several types of enriched l m fuzzy convergence spaces abstract in this paper several types of enriched l m fuzzy convergence spaces are introduced including enriched l m fuzzy kent convergence spaces enriched l m fuzzy limit spaces pretopological enriched l m fuzzy convergence spaces topological enriched l m fuzzy convergence spaces and enriched l m fuzzy choquet convergence spaces these concepts generalize the concepts of kent convergence spaces of limit spaces of pretopological convergence spaces of topological convergence spaces and of choquet convergence spaces in general topology to the setting of l m fuzzy topology also their categorical properties and their mutual categorical relations are investigated

a priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient abstract we consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient such as f x u d u d 2 u c x u m x d u d u h x in a bounded domain with a dirichlet boundary condition here r c h l p p n 1 c 0 and the matrix m satisfies 0 1 i m 2 i recently this problem was studied in the coercive case c 0 where uniqueness of solutions can be expected and it was conjectured that the solution set is more complex for noncoercive equations this conjecture was verified in 2015 by arcoya de coster jeanjean and tanaka for equations in divergence form by exploiting the integral formulation of the problem here we show that similar phenomena occur for general even fully nonlinear equations in nondivergence form we use different techniques based on the maximum principle we develop a new method to obtain the crucial uniform a priori bounds which permit to us to use degree theory this method is based on basic regularity estimates such as half harnack inequalities and on a vazquez type strong maximum principle for our kind of equations

w 2 p estimates for solutions to asymptotically elliptic equations in nondivergence form abstract we introduce asymptotically elliptic equations and establish the global w 2 p estimates for viscosity solutions to both asymptotically linear elliptic equations and asymptotically fully nonlinear elliptic equations in bounded domains our approach for the proof is based on an appropriate transformation which converts a given asymptotically elliptic equation to a suitable uniformly elliptic equation

calderon zygmund estimates in generalized orlicz spaces abstract we establish the w 2 solvability of the linear elliptic equations in non divergence form under a suitable essentially minimal condition of the generalized orlicz function x t by deriving calderon zygmund type estimates the class of generalized orlicz spaces we consider here contains as special cases classical lebesgue and orlicz spaces as well as non standard growth cases like variable exponent and double phase growth

to the theory of viscosity solutions for uniformly elliptic isaacs equations we show how a theorem about the solvability in c 1 1 c 1 1 of special isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate isaacs equations we apply it also to establish the c 1 c 1 regularity of viscosity solutions and show that finite difference approximations have an algebraic rate of convergence the main coefficients of the isaacs equations are supposed to be in c c with slightly less than 1 2

degrees of compactness in l m fuzzy convergence spaces and its applications based on completely distributive lattices l and m we define the degrees of compactness of l m fuzzy convergence spaces l m fuzzy topological spaces l m fuzzy pseudo quasi metric spaces and pointwise l m fuzzy quasi uniform spaces it is shown that 1 the tychonoff theorem with respect to the compactness degrees holds in l m fuzzy convergence spaces and l m fuzzy topological spaces 2 the compactness degrees of an l m fuzzy pseudo quasi metric space and a pointwise l m fuzzy quasi uniform space are equal to the compactness degrees of their induced l m fuzzy topological spaces respectively 3 an l m fuzzy pseudo quasi metric space can induce a pointwise l m fuzzy quasi uniform space and their compactness degrees are equal

the categorical topology approach to fuzzy topology and fuzzy convergence it is the purpose of this paper to have a look at fuzzy topology from the point of view of categorical topology we first look at interesting subcategories of fts and determine whether they are bireflective and or coreflective we also describe the set of all subcategories of fts which are at the same time bireflective and coreflective due to the fact that top is one of these subcategories of fts ffs itself can be neither cartesian closed nor extensional we therefore secondly construct a supercategory of bts which is a topological quasitopos the construction of this quasitopos is based on the theory of convergence in fts

new integrable generalizations of calogero moser quantum problem a one parameter deformation of calogero moser quantum problem is introduced it is shown that corresponding schrodinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

on l m fuzzy convergence spaces this paper presents a definition of l m fuzzy convergence spaces it is shown that the category l m fc of l m fuzzy convergence spaces which embeds the category l m ftop of l m fuzzy topological spaces as a reflective subcategory is a cartesian closed topological category further it is proved that the category of topological pretopological l m fuzzy convergence spaces is isomorphic to the category of topological l m fuzzy quasi coincident neighborhood spaces moreover the relations among l m fuzzy convergence spaces pretopological l m fuzzy convergence spaces and topological l m fuzzy convergence spaces are investigated in the categorical sense

