lipschitz regularity for minima without strict convexity of the lagrangian abstract we give in a non smooth setting some conditions under which some of the minimizers of f u x d x g x u x d x among the functions in w 1 1 that lie between two lipschitz functions are lipschitz we weaken the usual strict convexity assumption in showing that if just the faces of the epigraph of a convex function f r n r are bounded and the boundary datum u 0 satisfies a generalization of the bounded slope condition introduced by a cellina then the minima of f u x d x on u 0 w 0 1 1 whenever they exist are lipschitz a relaxation result follows

the existence of solutions to variational problems of slow growth abstract we consider the existence of solutions in the space w 1 1 to the problem minimize l v x d x on w 0 1 1 where l is of slow linear or at most quadratic growth we present a necessary and sufficient condition in order that for any smooth boundary datum and for any bounded with smooth boundary the minimum problem be solvable

variational problems for a class of functionals on convex domains abstract let be a bounded convex open subset of r n n 2 and let j be the integral functional j u f du x u x dx where f 0 r is a lower semicontinuous function possibly nonconvex and with linear growth we prove that the functional j admits a unique minimizer in the space of w1 10 functions that depend only on the distance from the boundary of provided that the ratio between the lebesgue measure of and the n 1 dimensional hausdorff measure of is strictly less than a constant related to the growth of f at infinity

existence of minima for nonconvex functionals in spaces of functions depending on the distance from the boundary we prove that some nonconvex functionals admit a unique minimum in a functional space of functions which depend only on the distance from the boundary of the plane domain where they are defined the domains considered are disks and regular polygons we prove that the sequence of minima of the functional on the polygons converges to the unique minimum on the circumscribed disk as the number of sides tends to infinity our method also allows us to determine the explicit form of the minima

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

locating multiple zeros interactively abstract we present a globally convergent algorithm which has been implemented interactively for the total or partial factorisation of a polynomial initially a region is found containing all the zeros of a polynomial a parallel search algorithm is then used to locate successively smaller regions containing zeros this continues until it is possible to estimate the multiplicities of the zeros reliably when a rapidly converging iteration function if takes over at this point the user using an interactive terminal can elect to investigate further all of the zeros or only those in certain regions the if is selected from a class of such functions which exhibit certain desirable mathematical and computational properties a synopsis of computational results is given

an improved sciljak s algorithm for solving polynomial equations converges quadratically to multiple zeros siljak s method provides a globally convergent algorithm for inclusion of polynomial zeros the solution procedure is formulated as a minimization process of a positive definite function involving the real and imaginary parts of the polynomial the main objective of this paper is to propose an improved version of siljak s algorithm which exploits the minimizing function to ensure a quadratic convergence to multiple zeros and at the same time determine their multiplicity time comparisons with other standard zero inclusion methods are provided to demonstrate the efficiency of the proposed improvement of the original algorithm

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

integral representation of abstract functionals of autonomous type in this work we extend the results of amar et al to a family of abstract functionals of autonomous type satisfying suitable locality and additivity properties and general integral growth conditions of superlinear type we single out a condition which is necessary and sufficient in order for a functional of this class to admit an integral representation and sufficient to have an integral representation for its lower semicontinuous envelope we also show that the integrand f x q satisfies some nice regularity properties in the q variable in particular a convexity type property along lines by adapting the reparametrization techniques introduced in an earlier work by the author to the case at issue we then prove that the family of integral functionals associated to integrands of this kind do meet the condition mentioned above in particular it is closed by vargamma convergence

on the minimum problem for a class of noncoercive nonconvex functionals we are concerned with the problem of existence of solutions to the variational problem min left int 0 r g t v t dt v in ac 0 r v r 0 right with only one fixed endpoint prescribed the map g colon 0 r times mathbb r to overline mathbb r is a normal integrand for which neither convexity nor superlinear growth conditions are assumed as an application we give an existence result for the radially symmetric variational problem min u in wuu b r int b r f pt mod x mod nabla u x a x u x dx where b r is the ball of mathbb r n centered at the origin and with radius r 0 the map f colon 0 r times 0 infty to overline mathbb r is a normal integrand and a in l 0 r again neither convexity nor superlinear growth conditions are made on f these kinds of problems with nonconvex lagrangians with respect to nabla u arise in different fields of mathematical physics such as optimal design and nonlinear elasticity

