequality

partial lipschitz regularity of minimizers of asymptotically convex functionals with general growth conditions abstract we consider the partial lipschitz continuity of minimizers of functionals of the form v f x v d v d x where r n is open and bounded the map x u f x u appearing in the above functional is assumed to be holder continuous with respect to its first two arguments of class c 2 in its third argument and asymptotically related to the more regular map x u a x u g as the main novelty is the allowance that even as the map f may retain full dependence on the minimizer u not only its gradient du and the spatial coordinate x

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

the group with grassmann structure uosp 1 2 the finite dimensional representations of the lie superalgebraosp 1 2 and the group with grassmann structureosp 1 2 have been studied the explicit expression of the projection operator of the superalgebraosp 1 2 has been found the operator permits an arbitrary finite dimensional representation to be expanded in the components multiple to the irreducible ones the clebsch gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained the matrix elements of the irreducible representations of the groupuosp 1 2 the analoque of the compact form of the grouposp 1 2 are studied the explicit form of these matrix elements the differential equations satisfied by them and the integral of their product have been found

on the blow up of heat flow for conformal 3 harmonic maps using a comparison theorem chang ding and ye 1992 proved a finite time derivative blow up for the heat flow of harmonic maps from d 2 a unit ball in r 2 to s 2 a unit sphere in r 3 under certain initial and boundary conditions we generalize this result to the case of 3 harmonic map heat flow from d 3 to s 3 in contrast to the previous case our governing parabolic equation is quasilinear and degenerate technical issues such as the development of a new comparison theorem have to be resolved

liouville theorems partial regularity and holder continuity of weak solutions to quasilinear elliptic systems this paper describes the connections between liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth it is shown that equivalence does in general hold only in some restricted sense a complete correspondence can be established in certain cases e g for small solutions and for minima of quadratic variational integrals

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

racah wigner calculus for the super rotation algebra i the symmetry properties and the pseudo orthogonality relations of the super rotation clebsh gordan coefficients for the tensor product of two irreducible representations of the super rotation algebra are derived the symmetric super rotation 3 j symbol and the symmetric and invariant super rotation 6 j symbol are defined their basic properties are described and their relations to the usual 3 j and 6 j symbols are given

removability of singular sets of harmonic maps it is proved that a harmonic map with small energy and the monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density moreover the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability this work generalizes the results in cl df lg12 which were proved under the assumption that the singular sets are isolated points or smooth submanifolds

second order elliptic equations with variably partially vmo coefficients abstract the solvability in w p 2 r d spaces is proved for second order elliptic equations with coefficients which are measurable in one direction and vmo in the orthogonal directions in each small ball with the direction depending on the ball this generalizes to a very large extent the case of equations with continuous or vmo coefficients

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

the superalgebra embedding of osp 1 2 in sl 1 2 and the grade star representations of sl 1 2 first it is shown that the superalgebra embedding of osp 1 2 in sl 1 2 allows to characterize the irreducible representation states of sl 1 2 without any missing label through this embedding the general nonpositive definite scalar product in the osp 1 2 representation space can be generalized to the sl 1 2 representation space then grade star representations of sl 1 2 are derived by considering this nonpositive scalar product

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

on fuzzy topological spaces abstract in this paper a few separation properties and some aspects of subspace fuzzy topology have been studied where both the crisp and the fuzzy elements have been taken into consideration since the conventional definition of compactness is not quite meaningful in hausdorff fuzzy spaces as introduced by us a new more natural definition of proper compactness is given and a few properties resulting from this are established

subelliptic harmonic maps we study a nonlinear harmonic map type system of subelliptic pde in particular we solve the dirichlet problem with image contained in a convex ball

schauder estimates for parabolic nondivergence operators of hormander type let x1 x2 xq be a system of real smooth vector fields satisfying hormander s rank condition in a bounded domain of rn let a aij t x i j 1q be a symmetric uniformly positive definite matrix of real functions defined in a domain u r for operators of kind h t i j 1qaij t x xixj i 1qbi t x xi c t x we prove local a priori estimates of schauder type in the natural parabolic ck u spaces defined by the vector fields xi and the distance induced by them namely for aij bi c ck u and u u we prove u ck 2 u c hu ck u u l u

vanishing viscosity limit for homogeneous axisymmetric no swirl solutions of stationary navier stokes equations abstract 1 homogeneous axisymmetric no swirl solutions of three dimensional incompressible stationary navier stokes equations which are smooth on the unit sphere minus the north and south poles have been classified in this paper we study the vanishing viscosity limit of sequences of these solutions as the viscosity tends to zero some sequences of solutions c l o c m converge to solutions of euler equations on the sphere minus the poles while for other sequences of solutions transition layer behaviors occur for every latitude circle there are sequences which c l o c m converge respectively to different solutions of the euler equations on the spherical caps above and below the latitude circle we give detailed analysis of these convergence and transition layer behaviors

interior w2 p x regularity for elliptic equations of nondivergence form abstract in this paper we consider the following elliptic equations of nondivergence form i j 1 n a i j 2 u x i x j f in r n where the coefficients a i j are real valued bounded measurable functions defined in r n belong to vmo loc see definition 2 5 and satisfy the uniformly elliptic condition under the assumption that p satisfies the log holder continuity condition we will establish the interior w 2 p x regularity for strong solutions to the elliptic equations of nondivergence form

on fuzzy d continuity and a near compactness in fuzzy topological spaces abstract the concept of fuzzy continuity was introduced by saha 18 mashhour et al have introduced the concept of near compactness in fuzzy setting in this paper we give a characterization of fuzzy continuity and fuzzy nearly compactness also we investigate the image of fuzzy nearly compact spaces under some types of functions finally a refinement is made to some results in 15 17

global minimizers for a p ginzburg landau type energy in r 2 given a p 2 we prove existence of global minimizers for a p ginzburg landau type energy over maps on r2 with degree d 1 at infinity for the analogous problem on the half plane we prove existence of a global minimizer when p is close to 2 the key ingredient of our proof is the degree reduction argument that allows us to construct a map of degree d 1 from an arbitrary map of degree d 1 without increasing the p ginzburg landau energy

super angular momentum and super spherical harmonics the usual quantum mechanics methods of defining angular momentum and spherical harmonics in the ordinary space e 3 are generalized to the real commuting super space e 3 2 the super angular momentum generators of the super rotation is realized as differential operators acting on functions defined on e 3 2 by solving the eigenvalue problem in spherical coordinates a set of pseudo orthonormalized functions is constructed the super spherical harmonics ylmj whose main properties are given

some regularity results for p harmonic mappings between riemannian manifolds abstract let m be a c 2 smooth riemannian manifold with boundary and n a complete c 2 smooth riemannian manifold we show that each stationary p harmonic mapping u m n whose image lies in a compact subset of n is locally c 1 for some 0 1 provided that n is simply connected and has non positive sectional curvature we also prove similar results for minimizing p harmonic mappings with image being contained in a regular geodesic ball moreover when m has non negative ricci curvature and n is simply connected with non positive sectional curvature we deduce a gradient estimate for c 1 smooth weakly p harmonic mappings from which follows a liouville type theorem in the same setting

functions and as closedness in fuzzy topological spaces abstract in this paper we discuss s closedness in fuzzy topological spaces and we give some characterizations of s closedness also we investigate the image and the inverse image of s closed spaces under some types of functions and we improve and generalize some results on s closed spaces in fuzzy setting

parabolic equations with bmo coefficients in lipschitz domains abstract in this paper we are concerned with certain natural sobolev type estimates for weak solutions of inhomogeneous problems for second order parabolic equations in divergence form the geometric setting is that of time independent cylinders having a space intersection assumed to be locally given by graphs with small lipschitz coefficients the constants of the operator being uniformly parabolic we prove the relevant l p estimates assuming that the coefficients are in parabolic bounded mean oscillation bmo and that their parabolic bmo semi norms are small enough

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

the regularity of harmonic maps into spheres and applications to bernstein problems we show the regularity of and derive a priori estimates for weakly harmonic maps from a riemannian manifold into a euclidean sphere under the assumption that the image avoids some neighborhood of a half equator the proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems we apply these results to the spherical and euclidean bernstein problems for minimal hypersurfaces obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in euclidean spaces are trivial

w 1 p regularity for elliptic equations with measurable coefficients in nonsmooth domains we establish global w 1 p estimates for second order elliptic equations in divergence form under the natural assumption that p is log holder continuous to this end we assume that the coefficients are measurable in one variable and have small bmo semi norms in the other variables and the boundary of the domain is reifenberg flat our work is an optimal and natural extension of w 1 p regularity for such equations with merely measurable coefficients beyond lipschitz domains

parabolic equations in reifenberg domains natural sobolev type estimates are proved for weak solutions of inhomogeneous parabolic equations in divergence form in a bounded cylinder 0 t which is reifenberg flat in the space direction the principal coefficients of the operator are assumed to be in bmo space with their bmo semi norms small enough

interior w 1 p x t regularity for parabolic divergence equations of hormander s type abstract let x 1 x 2 x q be a system of real smooth vector fields satisfying hormander s condition in a bounded domain r n we consider the following parabolic equations u t x i a i j x t x j u x i f i in t where x i is the formal adjoint of x i the coefficients a i j x t are real valued measurable functions defined in t 0 t satisfying the uniform parabolic condition 0 1 2 i j 1 q a i j x t i j 1 2 for almost every x t r n r every r q and some constant we prove the interior w 1 p x t regularity for weak solutions to the parabolic equations under the assumptions that p x t satisfies the strong log holder continuity condition and the coefficients a i j x t belong to the space v m o l o c l our method relies on a gagliardo nirenberg inequality constituted on hormander s vector fields and a certain vitali covering lemma

harnack s inequality for quasiminimizers with nonstandard growth conditions we prove harnack inequalities for quasiminimizers of the variable exponent dirichlet energy integral by employing the de giorgi method

mixed states for an allen cahn type equation this paper continues the recent study of an allen cahn model pde 1 by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions enabling us to obtain multibump solutions in a much more general setting as in 1 novel minimization arguments play a key role in finding solutions

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

holder continuity of weak solutions of p laplacian pdes with vmo coefficients abstract we consider solutions u w 1 p r n of the p laplacian pde a x d u p 2 d u 0 for x r n where is open and bounded more generally we consider solutions of the elliptic system a x g a x d u d u d u 0 x as well as minimizers of the functional g a x d u d x in each case the coefficient map a r is only assumed to be of class v m o l which means that it may be discontinuous without assuming that x a x has any weak differentiability we show that u c loc 0 for each 0 1 the preceding results are in fact a corollary of a much more general result which applies to the functional f x u d u d x in case f is only asymptotically convex

interior hw 1 p estimates for divergence degenerate elliptic systems in carnot groups abstract let x 1 x q be the basis of the space of horizontal vector fields on a homogeneous carnot group g r n q n we consider the following divergence degenerate elliptic system 1 n i j 1 q x i a i j x x j u i 1 q x i f i 1 2 n where the coefficients a i j are real valued bounded measurable functions defined in g satisfying the strong legendre condition and belonging to the space v m o l o c defined by the carnot caratheodory distance induced by the x i s we prove interior h w 1 p estimates 2 p for weak solutions to the system