aleksandrov bakelman pucci maximum principles for a class of uniformly elliptic and parabolic integro pde abstract we prove generalized aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes of hamilton jacobi bellman isaacs types whose pde parts are either uniformly elliptic or uniformly parabolic the proofs of these results are based on the classical aleksandrov bakelman pucci maximum principles for the elliptic and parabolic pdes and an iteration procedure using solutions of pucci extremal equations we also provide proofs of nonlocal versions of the classical aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes

extrapolation with weights rearrangement invariant function spaces modular inequalities and applications to singular integrals abstract we present an extrapolation theory that allows us to obtain from weighted l p inequalities on pairs of functions for p fixed and all a weights estimates for the same pairs on very general rearrangement invariant quasi banach function spaces with a weights and also modular inequalities with a weights vector valued inequalities are obtained automatically without the need of a banach valued theory this provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators in particular we obtain weighted and vector valued extensions of the classical theorems of boyd and lorentz shimogaki the key is to develop appropriate versions of rubio de francia s algorithm

higher order nondivergence elliptic and parabolic equations in sobolev spaces and orlicz spaces abstract in this paper we obtain the global regularity estimates of the solutions in sobolev spaces and orlicz spaces for higher order elliptic and parabolic equations of nondivergence form in the whole space we only need to focus on the parabolic case since the corresponding result in the elliptic case can be obtained as a corollary

categories isomorphic to l ftop in this paper our purpose is twofold first we deal with the local structure of l fuzzy topology hence modifying the idea of quasi coincident neighborhood in l topology we concretely create an axiomatic system of neighborhood in l fuzzy topology thus we find the category l fqn of topological l fuzzy quasi coincident neighborhood spaces and appropriate continuous maps which is isomorphic to the category l ftop where l is fuzzy lattice further we prove that the category l fqn is concretely bireflective in another category l prfqn constructed in the paper and l prfqn is topological category over set secondly to establish the relation between l ftop and l top we construct two categories l aitop and latqn it is proved that l aitop is isomorphic to l ftop in particular when l is fuzzy lattice with locally multiplicative property latqn is isomorphic to l ftop

i ftop is isomorphic to i fqn and i aitop in this paper we give two characterizations of i ftop the category of i fuzzy topological spaces and i fuzzy continuous mappings first we construct the category of i fuzzy quasi coincident neighborhood spaces and its continuous mappings denoted i fqn and then show that i ftop is isomorphic to i fqn second we construct the category of antichain i topological spaces and its continuous mappings denoted i aitop and show that i ftop is isomorphic to i aitop

diagonal conditions and continuous extension theorem abstract in this paper we obtain a generalization of kowalsky diagonal condition and that of fischer diagonal condition respectively namely kowalsky diagonal condition and fischer diagonal condition we show that our fischer diagonal condition assures a complete mv algebra valued convergence space proposed in this paper is strong l topological and kowalsky diagonal condition assures a principle or pretopological complete mv algebra valued convergence space is strong l topological also as applications we give a dual form of our fischer diagonal condition and obtain a concept of regular convergence space in addition we present an extension theorem for continuous maps from a dense subspace to a regular convergence space to show that our diagonal conditions works indeed

hessian estimates in weighted lebesgue spaces for fully nonlinear elliptic equations abstract we prove global regularity in weighted lebesgue spaces for the viscosity solutions to the dirichlet problem for fully nonlinear elliptic equations as a consequence regularity in morrey spaces of the hessian is derived as well

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

weighted regularity estimates in orlicz spaces for fully nonlinear elliptic equations abstract in this paper we develop a global w 2 p estimate for the viscosity solution of the dirichlet problem of fully nonlinear elliptic equations f d 2 u d u u x f x in u 0 on to a more general function space given an n function and a muckenhoupt weight w we prove that if f belongs to the associated weighted orlicz space l w then d 2 u l w and u satisfies a global w w 2 estimate under a minimal regularity requirement on f in the variable x and a basic geometric assumption on the correct condition on the couple and w is also addressed this result generalizes the w 2 p estimate caffarelli 1989 escauriaza 1993 winter 2009 of calderon and zygmund as well as an analogous one byun et al 2016 in the weighted l p setting

on the aleksandrov bakel man pucci estimate for some elliptic and parabolic nonlinear operators in this work we prove the aleksandrov bakel man pucci estimate for possibly degenerate nonlinear elliptic and parabolic equations of the form rm div left f left nabla u x right right f left x right quad rm in omega subset mathbb r n and u t x t rm div left f left nabla u x t right right f left x t right quad rm in q subset mathbb r n 1 for f a fancyscript c 1 monotone field under some suitable conditions examples of applications such as the p laplacian and the mean curvature flow are considered as well as extensions of the general results to equations that are not in divergence form such as the m curvature flow

convergence in i fuzzy topological spaces the purpose of this work is to combine the fuzzy q neighborhood system of fuzzy points introduced by sostak with the convergence structures of lee et al using the l filter approach of gahler and eklund

weighted estimates for nondivergence parabolic equations in orlicz spaces abstract we obtain the global weighted orlicz regularity of the maximum order derivatives of the solution to the dirichlet problem for nondivergence parabolic equations in a bounded domain we find correct conditions on the associated weight and young function the coefficients are assumed to have small bounded mean oscillation bmo seminorms

liouville property and regularity of a hessian quotient equation we are concerned with properties of convex solutions to the hessian quotient equation sn c d2 p 1 k n as our first main result we prove some regularity of strong solutions while the second states that for p 1 an entire convex solution with a quadratic growth bound must be a quadratic polynomial