the classical problem of the calculus of variations in the autonomous case relaxation and lipschitzianity of solutions we consider the problem of minimizing b a l x t x t dt x a a x b b under the assumption that the lagrangian l is continuous and satisfies a growth assumption that does not imply superlinear growth we provide a result on the relaxation of the functional and show that a solution to the minimum problem is lipschitzian

minimizing non convex multiple integrals a density result we consider variational problems of the form where is a bounded open set in r n f r n r is a possibly non convex lower semicontinuous function with p growth at infinity for some 1 p u 0 is any function in w 1 p assuming that the convex envelope f of f is affine on each connected component of the set f f we prove the existence of solutions to p for every continuous function g such that i g has no strict local minima and ii every convergent sequence of extremum points of g eventually belongs to an interval where g is constant thus showing that the set of continuous functions g that yield existence to p is dense in the space of continuous functions on r

existence and lipschitz regularity of solutions to bolza problems in optimal control in this paper we investigate the existence and lipschitz continuity of optimal trajectories for the autonomous bolza problem in control theory the main feature of our results is that they relax the usual fast growth condition for the lagrangian furthermore we show that optimal solutions do satisfy the maximum principle

variable projection methods for approximate greatest common divisor computations we consider the problem of finding for a given n tuple of polynomials real or complex the closest n tuple that has a common divisor of degree at least d extended weighted euclidean seminorm of the coefficients is used as a measure of closeness two equivalent representations of the problem are considered i direct parameterization over the common divisors and quotients image representation and ii sylvester low rank approximation kernel representation we use the duality between least squares and least norm problems to show that i and ii are closely related to mosaic hankel low rank approximation this allows us to apply to the approximate common divisor problem recent results on complexity and accuracy of computations for mosaic hankel low rank approximation we develop optimization methods based on the variable projection principle both for image and kernel representation these methods have linear complexity in the degrees of the polynomials for small and large d we provide a software implementation of the developed methods which is based on a software package for structured low rank approximation

a unified approach to computing the nearest complex polynomial with a given zero suppose we have a complex polynomial f z whose coefficients are inaccurate and a prescribed complex number such that f 0 we study the problem of computing a complex polynomial f z such that f 0 and the distance between f and f i e f f is minimal considering that previous works usually took the usual l p norm weighted l p norm and block wise norm as distance measures we first introduce a new defined synthetic norm that integrates all these norms then we propose a unified approach to study the proposed problem and succeed in giving explicit expressions of the nearest polynomial the effectiveness of our approach is illustrated by two examples one of which shows an extension of finding the nearest complex polynomial with a zero in a given domain finally as an application of the new defined norm we discuss a matrix valued optimization problem that is very common in machine learning this paper introduces a new synthetic block wise norm and proves its dual norm which is very important here this paper for the first time proposes a unified approach to compute the nearest complex polynomial with a given zero this paper discusses a matrix valued optimization problem that is very common in machine learning as an application of the introduced norm

relationships between order and efficiency of a class of methods for multiple zeros of polynomials the behavior of a class of high order methods for solving polynomial equations is examined it is shown that the number of iterations for local convergence to a multiple zero to the limits of attainable accuracy is bounded independent of the multiplicity of the zero and decreases as the order of the method increases for the higher order methods the number of iterations decreases as the multiplicity increases computational efficiency as a function of degree order and multiplicity is investigated and an effective choice of order is recommended numerical examples are provided

multiple zeros of nonlinear systems as an attempt to bridge between numerical analysis and algebraic geometry this paper formulates the multiplicity for the general nonlinear system at an isolated zero presents an algorithm for computing the multiplicity strucuture proposes a depth deflation method for accurate computation of multiple zeros and introduces the basic algebraic theory of the multiplicity furthermore this paper elaborates and proves some fundamental theorems of the multiplicity including local finiteness consistency perturbation invarance and depth deflatability the proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed

lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands the study of regularity properties for minimizers of problems of the form min u u dt u e iv q r u a a u b b 1 1 has been undertaken for a long time in calculus of variations and a number of results are by now available on this subject when f t s z is a smooth coercive function problem 1 1 was considered by tonelli see 18 19 who proved that every minimizer u t of the problem 1 1 is locally lipschitz on a relatively open set of full measure in a b under some additional assumptions on f it is also possible to prove global regularity theorems see for instance cesari 3 chap 2 sect 61 in recent years much attention has been devoted to problems of the form l l on the one hand in weakening the hypotheses of tonelli s regularity results and on the other in finding counterexamples of nonregular minimizers for coercive functionals with respect to this last subject we recall that it has been shown see ball and mizel l and clarke and vinter 6 that there exist polynomial functions f t s z such that for suitable c 0 and p

necessary conditions and non existence results for autonomous nonconvex variational problems abstract classical one dimensional autonomous lagrange problems are considered in absence of any smoothness convexity or coercivity condition on the energy density we prove a dubois reymond type necessary condition expressed as a differential inclusion involving the subdifferential of convex analysis as a consequence a non existence result is obtained

existence of global weak solutions for 3d degenerate compressible navier stokes equations in this paper we prove the existence of global weak solutions for 3d compressible navier stokes equations with degenerate viscosity the method is based on the bresch and desjardins commun math phys 238 211 223 2003 entropy conservation the main contribution of this paper is to derive the mellet and vasseur commun partial differ equ 32 431 452 2007 type inequality for weak solutions even if it is not verified by the first level of approximation this provides existence of global solutions in time for the compressible barotropic navier stokes equations the result holds for any gamma 1 in two dimensional space and for 1 gamma 3 in three dimensional space in both case with large initial data possibly vanishing on the vacuum this solves an open problem proposed by lions mathematical topics in fluid mechanics vol 2 compressible models 1998

about the barotropic compressible quantum navier stokes equations abstract in this paper we consider the barotropic compressible quantum navier stokes equations with a linear density dependent viscosity and its limit when the scaled planck constant vanishes following recent works on degenerate compressible navier stokes equations we prove the global existence of weak solutions by the use of a singular pressure close to vacuum with such singular pressure we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled planck constant denoted by e tends to 0

nonconvex minimization problems for functionals defined on vector valued functions abstract we consider the minimization problem min v w 1 1 0 b n r r m b n r f v x h v x dx where b n r is the ball of r n centered at the origin and with radius r 0 f is a lower semicontinuous function and h is a convex function we give sufficient conditions for the existence and uniqueness of minimizers our technique relies on a detailed knowledge of the properties of the solutions to the convexified problem obtained using the corresponding euler lagrange inclusions

polynomial roots the ultimate answer we show that it is always possible to transform the problem of tinding the mots of a generic polynomial to the problem of determining the eigenvalues of tridiagonal matrices having only simple eigenvalues since this problem is very efficiently solved for example with the familiar qr method it seems that the present veiy simple approach bas the potentiality to supplant the existing ones

an improved toeplitz algorithm for polynomial matrix null space computation in this paper we present an improved algorithm to compute the minimal null space basis of polynomial matrices a problem which has many applications in control and systems theory this algorithm takes advantage of the block toeplitz structure of the sylvester matrix associated with the polynomial matrix the analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well known pencil state space algorithms found in the literature

third order and fourth order iterative methods for finding multiple and distinct zeros of non linear equations in this paper we point out and analyse six two step iterative methods for finding multiple as well as distinct zeros of non linear equations we prove that the methods for multiple zeros have third order convergence where as the methods for distinct zeros have fourth order convergence the methods calculate the multiple as well as distinct zeros with high accuracy the numerical tests show their better performance in case of algebraic as well as non algebraic equations

iterative methods for multiple zeros of a polynomial by clustering abstract for the numerical determination of zeros of a complex polynomial by simultaneous iterative methods we show theorems on the fact that the approximations for a multiple zero gather around the exact zero when they become close approximations application of our theorems makes it possible to estimate the multiplicity of zeros dynamically as well as to improve the convergence to multiple zeros for the known methods e g durand kerner s or tanabe s methods

on minima of radially symmetric functionals of the gradient where b is the unit ball of ir and the map u g r v is lower semicontinuous but not necessarily convex problems of this kind arise in domains as different as nonlinear elasticity fluid dynamics and shape optimization and either for the problem of the existence of solutions or for the properties of the relaxed problem are considered in l 7 in particular the very same problem is considered in 7 our results present the following features a no smoothness on g or h is required g is either a normal integrand or a lower semicontinuous function b the case h 0 is allowed in this case the assumption on g reduces for the existence of solutions to g being lower semicontinuous and growing at infinity as is to be expected for the uniqueness in addition to g being strictly increasing as is also to be expected c the case h au is allowed for a 0 our theorems yield at once existence and uniqueness of solutions with no further assumptions on g besides lower semicontinuity and growth at infinity

existence results for noncoerecive variational problems the aim of this paper is to give an existence result for a class of one dimensional nonconvex noncoercive problems in the calculus of variations the main tools for the proof are an existence theorem in the convex case and the closure of the convex hull of the epigraph of functions strictly convex at infinity

existence of solutions for a class of non convex minimum problems p celada1 s perrotta2 g treu3 1 dipartimento di scienze matematiche universita degli studi di trieste p le europa 1 i 34127 trieste italia e mail celada univ trieste it 2 dipartimento di matematica pura ed applicata g vitali universita degli studi di modena via campi 213 b i 41100 modena italia e mail perrotta c220 unimo it 3 dipartimento di matematica ed informatica universita degli studi di udine v le delle scienze 206 i 33100 udine italia e mail treu dimi uniud it

newton s method for overdetermined systems of equations complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors in this paper we consider overdetermined systems where there are more equations than unknowns we study newton s method for such a system

minimizing nonconvex simple integrals of product type abstract we consider the problem of minimizing simple integrals of product type i e min t 0 g x t f x t dt x ac 0 t x 0 x 0 x t x t p where f r 0 is a possibly nonconvex lower semicontinuous function with either superlinear or slow growth at infinity assuming that the relaxed problem p obtained from p by replacing f with its convex envelope f admits a solution we prove attainment for p for every continuous positively bounded below the coefficient g such that i every point t r is squeezed between two intervals where g is monotone and ii g has no strict local minima this shows in particular that for those f such that the relaxed problem p has a solution the class of coefficients g that yield existence to p is dense in the space of continuous positive functions on r we discuss various instances of growth conditions on f that yield solutions to p and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp

convergence of the navier stokes poisson system to the incompressible navier stokes equations the quasineutral limit of the navier stokes poisson system in the whole space rd d 1 and in the torus td is studied in this paper it is shown that for the well prepared initial data the global weak solution of the navier stokes poisson system converges strongly to the strong solution of the incompressible navier stokes equations

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

another theorem of classical solvability in small for one dimensional variational problems in this paper we suggest a direct method for studying local minimizers of one dimensional variational problems which naturally complements the classical local theory this method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable the basis of these results is a regularity theory a priori estimates and compactness in c1 for solutions of obstacle problems with sufficiently close obstacles in these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory

helically symmetric solutions to the 3 d navier stokes equations for compressible isentropic fluids abstract we prove the existence of global weak solutions to the navier stokes equations for compressible isentropic fluids for any 1 when the cauchy data are helically symmetric where the constant is the specific heat ratio moreover a new integrability estimate of the density in any neighborhood of the symmetry axis the singularity axis is obtained

existence and regularity of minimizers of nonconvex functionals depending onuand u abstract we consider variational problems of the form min f u x g x u x dx u u 0 h 1 0 wheref r n 0 is a possibly nonconvex function with quadratic growth at infinity andg x u is lipschitz continuous and strictly increasing decreasing inu we prove the existence and local lipschitz regularity of solutions for every boundary datumu0 h1 l on the basis of the structure of the epigraph of the convex envelope off

existence of minimizers for nonconvex noncoercive simple integrals we consider the problem of minimizing autonomous simple integrals such as min left int 0 t f left x t x prime t right dt colon text x in ac 0 t x 0 x 0 x t x t right tag cal p where f mathbb r times mathbb r to 0 infty is a possibly nonconvex function with either superlinear or slow growth at infinity assuming that the relaxed problem cal p ast ast obtained from cal p by replacing f with its convex envelope f with respect to the derivative variable x prime admits a solution we prove attainment for cal p under mild regularity and growth assumptions on f and f we discuss various instances of growth conditions on f that yield solutions to the corresponding relaxed problem cal p ast ast and we present examples that show that the hypotheses on f and f considered here for attainment are essentially sharp

relaxation and regularity in the calculus of variations in this work we prove that if l t u is a continuous function in t and u borel measurable in with bounded non convex pieces in then any absolutely continuous solution u to the variational problem min abl t u t u t dt u w01 1 a b is quasi regular in the sense of tonelli i e u is locally lipschitz on an open set of full measure of a b under the further assumption that either l is lipschitz continuous in u locally uniformly in but not necessarily in t or l is invariant under a group of c1 transformations as in the noether s theorem without one of those further assumptions the solution could be not regular as shown by a recent example in gratwick and preiss 2010 13 our result is then optimal in this sense moreover we improve the standard hypothesis used so far in buttazzo et al 1998 1 clarke and vinter 1985 5 6 csornyei et al 2008 7 tonelli 1915 15 which have been the lipschitz continuity of l in u locally uniform in and t and some growth condition in we also show that the relaxed and the original problem have the same solutions without assuming any of the two further assumptions above this extends a result in mariconda and treu 2004 14 to the non autonomous case

radially symmetric critical points of non convex functionals we investigate critical points of the functionalover a ball in n here w is radially symmetric but not convex we embed the functional into a family of functionalswhere e0 0 u e u a global bifurcation analysis yields a branch of non trivial critical points depending on and positive e where we can set 0 the geometric properties preserved on that branch due to the maximum principle prove compactness such that the singular limit as e 0 exists under natural conditions on w and g the critical point obtained in this way is a minimizer of the original functional that plan can be carried out only under the restriction of radial symmetry since the maximum principle applies only to special elliptic equations of fourth order that restriction however is not essential since every minimizer of the functional is radially symmetric

global weak solutions to 3d compressible navier stokes poisson equations with density dependent viscosity abstract in this paper we study the global existence of weak solutions to the compressible navier stokes poisson n s p equations with density dependent viscosity and non monotone pressure in a three dimensional torus our approach is based on the faedo galerkin method and the compactness arguments motivated by vasseur yu 29 and 30 we construct the approximate solutions and the key estimates ling in the elementary energy estimates b d entropy and mellet vasseur type inequality for the weak solutions here we need the conditions that the adiabatic constant satisfies 4 3 3 for 1 or 1 3 for 1 where is a sign constant of poisson equation which determines the physical meaning of the n s p system

reparametrizations and approximate values of integrals of the calculus of variations abstract we prove an approximation result that implies the non occurrence of the lavrentiev phenomenon

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

iterative algorithms for computing the singular subspace of a matrix associated with its smallest singular values abstract many problems in science require the computation of only one singular vector or more generally a singular subspace of a matrix instead of computing the complete singular value decomposition iterative methods can be used to improve the computational speed in particular when a priori information about the solution is available this paper deals with the computation of a basis of a singular subspace of a matrix associated with its smallest singular values three iterative methods inverse chebyshev and inverse chebyshev iteration are compared by analyzing their convergence properties based on the convergence rate and the operation counts per iteration step it is shown in which problems a particular iterative algorithm is most efficient if the gap between the singular values associated with the desired and undesired singular subspace is large inverse iteration is clearly the best choice if not convergence can be accelerated by applying inverse chebyshev iteration provided sufficiently tight bounds delineating the undesired singular value spectrum are known the smaller the gap the larger the gain in speed unless the matrix is structured or sparse this method is shown to be always more efficient than ordinary chebyshev iteration

an indirect method in the calculus of variations this article presents a new approach to the issue of the existence of solutions to the basic problem in the calculus of variations the method is indirect and applies to certain classes of problems with slow or no growth in addition to those satisfying the traditional coercivity condition the proof hinges upon showing with the help of nonsmooth analysis that a certain value function is constant examples are given to illustrate the applicability of the results and the necessity of the hypotheses

on radially symmetric minima of nonconvex functionals abstract we study a minimization problem in the space w1 10 br where br is the ball of radius r with center at the origin the functional considered is not necessarily convex under suitable assumptions we prove the existence of a radially symmetric decreasing solution by strengthening the assumptions we obtain uniqueness results finally we study under which assumptions and in which sense the solutions found solve the corresponding euler equation the proofs are very direct and simple they only make use of the functions t n introduced by the author arch rational mech anal 1999

computing multiple roots of inexact polynomials we present a combination of two algorithms that accurately calculate multiple roots of general polynomials algorithm i transforms the singular root finding into a regular nonlinear least squares problem on a pejorative manifold and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations to fulfill the input requirement of algorithm i we develop a numerical gcd finder containing a successive singular value updating and an iterative gcd refinement as the main engine of algorithm ii that calculates the multiplicity structure and the initial root approximation while limitations exist in certain situations the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact this is perhaps the first blackbox type root finder with such capabilities to measure the sensitivity of the multiple roots a structure preserving condition number is proposed and error bounds are established according to our computational experiments and error analysis a polynomial being ill conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved and its multiple roots can be computed with high accuracy

existence and regularity in the small in the calculus of variations abstract a local existence theorem is proved for the basic problem in the calculus of variations that of minimizing l t x x dt over a class of functions x assuming given boundary conditions the lagrangian l is only assumed to be locally lipschitz and strictly convex in its x variable

computing the nearest singular univariate polynomials with given root multiplicities in this paper we derive explicit expressions for the nearest singular polynomials with given root multiplicities and its distance to the given polynomial these expressions can be computed recursively these results extend previous results of zhi et al 2004 10 and zhi and wu 1998 11

normality of the maximum principle for nonconvex constrained bolza problems we consider a bolza optimal control problem with state constraints it is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle in general the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls in the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate cf a v arutyanov s m aseev investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints siam j control optim 35 1997 930 952 f rampazzo r b vinter a theorem on existence of neighbouring trajectories satisfying a state constraint with applications to optimal control ima 16 4 1999 335 351 f rampazzo r b vinter degenerate optimal control problems with state constraints siam j control optim 39 4 2000 989 1007 our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle in particular we allow the initial condition to be fixed and the state constraints to be nonsmooth to prove normality we use j yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined in turn by linearization of the original set of constraints along an extremal trajectory

finding a multiple zero by transformations and newton like methods many methods for finding a multiple zero x of a function f are based on transforming f to a function t for which x is a simple zero we give and analyze a general transformation which i

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

on the flow of chemically reacting gaseous mixture abstract we consider the cauchy problem for the system of equations governing flow of isothermal reactive mixture of compressible gases our main contribution is to prove sequential stability of weak solutions when the state equation essentially depends on the species concentration and the viscosity coefficients vanish on vacuum moreover under additional assumption on the cold component of the pressure in the regions of small density we prove the existence of weak solutions for arbitrary large initial data

a new variational inequality in the calculus of variations and lipschitz regularity of minimizers abstract we consider a problem of the calculus of variations of the form p minimize i x a b t x t x t d t x a x b subject to x w 1 m a b r n x t c a e x t t a b where a b r n r n r is borel measurable c is a cone is a nonempty subset of r n and is an arbitrary extended valued function we prove that if t t x satisfies a nonsmooth version of cesari s condition s then any local minimizer of p in the ac norm is lipschitz whenever is coercive the proof is obtained via a new variational inequality formulated here that holds under the extended condition s just borel measurability if is autonomous for a e t in a b w t x t x t v v t x t x t p t v 1 v 0 where p t is an absolutely continuous arc the derivative of which belongs to a suitable subdifferential of x t x t a e on a b the proof of the directional weierstrass type condition w is based on clarke s nonsmooth recent versions of the maximum principle

existence of vector minimizers for nonconvex 1 dim integrals with almost convex lagrangian this paper proves new results of existence of minimizers for the nonconvex integral abl x x dt among the ac functions x a b rn with x a a x b b our lagrangian l is e g lsc with superlinear growth assuming values freely we replace convexity by almost convexity a hypothesis which in the radial superlinear case l s f s is automatically satisfied provided f s is convex at zero

global behavior of spherically symmetric navier stokes poisson system with degenerate viscosity coefficients in this paper we study a free boundary problem for compressible spherically symmetric navier stokes poisson equations with degenerate viscosity coefficients and without a solid core under certain assumptions that are imposed on the initial data we obtain the global existence and uniqueness of the weak solution and give some uniform bounds with respect to time of the solution moreover we obtain some stabilization rate estimates of the solution the results show that such a system is stable under small perturbations and could be applied to the astrophysics

radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients abstract in this paper we prove existence of radially symmetric minimizers u a x u a x having u a ac monotone and l u a 0 increasing for the convex scalar multiple integral b r l u x u x 1 x 2 x d x among those u in the sobolev space a w 0 1 1 b r here u x is the euclidean norm of the gradient vector and b r is the b a l l x r d x r while a is the boundary data besides being e g superlinear but no growth needed if is known to have minimum our lagrangian l r r 0 is just convex lsc and min l r 0 and l s is even s while 1 and 2 are borel bounded away from 0 and remarkably may also be seen as the calculus of variations reformulation of a distributed parameter scalar optimal control problem indeed state and gradient pointwise constraints are in a sense built in since l s v is freely allowed

necessary and sufficient conditions for optimality of nonconvex noncoercive autonomous variational problems with constraints we consider the classical autonomous constrained variational problem of minimization of a b v t v t dt in the class v w 1 1 a b v a v b v t 0 a e in a b where a x 0 r is a lower semicontinuous nonnegative integrand which can be nonsmooth nonconvex and noncoercive we prove a necessary and sufficient condition for the optimality of a trajectory v 0 in the form of a dubois reymond inclusion involving the subdifferential of convex analysis moreover we also provide a relaxation result and necessary and sufficient conditions for the existence of the minimum expressed in terms of an upper limitation for the assigned mean slope 0 b a applications to various noncoercive variational problems are also included