the singular set of solutions to non differentiable elliptic systems we estimate the hausdorff dimension of the singular set of solutions to elliptic systems of the type if the vector fields a and b are holder continuous with respect to the variable x with exponent then the hausdorff dimension of the singular set of any weak solution is at most n 2

on the singular set for lipschitzian critical points of polyconvex functionals abstract partial regularity is proved for lipschitzian critical points of polyconvex functionals provided d u l is small enough in particular the singular set for a lipschitzian critical point has hausdorff dimension strictly less than n when d u l is small enough model problems treated include u 2 det u 2 where u r 2 r 2 and u 2 u s ad u s det u s where u r 3 r 3 with s 2 moreover it is shown that the singular set of a lipschitzian global minimizer has hausdorff dimension strictly less than n

elimination and nonlinear equations of rees algebra abstract a new approach is established to computing the image of a rational map whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees in the case the map is everywhere defined this analysis provides free resolutions of graded parts of the rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra a surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion an additional point is that this contribution which of course corresponds to nonlinear equations of the rees algebra can be described in these degrees in terms of non koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by j herzog the third named author and w vasconcelos as a measure of the reach of this torsion analysis we could say that in the case of a general everywhere defined map half of the degrees where the torsion does not vanish are understood

from concentration to logarithmic sobolev and poincare inequalities abstract we give a new proof of the fact that gaussian concentration implies the logarithmic sobolev inequality when the curvature is bounded from below and also that exponential concentration implies poincare inequality under null curvature condition our proof holds on non smooth structures such as length spaces and provides a universal control of the constants we also give a new proof of the equivalence between dimension free gaussian concentration and talagrand s transport inequality

regularity in parabolic systems with continuous coefficients under the only assumption of continuous coefficients we prove a partial holder continuity result for solutions to parabolic systems with polynomial growth a key component throughout the argument is the use of dibenedetto s intrinsic geometry degenerate parabolic equations universitext springer new york 1993 to accommodate the inhomogeneity in the system a main technical point is that although we are proving the holder continuity of solutions we employ the intrinsic geometry using cylinder stretched according to the size of the spatial gradient in an iteration scheme that does not necessarily imply the boundedness of the gradient itself

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div 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 p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

asymptotically regular problems i higher integrability abstract we consider weak solutions u of non linear systems of partial differential equations assuming that the system exhibits a certain kind of elliptic behavior near infinity we prove higher integrability results for the gradient du in particular we establish holder continuity of u in low dimensions moreover we obtain analogous results for vectorial minimizers of multi dimensional variational integrals finally we discuss an extension to minimizing sequences and applications to generalized minimizers

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

regularity of relaxed minimizers of quasiconvex variational integrals with p q growth we consider autonomous integrals f u int omega f du rm d x quad rm for u mathbb r n supset omega to mathbb r n in the multidimensional calculus of variations where the integrand f is a strictly quasiconvex c2 function satisfying the p q growth conditions gamma a p leqq f a leqq gamma 1 a q quad rm for every a in mathbb r nn with exponents 1 p leqq q infty

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

the singular set of lipschitzian minima of multiple integrals the singular set of any lipschitzian minimizer of a general quasiconvex functional is uniformly porous and hence its hausdorff dimension is strictly smaller than the space dimension

interior c0 γ regularity for vector valued minimizers of quasilinear functionals abstract we consider weak minimizers of variational integrals whose integrands are quasilinear under suitable conditions on the integrand we obtain results on interior everywhere h lder regularity for the weak minimizers more precise results on the partial regularity can be deduced from our main result too some illustrating examples are given at the end of the paper

quasilinear elliptic equations on noncompact riemannian manifolds abstract the existence of solutions to a class of quasilinear elliptic problems on noncompact riemannian manifolds with finite volume is investigated boundary value problems with homogeneous neumann conditions in possibly irregular euclidean domains are included as a special instance a nontrivial solution is shown to exist under an unconventional growth condition on the right hand side which depends on the geometry of the underlying manifold the identification of the critical growth is a crucial step in our analysis and entails the use of the isocapacitary function of the manifold a condition involving its isoperimetric function is also provided

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

on the lagrangian structure of transport equations the vlasov poisson system the vlasov poisson system is an important nonlinear transport equation used to describe the evolution of particles under their self consistent electric or gravitational field the existence of classical solutions is limited to dimensions d 3 under strong assumptions on the initial data whereas weak solutions are known to exist under milder conditions however in the setting of weak solutions it is unclear whether the eulerian description provided by the equation physically corresponds to a lagrangian evolution of the particles in this article we develop several general tools concerning the lagrangian structure of transport equations with nonsmooth vector fields and we apply these results to show that weak renormalized solutions of vlasov poisson are lagrangian and actually that the concepts of renormalized and lagrangian solutions are equivalent as a corollary we prove that finite energy solutions in dimension d 4 are transported by a global flow in particular they preserve all the natural casimir invariants and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data

the singular set of ω minima we consider minima open image in new window of convex variational integrals in the vectorial case n n 2 and we provide estimates for the hausdorff dimension of their singular sets

gradient lq theory for a class of non diagonal nonlinear elliptic systems abstract we consider a class of nonlinear non diagonal elliptic systems with p growth and establish the l q integrability for all q p p 2 of any weak solution provided the corresponding right hand side belongs to the corresponding lebesgue space and the involved elliptic operator asymptotically satisfies the p uniform ellipticity the so called splitting condition and it is continuous with respect to the spatial variable for operators satisfying the uniform p ellipticity condition the higher integrability is known for q p d p d 2 and for operators having the so called uhlenbeck structure the theory is valid for all q p the key novelty of the paper is twofold first the statement uses only the information coming from the asymptotic operator and second and more importantly by using the splitting condition we are able to extend the range of possible q s significantly whenever p d 2

harmonic type approximation lemmas we give a survey of known and not known harmonic type approximation lemmas which are descendants of the classical de giorgi s one and we outline some of their recent or possible applications

partial regularity for minimizers of discontinuous quasi convex integrals with degeneracy abstract in this paper we are concerned with the regularity of minimizers u w 1 p r n of quasi convex integral functionals of the type f u f x u d u d x the crucial point here is that the integrand f admits very weak regularity properties with respect to the gradient variable it satisfies degenerate singular p growth conditions without necessarily possessing a quasi diagonal uhlenbeck type structure and with respect to the x variable the integrand might be even discontinuous it is only assumed that a certain vmo condition holds under those assumptions we prove partial holder continuity of minimizers i e holder continuity of u for any holder exponent 0 1 outside a set of measure zero under such weak assumptions regularity results for the gradient of minimizers is not expected to hold since even in the scalar case counterexamples to c 1 regularity are known

partial holder regularity for elliptic systems with non standard growth abstract we study partial holder regularity for elliptic systems with non standard growth we consider general systems with orlicz growth and discontinuous coefficient factor for which we prove that their weak solutions are partially holder continuous for any holder exponents in addition we also obtain a similar result for systems with double phase growth

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

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

a trace preserving operator and applications abstract we construct a trace preserving operator which improves the integrability of functions in sobolev classes refining the ones available in literature as applications we prove a c 1 partial regularity result for local minimizers of quasiconvex integral functionals satisfying non standard p q growth conditions in the borderline case p n 1 and q n and a global integrability result for weak solutions to a nonlinear elliptic system

partial regularity for subquadratic homogeneity elliptic system with vmo coefficients abstract in this paper we are concerned with subquadratic homogeneity elliptic problems with vmo coefficients in divergence form we obtain that the weak solution u is locally holder continuous besides a singular set by using a harmonic approximation where the holder exponent 0 1 the hausdorff dimension of the singular set is less than n p

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

gradient continuity for p laplace systems abstract we prove the continuity of the gradient of weak solutions to nonhomogeneous p laplace systems with coefficients we assume that the nonhomogeneous term belongs to the lorentz space l n 1 the coefficient is dini continuous and the variable exponent p is logarithmic dini type continuous

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

calderon zygmund estimates for higher order systems with p x growth for weak solutions u in w m 1 omega mathbb r n of higher order systems of the type int omega left dx int omega left dx for all phi in c infty c omega mathbb r n m 1 with variable growth exponent p 1 we prove that if f p cdot in l q loc omega with 1 0 and in the degenerate case 0

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

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

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

marcinkiewicz estimates for degenerate parabolic equations with measure data we consider solutions to degenerate parabolic equations with measurable coefficients having on the right hand side a measure satisfying a suitable density condition we prove integrability results for the gradient in the marcinkiewicz scale

from u bounds to isoperimetry with applications to h type groups abstract in this paper we study u bounds in relation to l 1 type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space r n d we prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework the usefulness of our approach is illustrated by an application to the setting of h type groups and an extension to infinite dimensions

variable lorentz estimate for conormal derivative problems of stationary stokes system with partially bmo coefficients abstract we prove a global calderon zygmund type estimate in the framework of lorentz spaces for the variable power of the gradients to the solution pair u p of the conormal derivative problems of stationary stokes system it is mainly assumed that the leading coefficients are merely measurable in one of the spatial variables and have sufficiently small bounded mean oscillation seminorm in the other variables the boundary of domain belongs to the reifenberg flatness and the variable exponents p x satisfy the log holder continuity

gradient regularity for elliptic equations in the heisenberg group abstract we give dimension free regularity conditions for a class of possibly degenerate sub elliptic equations in the heisenberg group exhibiting super quadratic growth in the horizontal gradient this solves an issue raised in j j manfredi g mingione regularity results for quasilinear elliptic equations in the heisenberg group math ann 339 2007 485 544 where only dimension dependent bounds for the growth exponent are given we also obtain explicit a priori local regularity estimates and cover the case of the horizontal p laplacean operator extending some regularity proven in a domokos j j manfredi c 1 regularity for p harmonic functions in the heisenberg group for p near 2 in contemp math vol 370 2005 pp 17 23 in turn using some recent techniques of caffarelli and peral l caffarelli i peral on w 1 p estimates for elliptic equations in divergence form comm pure appl math 51 1998 1 21 the a priori estimates found are shown to imply the suitable local calderon zygmund theory for the related class of non homogeneous possibly degenerate equations involving discontinuous coefficients these last results extend to the sub elliptic setting a few classical non linear euclidean results t iwaniec projections onto gradient fields and l p estimates for degenerated elliptic operators studia math 75 1983 293 312 e dibenedetto j j manfredi on the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems amer j math 115 1993 1107 1134 and to the non linear case estimates of the same nature that were available in the sub elliptic setting only for solutions to linear equations

survey on the theory and applications of μ bases for rational curves and surfaces abstract bases are new representations for rational curves and surfaces which serve as a bridge between their parametric forms and implicit forms geometrically bases are represented by moving lines or moving planes while their algebraic counterparts are special syzygies of the parametric equations of rational curves or surfaces bases have been proven to be significant in solving many important problems in geometric modeling such as fast implicitization singularity computation reparametrization as well as providing easy inversion formulas for points we review the state of the art results in bases theory and applications for rational curves and surfaces and raise unsolved problems for future research

global higher integrability of minimizers of variational problems with mixed boundary conditions abstract we consider integral functionals with densities of p growth with respect to gradients on a lipschitz domain with mixed boundary conditions the aim of this paper is to prove that under uniform estimates within certain classes of p growth and coercivity assumptions on the density the minimizers are of higher integrability order meaning that they belong to the space of first order sobolev functions with an integrability of order p for a uniform 0 the results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys

on the closed image of a rational map and the implicitization problem in this paper we investigate some topics around the closed image s of a rational map lambda given by some homogeneous elements f 1 ldots f n of the same degree in a graded algebra a we first compute the degree of this closed image in case lambda is generically finite and f 1 ldots f n define isolated base points in proj a we then relate the definition ideal of s to the symmetric and the rees algebras of the ideal i f 1 ldots f n subset a and prove some new acyclicity criteria for the associated approximation complexes finally we use these results to obtain the implicit equation of s in case s is a hypersurface proj a pp n 2 k with k a field and base points are either absent or local complete intersection isolated points

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

regularity results for a class of non differentiable obstacle problems abstract in this paper we prove the higher differentiability in the scale of besov spaces of the solutions to a class of obstacle problems of the type min f x z d z z k here is an open bounded set of r n n 2 is a fixed function called obstacle and k is set of admissible functions z w 1 p such that z a e in we assume that the gradient of the obstacle belongs to a suitable besov space the main novelty here is that we are not assuming any differentiability on the partial maps x f x z d z and z f x z d z but only their holder continuity

a new partial regularity result for non autonomous convex integrals with non standard growth conditions abstract we establish c 1 partial regularity of minimizers of non autono mous convex integral functionals of the type f u f x d u d x with non standard growth conditions into the gradient variable 1 l p f x l 1 q for a couple of exponents p q such that 1 p q min p n n 1 p 1 and holder continuous dependence with respect to the x variable the significant point here is that the distance between the exponents p and q is independent of moreover this bound on the gap between the growth and the coercitivity exponents improves previous results in this setting

weak solution for neumann p q laplacian problem on riemannian manifold abstract we prove the existence of a nontrivial solution for a nonlinear p q laplacian problem with neumann boundary condition on a non compact riemannian manifold the idea is to reduce the problem in variational form which means to consider the critical points of the corresponding euler lagrange functional in an orlicz sobolev space

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

rees algebras and almost linearly presented ideals abstract consider a grade 2 perfect ideal i in r k x 1 x d which is generated by forms of the same degree assume that the presentation matrix is almost linear that is all but the last column of consist of entries which are linear for such ideals we find explicit forms of the defining ideal of the rees algebra r i we also introduce the notion of iterated jacobian duals

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

a generalization of caffarelli s contraction theorem via reverse heat flow a theorem of l caffarelli implies the existence of a map pushing forward a source gaussian measure to a target measure which is more log concave than the source one which contracts euclidean distance in fact caffarelli showed that the optimal transport brenier map topt is a contraction in this case we generalize this result to more general source and target measures using a condition on the third derivative of the potential by providing two different proofs the first uses a map t whose inverse is constructed as a flow along an advection field associated to an appropriate heat diffusion process the contraction property is then reduced to showing that log concavity is preserved along the corresponding diffusion semi group by using a maximum principle for parabolic pde in particular caffarelli s original result immediately follows by using the ornstein uhlenbeck process and the prekopa leindler theorem the second uses the map topt by generalizing caffarelli s argument employing in addition further results of caffarelli as applications we obtain new correlation and isoperimetric inequalities

poincare modified logarithmic sobolev and isoperimetric inequalities for markov chains with non negative ricci curvature abstract we study functional inequalities for markov chains on discrete spaces with entropic ricci curvature bounded from below our main results are that when curvature is non negative but not necessarily positive the spectral gap the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space with respect to a suitable metric these estimates are discrete analogues of classical results of riemannian geometry obtained by li and yau buser and wang

minimization of a quasi linear ginzburg landau type energy abstract let g be a smooth bounded domain in r 2 consider the functional e e u 1 2 g p 0 t x k u l u 2 1 4 e 2 g 1 u 2 2 on the set h g 1 g c u h 1 g c u g on g where g is a given boundary data with degree d 0 in this paper we will study the behavior of minimizers u e of e e and we will estimate the energy e e u e

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

the regularity of minima for the dirichlet problem on bd we establish that the dirichlet problem for linear growth functionals on text bd the functions of bounded deformation gives rise to the same unconditional sobolev and partial text c 1 alpha regularity theory as presently available for the full gradient dirichlet problem on text bv functions of bounded deformation play an important role in for example plasticity however by ornstein s non inequality contain text bv as a proper subspace thus techniques to establish regularity by full gradient methods for variational problems on bv do not apply here in particular applying to all generalised minima that is minima of a suitably relaxed problem despite their non uniqueness and reaching the ellipticity ranges known from the text bv case this paper extends previous sobolev regularity results by gmeineder and kristensen in j calc var 58 56 2019 in an optimal way

the gehring lemma dimension free estimates abstract we present a dimension free estimate for the exponent of higher integrability in gehring s lemma for an inhomogeneous weak reverse holder inequality

quasiconvexity and partial regularity in the calculus of variations we prove partial regularity of minimizers of certain functionals in the calculus of variations under the principal assumption that the integrands be uniformly strictly quasiconvex this is of interest since quasiconvexity is known in many circumstances to be necessary and sufficient for the weak sequential lower semicontinuity of these functionals on appropriate sobolev spaces examples covered by the regularity theory include functionals with integrands which are convex in the determinants of various submatrices of the gradient matrix

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

on coercive variational integrals abstract it is well known that sequential weak lower semicontinuity of a variational integral f u f u x d x on the sobolev space w 1 p r n under a p growth condition on the integrand f is equivalent to quasiconvexity in the sense of morrey we show that coercivity on dirichlet classes likewise is equivalent to a quasiconvexity condition we also discuss some examples and extend a sequential weak lower semicontinuity result to the case of signed integrands

non lipschitz minimizers of smooth uniformly convex functionals we construct non lipschitz minimizers of smooth uniformly convex functionals of type i u f du x dx our method is based on the use of null lagrangians

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

partial holder continuity for solutions of subquadratic elliptic systems in low dimensions abstract we consider weak solutions of second order nonlinear elliptic systems in divergence form under standard subquadratic growth conditions with boundary data of class c 1 in dimensions n 2 3 we prove that u is locally holder continuous for every exponent 0 1 n 2 p outside a singular set of hausdorff dimension less than n p this result holds up to the boundary both for non degenerate and degenerate systems in the proof we apply the direct method and classical morrey type estimates introduced by campanato

fractional differentiability for solutions of a class of parabolic systems with l1 θ data abstract we study the higher differentiability of the very weak solution of the cauchy dirichlet problem associated to the linear parabolic system of the form u t div a x t d u in 0 t u 0 on 0 t u 0 on 0 under a weak holder continuity condition on the coefficients and the right hand side belonging to the morrey space l 1 in such a way we extend to the vectorial parabolic setting the result by di castro and palatucci 2013

calderon zygmund estimates for parabolic measure data equations abstract we consider parabolic equations of the type u t div a x t d u having a radon measure on the right hand side and prove fractional integrability and differentiability results of calderon zygmund type for weak solutions we extend some of the integrability results for elliptic equations achieved by g mingione 2007 24 to the parabolic setting and locally recover the integrability results of l boccardo a dall aglio t gallouet and l orsina 1997 in 5

regularity in oscillatory nonlinear elliptic systems we give uniform bounds in morrey sobolev spaces h1 for weak solutions ue to oscillatory nonlinear elliptic systems the main novelty is in the precise qualitative description of the e uniform morrey exponent in terms of the ellipticity ratio of the elliptic systems this is achieved by showing that weak solutions of general nonlinear elliptic systems with lipschitz coefficients are of class w2 p locally or globally depending on assumptions for an exponent p 2 that is independent of dimensions and for which p 2 is inversely proportional to the ellipticity ratio