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

harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations abstract we present a new harnack inequality for non negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices this estimate applies to all supersolutions and has the harnack constant depending on the graph distance on lattices for the proof we modify the proof of the weak harnack inequality applying the same idea to elliptic equations in a euclidean space we also derive a harnack type inequality for non negative viscosity supersolutions

a class of baker akhiezer arrangements we study a class of arrangements of lines with multiplicities on the plane which admit the chalykh veselov baker akhiezer function these arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities we describe all the baker akhiezer arrangements when at most one line has multiplicity higher than 1 we study associated algebras of quasi invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized calogero moser operators we compute the hilbert series of these algebras and we conclude that the algebras are gorenstein we also show that there are no other arrangements with gorenstein algebras of quasi invariants when at most one line has multiplicity bigger than 1

on lp viscosity solutions of bilateral obstacle problems with unbounded ingredients the global equi continuity estimate on l p viscosity solutions of bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous the existence of l p viscosity solutions is established via an approximation of given data the local holder continuity estimate on the first derivative of l p viscosity solutions is shown when the obstacles belong to c 1 beta and p n

bolza problems with discontinuous lagrangians and lipschitz continuity of the value function we study the local lipschitz continuity of the value function v associated with a bolza problem in the presence of a lagrangian l x q convex and uniformly superlinear in q but only borel measurable in x under these assumptions the associated integral functional is not lower semicontinuous with respect to the suitable topology which ensures the existence of minimizers so all results known in the literature fail to apply yet the lipschitz regularity of v does not depend on the existence of minimizers in fact it is enough to control the derivatives of quasi minimal curves but the problem is nontrivial due to the general growth conditions assumed here on l x cdot we propose a new approach based on suitable reparameterization arguments to obtain suitable a priori estimates on the lipschitz constants of quasi minimizers as a consequence of our analysis we derive the lipschitz continuity of v and a compactness result for value functions associated with sequences of locally equibounded discontinuous lagrangians

regularity of the free boundary in two phase problems for linear elliptic operators abstract in this paper we complete the study of the regularity of the free boundary in two phase problems for linear elliptic operators started in m c cerutti f ferrari s salsa two phase problems for linear elliptic operators with variable coefficients lipschitz free boundaries are c 1 arch ration mech anal 171 2004 329 348 in particular we prove that lipschitz and flat free boundaries in a suitable sense are smooth as byproduct we prove that lipschitz free boundaries are smooth in the case of quasilinear operators of the form div a x u u with lipschitz coefficients

generalised filters 1 we continue the study started in 8 of generalised filters prime prefilters have played a central role in the theory of lowen fuzzy uniform spaces and lowen discovered a characterisation of the set of all minimal prime prefilters finer than a given prefilter in terms of ultrafilters we define the notion of a prime generalised filter and describe the set of all minimal prime g filters finer than a given g filter in terms of ultrafilters the relationship between prime prefilters and prime g filters is revealed the behaviour of the images and preimages of g filters are investigated

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

on the reflective and coreflective subcategory of stratified l cech closure spaces abstract in this paper it is shown that the category of cech closure spaces can be embedded in the category of stratified l cech closure spaces as a coreflective full subcategory furthermore if the underlying lattice of the complete residuated lattice l is completely distributive the embedding is also reflective finally some related commutative diagrams are obtained

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

multidimensional baker akhiezer functions and huygens principle a notion of the rational baker akhiezer ba function related to a configuration of hyperplanes in c n is introduced it is proved that the ba function exists only for very special configurations locus configurations which satisfy a certain overdetermined algebraic system the ba functions satisfy some algebraically integrable schrodinger equations so any locus configuration determines such an equation some results towards the classification of all locus configurations are presented this theory is applied to the famous hadamard problem of description of all hyperbolic equations satisfying huygens principle we show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the ba functions

fuzzy convergence structures abstract we define a fuzzy convergence structure which is a correspondence between the prefilters on a given set x and the fuzzy subsets of x the notion of fuzzy convergence structure is shown to be a generalization of a fuzzy topology

estimates in the generalized campanato john nirenberg spaces for fully nonlinear elliptic equations we establish estimates in bmo and campanato john nirenberg spaces bmo for the second derivatives of solutions to the fully nonlinear elliptic equation f d2u x f x

hypersurfaces with mean curvature given by an ambient sobolev function we consider n hypersurfaces j with interior ej whose mean curvature are given by the trace of an ambient sobolev function uj w 1 p r h j uj ej on j 0 1 where ej denotes the inner normal of j we investigate 0 1 when j weakly as varifolds and prove that is an integral n varifold with bounded first variation which still satisfies 0 1 for uj u ej e p has to satisfy

subcategories of lattice valued convergence spaces we study several subcategories of heyting algebra valued convergence spaces these categories generalize the well known categories of kent convergence spaces of limit spaces of pretopological spaces of pseudo topological spaces and of topological limit spaces to the many valued setting moreover a functorial mechanism is described for changing the basis lattice

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

the general fuzzy filter approach to fuzzy topology ii in this and a subsequent paper by means of an extension of the notion of fuzzy filter a new approach to fuzzy topology is presented whereas this paper is devoted to the general theory of fuzzy filters in part ii this theory is applied on fuzzy topology

baker akhiezer functions and generalised macdonald mehta integrals for the rational baker akhiezer functions associated with special arrangements of hyperplanes with multiplicities we establish an integral identity which may be viewed as a generalisation of the self duality property of the usual gaussian function with respect to the fourier transformation we show that the value of properly normalised baker akhiezer function at the origin can be given by an integral of macdonald mehta type and explicitly compute these integrals for all known baker akhiezer arrangements we use the dotsenko fateev integrals to extend this calculation to all deformed root systems related to the non exceptional basic classical lie superalgebras

w 2 p cdot regularity for elliptic equations in nondivergence form with bmo coefficients we establish an optimal w 2 p cdot estimate to the dirichlet problem for an elliptic equation in nondivergence form with discontinuous coefficients on a c 1 1 bounded domain for every variable exponent p cdot with log holder continuity the matrix of the coefficients is assumed to have a small bmo semi norm depending on the exponent the boundary of the domain and the matrix itself

basic notions for fuzzy topology 1 abstract a new notion of fuzzy real line is presented applications are obtained when solving the problem of generating fuzzy topologies by fuzzy metrics the results provide a fundamental example as a basis for a convergence based approach to fuzzy topology

nondivergence elliptic and parabolic problems with irregular obstacles we prove the natural weighted calderon and zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles we also obtain morrey regularity results for the hessian of the solutions and holder continuity of the gradient of the solutions

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

l fuzzy q convergence structures abstract in this paper we construct a topological category of pretopological l fuzzy q convergence spaces which contains the category of topological l fuzzy q convergence spaces as a bireflective full subcategory considering the connections with l fuzzy topology it is proved that the category of topological l fuzzy q convergence spaces is isomorphic to the category of topological l fuzzy quasi coincident neighborhood spaces and the latter is isomorphic to the category of l fuzzy topological spaces moreover we find that our pretopological l fuzzy q convergence spaces can be characterized as a kind of l fuzzy quasi coincident neighborhood spaces which is called strong l fuzzy quasi coincident neighborhood space

inhomogeneous hopf oleĭnik lemma and regularity of semiconvex supersolutions via new barriers for the pucci extremal operators abstract in this paper we construct new barriers for the pucci extremal operators with unbounded rhs the geometry of these barriers is given by a harnack inequality up to the boundary type estimate under the possession of these barriers we prove a new quantitative version of the hopf ole nik lemma for quasilinear elliptic equations with g laplace type growth finally we prove sharp regularity for semiconvex supersolutions for some nonlinear pdes these results are new even for second order linear elliptic equations in nondivergence form moreover these estimates extend and improve a classical a priori estimate proven by l caffarelli j j kohn j spruck and l nirenberg in 13 in 1985 as well as a more recent result on the c 1 1 regularity for convex supersolutions obtained by c imbert in 33 in 2006

monoidal closedness of the category of stratified l semiuniform convergence spaces abstract for an integral commutative unital quantale l we show that there exists a tensor product between stratified l semiuniform convergence spaces and with respect to this tensor product that the category of stratified l semiuniform convergence spaces and uniformly continuous maps is a symmetric monoidal category as a result we obtain the monoidal closedness of the category of stratified l semiuniform convergence spaces and uniformly continuous maps

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

neighborhood structures of smooth topological spaces abstract smooth topological spaces and their fundamental concepts have been discussed in the literature 2 4 7 in this paper we present the notions of base and two sorts of neighborhood structure of smooth topological spaces and give some of their properties which are results in 3 extended to smooth topological spaces

global weighted estimates in orlicz spaces for second order nondivergence parabolic equations abstract in this paper we prove the following global weighted estimates in orlicz spaces f l w t u d u d 2 u u t l w t for second order parabolic equations of nondivergence form with small bmo coefficients u t a i j x t u x i x j f in t u x t 0 on p t

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations