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

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

on existence theorems of positive solutions to nonlinear singular differential equations abstract suppose i f t u 0 1 0 0 is continuous and is increasing on u ii there exists a function g 1 0 g b b and g b b 2 is integrable on 1 such that f t bu g b f t u t u 0 1 0 consider the singular boundary value problem u t f t u t 0 0 t 1 u 0 au u 1 0 then a necessary and sufficient condition for eq to have c 0 1 positive solutions is that 0 0 1 s 1 s f s 1 d s a necessary and sufficient condition for eq to have c 1 0 1 positive solutions is that 0 0 1 f s 1 s d s also the uniqueness iterative methods and computational methods of c 1 0 1 positive solutions have been studied our nonlinearity may be singular at t 0 and or t 1

a regularity theorem for minimizers of quasiconvex integrals we prove c1 partial regularity for minimizers of functionals with quasiconvex integrand f x u du depending on vector valued functions u the integrand is required to be twice continuously differentiable in du and no assumption on the growth of the derivatives of f is made a polynomial growth is required only on f itself

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

partial regularity of strong local minimizers in the multi dimensional calculus of variations let n be a bounded domain and f a given strongly quasiconvex integrand of class c2 satisfying the growth condition f xi le c 1 xi p for some c 0 and 2 p consider the multiple integral i u int omega f nabla u where u w1 p n the main result of the paper is the proof that any strong local minimizer of i is of class c1 loc for any 0 1 on an open set of full n dimensional measure in the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large without such an assumption weak local minimizers need not be partially regular as we show by a class of examples we also briefly discuss the question of existence of strong local minimizers for i and connections of our results to extensions of weierstrass sufficiency theorem to the multi dimensional setting

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

real analytic approximation of lipschitz functions on hilbert space and other banach spaces let x be a separable banach space with a separating polynomial we show that there exists c 1 depending only on x such that for every lipschitz function f x r and every epsilon 0 there exists a lipschitz real analytic function g x r such that vertical bar f x g x vertical bar epsilon e and lip g c lip f this result is new even in the case when x is a hilbert space furthermore in the hilbertian case we also show that c can be assumed to be any number greater than i

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

weakly continuous mappings on banach spaces abstract it is shown that every n homogeneous continuous polynomial on a banach space e which is weakly continuous on the unit ball of e is weakly uniformly continuous on the unit ball of e applications of the result to spaces of polynomials and holomorphic mappings on e are given

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

smooth approximations without critical points of continuous mappings between banach spaces and diffeomorphic extractions of sets abstract let e f be separable hilbert spaces and assume that e is infinite dimensional we show that for every continuous mapping f e f and every continuous function e e 0 there exists a c mapping g e f such that f x g x e x and d g x e f is a surjective linear operator for every x e we also provide a version of this result where e can be replaced with a banach space from a large class including all the classical spaces with smooth norms such as c 0 l p or l p 1 p and f can be taken to be any banach space such that there exists a bounded linear operator from e onto f in particular for such e f every continuous mapping f e f can be uniformly approximated by smooth open mappings part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from banach spaces or hilbert manifolds

normal forms and invariant manifolds for nonlinear non autonomous pdes viewed as odes in infinite dimensions abstract we prove that a general class of nonlinear non autonomous ode s in frechet spaces are close to ode s in a specific normal form where closeness means that solutions of the normal form ode satisfy the original ode up to a residual that vanishes up to any desired order in this normal form the centre stable and unstable coordinates of the ode are clearly separated which allows us to define invariant manifolds of such equations in a robust way the main motivation is the case where the frechet space in question is a suitable function space and the maps involved in an ode in this space are defined in terms of derivatives of the functions so that the infinite dimensional ode is a finite dimensional pde we show that our methods apply to a relevant class of nonlinear non autonomous pde s in this way

smooth negligibility of compact sets in infinite dimensional banach spaces with applications this article deals with smooth removability of compact sets in infinite dimensional banach spaces the main result states that ifx is an infinite dimensional banach space which has a not necessarily equivalent cp smooth norm and k is a compact subset of x then x and x r k are cp diffeomorphic the proof relies on the construction of a deleting path through a nontrivial refinement of bessaga s incomplete norm technique however norms are not at present available and the construction requires the use of asymmetric functionals the noncompleteness of such functionals relies in turn on james theorem on existence of linear functionals which do not attain their norm on every nonreflexive space applications are given which show that several important theorems on finite dimensional spaces completely fail in the infinite dimensional case for instance on any banach space isomorphic to its cartesian square and for any natural number n 2 there exists a c1 diffeomorphism of pure period n with no fixed point this work opens the way to several interesting open questions on nonseparable banach spaces does every banach space with a c1 smooth norm admit a nonequivalent c1 smooth norm in which banach spaces is every compact subset the set where a certain c1 real valued function vanishes

extensions of smooth mappings into biduals and weak continuity abstract we study properties of uniformly differentiable mappings between real banach spaces among our main results are generalizations of a number of classical results for linear operators on l spaces into the setting of uniformly differentiable mappings denote by b x the closed unit ball of a banach space x let x be a l space 1 and let y be a banach space let t b x y be a continuous mapping which is uniformly differentiable in the open unit ball of x assuming that t is weakly compact then t can be extended preserving its best smoothness properties into the mapping from the 1 multiple of the unit ball of any superspace of the domain space x into the same range space y we also show that t maps weakly cauchy sequences from b x into norm convergent sequences in y this is a uniformly smooth version of the dunford pettis property for the l spaces we also show that a uniformly differentiable mapping t which is not necessarily weakly compact still maps weakly cauchy sequences from b x into norm convergent sequences in y provided y does not contain an isomorphic copy of c 0 we prove that for certain pairs of banach spaces the completion of the space of polynomials equipped with the topology of uniform convergence on the bounded sets of the functions and their derivatives up to order k coincides with the space of uniformly differentiable up to order k mappings our work is based on a number of tools that are of independent interest we prove for every pair of banach spaces x y that any continuous mapping t b x y which is uniformly differentiable of order up to k in the interior of b x can be extended preserving its best smoothness into a bidual mapping t b x y another main tool is a result of zippin s type we show that weakly cauchy sequences in x c k can be uniformly well approximated by weakly cauchy sequences from a certain c 0 is a countable ordinal subspace of x

polynomial algebras and smooth functions in banach spaces abstract let a n x be the algebra of polynomials on a real banach space x which is generated by all continuous polynomials of degree not exceeding n let m be the minimal integer such that there is a non compact m homogeneous polynomial p p x m l 1 then n m implies that the uniform closure of a n x does not contain all polynomials of degree n 1 and hence the chain of closures a n x n m is strictly increasing in the rest of the note we give solutions to three problems concerning the behaviour of smooth functions on banach spaces posed in the literature in particular we construct an example of a uniformly differentiable real valued function f on the unit ball of a certain banach space x such that there exists no uniformly differentiable function g on b x for any 1 which coincides with f in some neighbourhood of the origin

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

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