partial regularity for holonomic minimisers of quasiconvex functionals we prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals over a class of sobolev mappings into a compact riemannian manifold to which such mappings are said to be holonomically constrained our approach uses the lifting of sobolev mappings to the universal covering space the connectedness of the covering space an application of ekeland s variational principle and a certain tangential mathbb a harmonic approximation lemma obtained directly via a lipschitz approximation argument this allows regularity to be established directly on the level of the gradient several applications to variational problems in condensed matter physics with broken symmetries are also discussed in particular those concerning the superfluidity of liquid helium 3 and nematic liquid crystals

solutions of nonlinear dirac equations abstract we study the dirac equation i t i c k 1 3 k k m c 2 g x and obtain existence and multiplicity results of stationary solutions for several classes of nonlinearities g r 3 c 4 r modeling various types of interaction a typical result states that if g x u depends periodically on x and is even in u the problem has infinitely many geometrically different localized solutions the arguments are variational the associated lagrangian functional is strongly indefinite and the palais smale condition does not hold we apply some recently developed critical point theorems

ground state solutions for nonperiodic dirac equation with superquadratic nonlinearity this paper is concerned with the following nonperiodic dirac equation i k 13 k ku a u m x u fu x u in r3 where m x is a vector potential under weak superquadratic condition on the nonlinearity we establish the existence of ground state solutions of nehari type by using the generalized nehari manifold method developed recently by szulkin and weth for above problem

rank one property for derivatives of functions with bounded variation when u is a function of bounded variation of the open set r into r du is a measure on which takes values in the space rm n of all m n matrices such that du i j ui xj for every i j with 1 i m 1 j n and we denote by dsu the part of the measure du which is singular with respect to lebesgue measure we recall that the rank of a matrix a rm n is dimension of the space spanned by its columns in r or by its rows in r and in particular the rank of a is one if and only if there exist e r and r with e 6 0 6 0 so that a e i e ai j ei j for all i j cf remark 1 7 when is a measure which takes values in the space rm n and d d denotes the radon nikodym derivative of with respect to the total variation we say that has rank one when the rank of the matrix d d x is one for almost every x in this paper we prove we prove the following statement see theorem 4 5 corollary 4 6 and remark 4 9

on rank one convex functions that are homogeneous of degree one we show that positively 1 homogeneous rank one convex functions are convex at 0 and at matrices of rank one the result is a special case of an abstract convexity result that we establish for positively 1 homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space from these results we derive a number of consequences including various generalizations of the ornstein l1 non inequalities most of the results were announced in c r acad sci paris ser i 349 407 409 2011

on global solutions of the maxwell dirac equations we prove for the maxwell dirac equations in 1 3 dimensions that modified wave operators exist on a domain of small entire test functions of exponential type and that the cauchy problem inr r3 has a unique solution for each initial condition att 0 which is in the image of the wave operator the modification of the wave operator which eliminates infrared divergences is given by approximate solutions of the hamilton jacobi equation for a relativistic electron in an electromagnetic potential the modified wave operator linearizes the maxwell dirac equations to their linear part

semi classical ground states concentrating on the nonlinear potential for a dirac equation abstract we study the semi classical limit of the least energy solutions to the nonlinear dirac equation i e k 1 3 k k u a u p x u p 2 u for x r 3 since the dirac operator is unbounded from below and above the associate energy functional is strongly indefinite and since the problem is considered in the global space r 3 the palais smale condition is not satisfied new phenomena and mathematical interests arise in the use of the calculus of variations we prove that the equation has the least energy solutions for all e 0 small and additionally these solutions converge to the least energy solutions of the associate limit problem and concentrate to the maxima of the nonlinear potential p x in certain sense as e 0

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

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

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

solutions of a nonlinear dirac equation with external fields we study the stationary dirac equation ic hbar sum 3 k 1 alpha k partial k u mc 2 beta u m x u r u x u where m x is a matrix potential describing the external field and r x u stands for an asymptotically quadratic nonlinearity modeling various types of interaction without any periodicity assumption for fixed our discussion includes the coulomb potential as a special case and for the semiclassical situation 0 we handle the scalar fields we obtain existence and multiplicity results of stationary solutions via critical point theory

positive solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x u p 1 u x r 3 k x u 2 x r 3 with p 3 5 assuming that a r 3 r and k r 3 r are nonnegative functions such that lim x a x a 0 lim x k x 0 and satisfying suitable assumptions but not requiring any symmetry property on them we prove the existence of positive solutions

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

analysis of a class of multi stage multistep runge kutta methods abstract in this paper a class of methods that numerically solve initial value problems for second order ordinary differential equations of the form y f x y x is investigated methods in this class are two step implicit runge kutta methods with s internal stages that do not require an update of y there are many examples in the literature of methods which conform to our format using a type of nystrom tree and a corresponding special type of nystrom series the order conditions for this method are developed with this technique of putting order conditions in terms of trees we obtain a set of simplifying conditions that serve as a framework for generating and analyzing higher order methods our analysis affords the development of a two parameter family of eight order methods the issue of maximum obtainable order for unconditionally stable s stage methods is investigated for s 1 2 when implemented these methods in general require at each step the solution of an algebraic equation of the form y m im f y y r n where m is an s 1 s 1 matrix to facilitate solving this equation we develop a method where m is lower triangular

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

stationary solutions of non autonomous maxwell dirac systems abstract the maxwell dirac system describes the interaction of a particle with its self generated electromagnetic field in this paper we study the existence of least energy stationary solutions for non autonomous maxwell dirac system with a nonlinear term in 3 1 minkowski space time via variational methods this problem is strongly indefinite and presents a lack of compactness to overcome these difficulties we will use the linking and concentration compactness arguments

high order p stable multistep methods abstract recently numerical integration of the special ivp y f t y whose solutions are of periodic type have been considered of great interest p stability is the appropriate requirement for numerical methods integrating periodic problems as established by lambert and watson 1976 a p stable linear multistep method has order no greater than two in this paper two step multi stage linear methods are proposed for numerical integration of periodic problems and p stable methods of this type obtained in a simple form up to order eight are shown finally those methods are successfully tested

infinitely many solutions for a class of nonlinear dirac equations without symmetry abstract this paper is concerned with the multiplicity of solutions for the nonlinear dirac equations i k 1 3 k k u v x a u u f u x u without symmetry assumption on f we establish the existence of infinitely many geometrically distinct solutions for superquadratic as well as asymptotically quadratic nonlinearities via variational approach

partial regularity for bv minimizers we establish an varepsilon regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth and as a consequence the partial regularity of such bv minimizers this result extends the regularity theory for minimizers of quasiconvex integrals on sobolev spaces to the context of maps of bounded variation previous partial regularity results for bv minimizers in the linear growth set up were confined to the convex situation

partial regularity of local minimizers of quasiconvex integrals with sub quadratic growth we prove partial regularity for minimizers of quasiconvex integrals of the form int limits omega f left du x right dx where the integral f has subquadratic growth ie left f xi right le l 1 xi p

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

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

existence of stationary states for nonlinear dirac equations abstract we generalize the previous result of cazenave and vasquez on the existence of stationary states for nonlinear dirac equations of the form i 3 0 m l 0 we seek solutions which are separable in spherical coordinates and we then make use of a shooting method to solve the associated problem for ordinary differential equations

ground states for nonlinear maxwell dirac system with magnetic field abstract in this paper we study the following maxwell dirac system with magnetic field and general nonlinearity k 1 3 k i k a k u a u u a 0 u f u x u a 0 4 u u 4 u 2 a k 4 k u u k 1 2 3 for x r 3 since this problem is strongly indefinite we use linking and concentration compactness arguments to establish the existence of ground state solutions for this system

a mono implicit runge kutta nystrom modification of the numerov method we present two two parameter families of fourth order mono implicit runge kutta nystrom methods each member of these families can be considered as a modification of the numerov method we analyze the stability and periodicity properties of these methods it is shown that i within one of these families there exist a stable even l stable and p stable methods and ii in both families there exist methods with a phase lag of order six

multiple solutions for a class of nonperiodic dirac equations with vector potentials abstract this paper is concerned with the nonperiodic dirac equations i k 1 3 k k u a u m x u f u x u in r 3 u x 0 as x where m x is a vector potential we introduce a new assumption on the potential m and prove that the corresponding dirac operator has only a point spectrum moreover by applying a generalized linking theorem for strongly indefinite functionals we establish the existence of multiple solutions for asymptotically quadratic nonlinearity as well as the existence of infinitely many solutions for superquadratic nonlinearity

on the relaxation in bv ω rm of quasi convex integrals abstract given a quasi convex function f with linear growth we find the integral representation in bv r m of the functional f arising from the relaxation of f u f u dx u c 1 r m in the l loc 1 r m topology

the regularity of critical points of polyconvex functionals in this paper we are concerned with the question of regularity of critical points for functionals of the type eq1 we construct a smooth strongly polyconvex eq2 and lipschitzian weak solutions eq3 to the corresponding euler lagrange system which are nowhere c1 moreover we show that f can be chosen in such a way that these irregular weak solutions are weak local minimisers

an improved trigonometrically fitted p stable obrechkoff method for periodic initial value problems in this paper we present an improved p stable trigonometrically fitted obrechkoff method with phase lag frequency distortion infinity compared with the previous p stable trigonometrically fitted obrechkoff method developed by simos our new method is simpler in structure and more stable in computation we have also improved the accuracy of the first order derivative formula from the numerical illustration presented we can show that the new method is much more accurate than the previous methods

families of two step fourth order p stable methods for second oder differential equations this paper deals with a class of symmetric hybrid two step fourth order p stable methods for the numerical solution of special second order initial value problems such methods were proposed independently by cash 1 and chawla 3 and normally require three function evaluations per step the purpose of this paper is to point out that there are some values of the free parameters available in the methods proposed which can reduce this work we study two classes of such methods the first is the class of economical methods see definition 3 1 which reduce this work to two function evaluations per step and the second is the class of efficient methods see definition 3 2 which reduce this work with respect to implementation for nonlinear problems we report numerical experiments to illustrate the order acuracy and implementational aspects of these two classes of methods

characterization of generalized gradient young measures generated by sequences in w1 1 and bv generalized young measures as introduced by diperna and majda commun math phys 108 667 689 1987 provide a quantitative tool for studying the one point statistics of oscillation and concentration in sequences of functions in this work after developing a functional analytic framework for such measures including a compactness theorem and results on the generation of such young measures by l1 bounded sequences or even by sequences of bounded radon measures we turn to investigation of those young measures that are generated by bounded sequences of w1 1 gradients or bv derivatives we provide several techniques to manipulate such measures including shifting averaging and approximation by piecewise homogeneous young measures and then establish the main new result of this work the duality characterization of the set of bv or w1 1 gradient young measures in terms of jensen type inequalities for quasiconvex functions with linear growth at infinity this result is the natural generalization of the kinderlehrer pedregal theorem arch ration mech anal 115 329 365 1991 j geom anal 4 59 90 1994 for classical young measures to the w1 1 and bv case and contains its version for weakly converging sequences in w1 1 as a special case finally we give an application to a new lower semicontinuity theorem in bv

new conditions on nonlinearity for a periodic schrodinger equation having zero as spectrum abstract we consider the semilinear schrodinger equation u v x u f x u for x r n u x 0 as x where f is a superlinear and subcritical nonlinearity we mainly study the case when both v and f are periodic in x and 0 is a boundary point of a spectral gap of v we extend a linking theorem of kryszewski and szulkin 15 and establish a new variational setting which is more suitable to the above case we obtain two theorems on the existence of ground state solutions with mild assumptions on f

existence and multiplicity of stationary solutions for a class of maxwell dirac system abstract this paper is concerned with the following maxwell dirac system i k 1 3 k k u a u m x u k x u f u x u 4 k x u 2 in r 3 where m x is a external potential and f x u is an asymptotically quadratic nonlinearity modeling various types of interactions in view of the effects of the nonlocal term we use some special techniques to deal with the nonlocal term moreover existence and multiplicity of stationary solutions are obtained for system without any periodicity assumption via variational methods

defect correction for two point boundary value problems on nonequidistant meshes new finite difference formulae of arbitrary order are derived for the speciai class of second order two point boundary value problems y f x y x a x b variable mesh spacing is possible and the required accuracy is achieved under a very mild mesh condition a naturai defect correction framework is set up to compute the higher order approximations

an efficient implicit runge kutta method for second order systems abstract we will consider the efficient implementation of a fourth order two stage implicit runge kutta method to solve periodic second order initial value problems to solve the resulting systems we will use the factorization of the discretized operator such proposed factorization involves both complex and real arithmetic the latter case is considered here the resulting system will be efficient and small in size it is one fourth the size of systems using normal implicit runge kutta method numerical details and examples will also be presented to demonstrate the efficiency of the method

boundary regularity and sufficient conditions for strong local minimizers abstract in this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning c 1 extremals at which the second variation is strictly positive the results are presented in the quasiconvex setting in accordance with the original statement by grabovsky and mengesha 31 the strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts as well as on a sufficiency result according to which smooth extremals are spatially local minimizers furthermore we prove partial regularity up to the boundary for strong local minimizers in the non homogeneous case and a full regularity result for lipschitz extremals with gradient of vanishing mean oscillation as a consequence we also establish a sufficiency result for this class of extremals the regularity results are established via a blow up argument

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

semiclassical solutions of perturbed p laplacian equations with critical nonlinearity abstract we consider the perturbed p laplacian equation e p p u v x u p 2 u k x u p 2 u f x u u w 1 p r n where p u div u p 2 u is the p laplacian operator with 1 p n p p n n p denotes the sobolev critical exponent k x is a bounded positive function under some mild conditions on v and f we show that the equation has at least one nontrivial solution provided that e e 0 where the bound e 0 is formulated in terms of p n v k and f

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

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

relaxation of quasiconvex functional in bv ω ℝp for integrands f x u u in this paper it is shown that if p x u is a quasiconvex function with linear growth then the relaxed functional in bv p of u to int limits omega f x u x nabla u x dx with respect to the l1 topology has an integral representation of the form begin gathered mathfrak f u int limits omega f x u x nabla u x dx int limits sigma u k x u x u x v x dh n 1 x hfill int limits omega f infty x u x dc u hfill end gathered where du u dx u u v dhn 1l u c u the proof relies on a blow up argument introduced by fonseca muller in the case where u w1 1 and on a recent result by alberti showing that the cantor part c u is rank one valued

efficient iterations for gauss methods on second order problems we consider some important aspects about the implementation of high order implicit formulas specially the gauss methods for solving second order differential systems having high frequencies and small amplitudes superimposed the choice of an appropriate iterative scheme is discussed in detail important topics about the predictors initial guesses are analyzed and a variable order strategy to select the best predictor at each integration step is supplied a few numerical experiments on some standard test problems confirm the theory presented

existence of localized solutions for a classical nonlinear dirac field we prove the existence of stationary states for nonlinear dirac equations of the form i gamma mu partial mu psi m psi f bar psi psi psi 0 we seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations