local behavior of solutions of quasilinear elliptic equations with general structure this paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form diva x u vu b x u vu where a and b are borel measurable are solutions to the equation diva x u vu b x u vu u for some nonnegative radon measure i among other things it is shown that if u is a holder continuous solution to this equation then the measure u satisfies the growth property j b x r 1 is given by the structure of the differential operator conversely if u is assumed to satisfy this growth condition then it is shown that u satisfies a harnack type inequality thus proving that u is locally bounded under the additional assumption that a is strongly monotonic it is shown that u is holder continuous

higher differentiability for solutions of nonhomogeneous elliptic obstacle problems abstract this paper focuses on weak solutions to the following variational inequality a x d u d u d x f p 2 f d u d x for all a w w 1 p w a e in where the vector field a satisfies some growth and ellipticity conditions we derive that the higher differentiability property of the weak solution u is related to the regularity of the assigned obstacle and the nonhomogeneous term f under a suitable integer or fractional differentiability assumption on a x with respect to x

gradient estimates for heat kernels and harmonic functions abstract let x d be a doubling metric measure space endowed with a dirichlet form e deriving from a carre du champ assume that x d e supports a scale invariant l 2 poincare inequality in this article we study the following properties of harmonic functions heat kernels and riesz transforms for p 2 i g p l p estimate for the gradient of the associated heat semigroup ii r h p l p reverse holder inequality for the gradients of harmonic functions iii r p l p boundedness of the riesz transform p iv g b e a generalised bakry emery condition we show that for p 2 i ii iii are equivalent while for p i ii iv are equivalent moreover some of these equivalences still hold under weaker conditions than the l 2 poincare inequality our result gives a characterisation of li yau s gradient estimate of heat kernels for p while for p 2 it is a substantial improvement as well as a generalisation of earlier results by auscher coulhon duong hofmann 7 and auscher coulhon 6 applications to isoperimetric inequalities and sobolev inequalities are given our results apply to riemannian and sub riemannian manifolds as well as to non smooth spaces and to degenerate elliptic parabolic equations in these settings

gradient estimates for problems with orlicz growth abstract we study a general nonlinear elliptic equation in the orlicz setting with data not belonging to the dual of the energy space we provide several lorentz type and morrey type estimates for the gradients of solutions under various conditions on the data

on the relation between generalized morrey spaces and measure data problems abstract we consider measure data problems of p laplacian type the measure on the right hand side has the property that the total variation of a generic ball decays in terms of generic functions of the radius we show that this condition has a natural relation with gradient integrability properties and we get as corollary borderline cases of classic results

input impedance of a probe fed stacked circular microstrip antenna the input impedance of a microstrip antenna consisting of two circular microstrip disks in a stacked configuration driven by a coaxial probe is investigated a rigorous analysis is performed using a dyadic green s function formulation whereby the mixed boundary value problem is reduced to a set of coupled vector integral equations using the vector hankel transform galerkin s method is used in the spectral domain using two sets of disk current expansions one set is based on the complete set of orthogonal modes of the magnetic cavity and the other uses chebyshev polynomials with the proper edge condition for the disk currents an additional term is added to the disk current expansion to model the current properly in the vicinity of the probe disk junction the input impedance of the antenna including the probe self impedance is calculated as a function of the layered parameters and the ratio of the two disk radii disk current distributions and radiation patterns are presented the calculated results are shown to be in good agreement with experimental data

on the existence of weak solutions to a class of nonlinear elliptic systems with drift term abstract we study the existence of a weak solution u of the following nonlinear vectorial dirichlet problem u w 0 1 2 r n i 1 n d i a i x u d u i 1 n d i j 1 n e i j x u j f x x where is a bounded subset of r n n 3 u and f for any 1 2 n are the th components of the vectors u and f respectively and the tensors a x s and e x satisfy suitable structural assumptions

regularity results for solutions to obstacle problems with sobolev coefficients abstract we establish the higher differentiability of solutions to a class of obstacle problems of the type min f x d v x d x v k where is a fixed function called obstacle k v w loc 1 p r v a e in and the convex integrand f satisfies p growth conditions with respect to the gradient variable we derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned under a suitable sobolev assumption on the partial map x d f x the main novelty is that such assumption is independent of the dimension n and this in the case p n 2 allows us to manage coefficients in a sobolev class below the critical one w 1 n

pointwise gradient estimates abstract we survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear possibly degenerate elliptic and parabolic equations

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

singular parabolic equations measures satisfying density conditions and gradient integrability abstract we consider solutions to singular parabolic equations with measurable dependence on the x t variables and having on the right hand side a measure satisfying a density condition we prove that the less the measure is concentrated the more the gradient is regular in the marcinkiewicz scale we provide local estimates and recover some classic results

morrey estimates for some classes of elliptic equations with a lower order term abstract we will browse a series of results on elliptic equations and systems with particular nonlinear lower order terms and measure right hand side obtained in the last years in the papers cirmi and leonardi 2014 cianci et al 2017 cirmi et al 2014 2017 2018 namely we take into account equations whose prototypes are u u d u 2 f or u d u 2 u f 0 1 where the right hand side f belongs to a suitable morrey space for instance to l 1 0 n 2 and we prove corresponding morrey estimates for the gradient of a solution

calderon zygmund estimates for general elliptic operators with double phase abstract we derive calderon zygmund type estimates for the gradient of bounded distributional solutions to the problem div b x u d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where 1 p q and 0 a c the coefficient b x z is supposed to be small bmo with respect to x and merely continuous in z

global gradient estimates for spherical quasi minimizers of integral functionals with p x growth abstract we investigate spherical quasi minimizers or so called spherical q minimizers u for integral functionals with p x growth of f u f x d u g p x 2 g d u d x in particular we find minimal regularity requirements on the integrand f x the variable exponent p x the boundary of the bounded domain as well as q under which g p l q d u p l q for every q 1 with the desired global calderon zygmund estimate

universal potential estimates abstract we prove a class of endpoint pointwise estimates for solutions to quasilinear possibly degenerate elliptic equations in terms of linear and nonlinear potentials of wolff type of the source term such estimates allow to bound size and oscillations of solutions and their gradients pointwise and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients in particular local estimates in holder lipschitz morrey and fractional spaces as well as calderon zygmund estimates follow as a corollary in a unified way moreover estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied the classical wolff potential estimate by kilpelainen mal and trudinger wang as well as recent wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases

lorentz estimates for degenerate and singular evolutionary systems we prove estimates of calderon zygmund type for evolutionary p laplacian systems in the setting of lorentz spaces we suppose the coefficients of the system to satisfy only a vmo condition with respect to the space variable our results hold true mutatis mutandis also for stationary p laplacian systems

gradient estimates via non linear potentials we present pointwise gradient bounds for solutions to p laplacean type non homogeneous equations employing non linear wolff type potentials and then prove similar bounds via suitable caloric potentials for solutions to parabolic equations

weighted gradient estimates for the parabolic p laplacian equations abstract in this paper we obtain the weighted l q estimates of the gradients of weak solutions for the quasilinear parabolic equations of p laplacian type with small bmo coefficients for every e 0 q 1 if u p l w l o c 1 e t and f p l w l o c q t then we prove that u p l w l o c q t our results improve the known results for such equations using a harmonic analysis free technique

the p laplacian with respect to measures abstract we introduce a definition for the p laplace operator with respect to positive and finite borel measures that satisfy an adams type embedding condition

potential estimates of superquadratic elliptic systems with vmo coefficients in reifenberg domains abstract the purpose of this paper is to study an inhomogeneous dirichlet problem of a superquadratic nonlinear elliptic system with vmo coefficients in reifenberg flat domains we derive pointwise potential estimates for weak solution to the dirichlet problem as a consequence we derive global integrability estimates for solutions with respect to lorentz and morrey norms

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

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

an end point orlicz type estimate for nonlinear elliptic equations abstract we study nonlinear elliptic equations of p laplacian type in divergence form to establish a natural calderon zygmund type theory of an orlicz space type where the lebesgue space is the special case with t t q p f p l 1 q 2 r d u p l 1 q r in the previous results the estimates obtained were strictly above the natural exponents and the function such as t t log 1 t was ruled out for the candidate of but with our approach can be selected up to the end point case of the estimates and the functions t t and t t log 1 t can be additionally selected for

nonlinear calderon zygmund theory in the limiting case we prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems specifically we deal with the limiting case of the classical theory of calderon and zygmund in the setting of nonlinear possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions a prototype of the results obtained here tells for instance that if rm div du p 2 du mu quad mbox in omega subset mathbb r n with mu being a borel measure with locally finite mass on the open subset omega subset mathbb r n and p 2 1 n then du p 2 du in w sigma 1 rm loc omega quad mbox for every sigma in 0 1 the case sigma 1 is obviously forbidden already in the classical linear case of the poisson equation triangle u mu

singular quasilinear and hessian equations and inequalities we solve the existence problem in the renormalized or viscosity sense and obtain global pointwise estimates of solutions for quasilinear and hessian equations with measure coefficients and data including the following model problems pu uq fk u uq u 0 on rn or on a bounded domain rn here p is the p laplacian defined by pu div u u p 2 and fk u is the k hessian i e the sum of the k k principal minors of the hessian matrix d2u k 1 2 n and are general nonnegative measurable functions or measures on

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

nonlinear parabolic problems with lower order terms and related integral estimates abstract we deal with the solutions to nonlinear parabolic equations of the form u t div a x t d u g x t u f x t on t t 0 under standard growth conditions on g and a with f only assumed to be integrable to the power 1 we prove general local decay estimates for level sets of the solutions u and the gradient d u which imply very general estimates in rearrangement function spaces lebesgue orlicz lorentz and non rearrangement ones up to lorentz morrey spaces

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

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

lorentz estimates for obstacle parabolic problems we prove that the spatial gradient of variational solutions to parabolic obstacle problems of p laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of lorentz spaces

sharp gradient estimates for quasilinear elliptic equations with p x growth on nonsmooth domains in this paper we study quasilinear elliptic equations with the nonlinearity modelled after the p x p x laplacian on nonsmooth domains and obtain sharp calderon zygmund type estimates in the variable exponent setting in a recent work of 12 the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p x p x see 1 3 and 1 4 here we bridge this gap to obtain the end point case of the estimates obtained in 12 see 1 5 in order to do this we have to obtain significantly improved a priori estimates below p x p x which is the main contribution of this paper we also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature

smallness and cancellation in some elliptic systems with measure data abstract in a bounded open subset r n we study dirichlet problems with elliptic systems involving a finite radon measure on r n with values into r n defined by div a x u x d u x in u 0 on where a i x y 1 n j 1 n a i j x y j with 1 n the equation index we prove the existence of a distributional solution u r n obtained as the limit of approximations by assuming i that coefficients a i j are bounded caratheodory functions ii ellipticity of the diagonal coefficients a i j and iii smallness of the quadratic form associated to the off diagonal coefficients a i j i e verifying a r staircase support condition with r 0 such a smallness condition is satisfied for instance in each one of these cases a a i j a j i skew symmetry b a i j is small c a i j may be decomposed into two parts the first enjoying skew symmetry and the second being small in absolute value we give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works a brezis s type nonexistence result is also given for general smooth elliptic hyperbolic systems

quasilinear elliptic systems with measure data abstract we study the existence of solutions of quasilinear elliptic systems involving n equations and a measure on the right hand side with the form i 1 n x i 1 n j 1 n a i j x u x j u in u 0 on where 1 n is the equation index is an open bounded subset of r n u r n and is a finite randon measure on r n with values in r n existence of a solution is proved for two different sets of assumptions on a examples are provided that satisfy our conditions but do not satisfy conditions required on previous works on this matter

the p laplace system with right hand side in divergence form inner and up to the boundary pointwise estimates abstract in this note we collect some very recent pointwise bounds for the gradient of solutions and for the solutions themselves to the p laplace system with right hand side in divergence form both estimates inside the domain for local solutions and global estimates for solutions to boundary value problems are discussed their formulation involves sharp maximal operators whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic analytic framework as a consequence a flexible comprehensive approach to estimates for solutions to the p laplace system for a broad class of norms is derived in particular global estimates under minimal boundary regularity are presented

on fractional regularity methods for a class of nonlocal problems abstract in the past years the phenomenon of fractional regularity has been addressed for a large class of linear and or quasilinear differential operators mostly in terms of certain besov spaces as it turned out for equations governed by the p laplacian in general the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness moreover despite its own interest fractional regularity methods may be used as a tool for the investigation of some partial differential equations which are not usually addressed in this manner thus the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the p laplacian this is done by means of explicit a priori estimates regarding lebesgue and nikolskii spaces which are part of the present contribution as a consequence this approach allows a relaxation on some of the standard conditions employed in this class of problems

riesz potential type estimates for nonlinear elliptic equations abstract for p 2 we study nonlinear elliptic equations of p laplacian type with measure data div a d u u x in when the nonlinearity depends on the variable of the possibly unbounded solution itself we obtain the point wise gradient bound and the gradient continuity in terms of a certain riesz potential of the right hand side measure

on main conjectures in non commutative iwasawa theory and related conjectures the main conjecture of non commutative iwasawa theory of coates fukaya kato sujatha and venjakob is shown to be equivalent modulo an appropriate torsion hypothesis to a family of main conjectures over cyclotomic zp extensions together with suitable compatibility relations between the associated abelian p adic l functions this result combines with techniques of ritter and weiss and of kakde to give a concrete strategy for deducing main conjectures for non abelian compact p adic lie extensions of arbitrarily large rank from main conjectures in the classical sense as an explicit example of this approach we combine it with a recent result of ritter and weiss to prove modulo an appropriate vanishing hypothesis on invariants the main conjecture of non commutative iwasawa theory for tate motives over compact p adic lie extensions of totally real fields of arbitrarily large rank we then deduce several interesting consequences of this result concerning a range of related conjectures including the equivariant tamagawa number conjecture 1991 mathematics subject classification primary 11s40 secondary 11r29 20c05

nonlinear potential theory of elliptic systems abstract we give a summary of recent results from nonlinear potential theory focusing on linear and nonlinear potential estimates of solutions to non homogeneous equations and systems we start with the cases of quasilinear possibly degenerate equations of p laplacian type and passing through fully nonlinear elliptic equations finally move to systems in this last case we describe recent results implying potential estimates for the p laplacian system finally we describe results bridging nonlinear potential theory and classical partial regularity theory for general elliptic systems the main outcomes are new e regularity criteria involving both excess functionals and nonlinear potentials

regularity for quasilinear equations on degenerate singular sets we prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy fancyscript s u x d u x 0 our main theorem reveals that along fancyscript s u u is asymptotically as regular as solutions to constant coefficient equations in particular along the critical set fancyscript s u du enjoys a modulus of continuity much superior than the possibly low continuity feature of the coefficients the results are new even in the context of linear elliptic equations where it is herein shown that h 1 weak solutions to text div left a ij x du right 0 with a ij elliptic and dini continuous are actually c 1 1 along fancyscript s u the results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations beyond typical elliptic regularity estimates precisely where the diffusion attributes of the equation collapse

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

gradient estimates for nonlinear elliptic double obstacle problems abstract we study a nonlinear elliptic double obstacle problem with irregular data and establish an optimal calderon zygmund theory the partial differential operator is of the p laplacian type and includes merely measurable coefficients in one variable we prove that the gradient of a weak solution is as integrable as both the gradient of assigned two obstacles and the nonhomogeneous divergence term under a small bmo semi norm assumption on the coefficients in the other variables

higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients abstract we study the local regularity of vectorial minimizers of integral functionals with standard p growth we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity under a w 1 n sobolev dependence on the spatial variable of the integrand n being the space dimension we show that the minimizers have the gradient locally in l q for every q p as a consequence they are locally holder continuous for every 1

weighted gradient estimates for the general elliptic p x laplacian equations abstract in this paper we obtain the weighted l q estimates of the gradients of weak solutions for the general elliptic p x laplacian equations div a x u div f p x 2 f under some proper assumptions on a p f we prove that f p x l w loc q u p x l w loc q for any q 1

gradient estimates via linear and nonlinear potentials we prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems involving possibly degenerate quasilinear operators whose prototype is given by pu in particular no matter the nonlinearity of the equations considered we show that in the case p 2 a pointwise gradient estimate is possible using standard linear riesz potentials the proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem and on the other allows to encode the weaker coercivity properties of the operators considered in the case p 2 in the case p 2 we prove a new gradient estimate employing nonlinear potentials of wolff type

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

good lambda and muckenhoupt wheeden type bounds in quasilinear measure datum problems with applications weighted good lambda type inequalities and muckenhoupt wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data such results are obtained globally over sufficiently flat domains in mathbb r n in the sense of reifenberg the principal operator here is modeled after the p laplacian where for the first time singular case frac 3n 2 2n 1 p le 2 frac 1 n is considered those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data as an application sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear riccati type equation having a gradient source term with linear or super linear power growth

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order