compact isospectral sets of plane domains abstract any isospectral family of two dimensional euclidean domains is shown to be compact in the c topology previously melrose using heat invariants was able to establish the c compactness of the curvature of the boundary curves the additional ingredient used in this paper to obtain the compactness of the domains is the behavior of the determinant of the laplacian near the boundary of the moduli space

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

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

submultiplicativity and the hanna neumann conjecture in this article we dene submultiplicativity of 2 numbers in the category of complexes over a given complex x which generalizes the statement of the strengthened hanna neumann conjecture shnc in the case when is a left orderable group and x is a free complex we prove submultiplicativity for the subcategory consisting of ordered leafages over x with an additional analytic assumption called the deep fall property we show that the deep fall property is satised for graphs this implies shnc

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

on the intersection of subgroups of a free group the hanna neumann conjecture says that the intersection of subgroups of rankn 1 andm 1 of a free group has rank at mostnm 1 this paper proves the conjecture for the casem 1 see theorem 1 our methods imply that the strengthened hanna neumann conjecture is also true in this case theorem 2

on higher spin partition functions we observe that the partition function of the set of all free massless higher spins s 0 1 2 3 in flat space is equal to one the ghost determinants cancel against the physical ones or equivalently the regularized total number of degrees of freedom vanishes this reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory the z 1 property extends also to the ads background i e the 1 loop vacuum partition function of vasiliev theory is equal to 1 assuming a particular regularization of the sum over spins this was noticed earlier as a consistency requirement for the vectorial ads cft duality we find that z 1 is true also in the conformal higher spin theory with higher derivative kinetic terms expanded near flat or conformally flat s4 background we also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2 derivative kinetic term but only scalar gauge invariance in flat 4d space this non unitary theory has weyl invariant action in curved background and it corresponds to partially massless field in ads5 we discuss in detail the special case of s 2 or conformal graviton compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions

the zeta functional determinants on manifolds with boundary abstract in this paper we derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on riemannian 3 and 4 manifolds with smooth boundary

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

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

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

regularity and necessary conditions for a bolza optimal control problem abstract we consider a bolza optimal control problem whose lagrangian possibly extended valued may be discontinuous in the state and control variable so that optimal solutions are not supposed to necessarily satisfy the maximum principle given an optimal trajectory control pair we prove that it satisfies a new erdmann du bois reymond type condition and show that from this condition it is possible to derive boundedness properties of the optimal control and a lipschitz regularity result for the optimal state arc just imposing general growth assumptions allowing some almost linear growth behaviors

existence and non existence results for non coercive variational problems and applications in ecology abstract variational problems without coercivity assumption appear in optimal foraging theory models in behavioural ecology using direct methods of the calculus of variations rearrangement techniques and relaxation theorems we show that they possess one or no solution in ac spaces depending on the prescribed boundary conditions

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

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

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

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 scalar minimizers for nonconvex simple integrals of sum type abstract one usually asks as sufficient condition to guarantee the existence of minimizersx a b r in ac for the integral ba h x x dt withh having superlinear growth at and being any lower semicontinuous function bounded below thath be convex that is h h at all in r the purpose of this paper is to show that the conditionh 0 h 0 is actually enough to guarantee the existence of minimizers

equivalence of the strengthened hanna neumann conjecture and the amalgamated graph conjecture we show that walter neumann s strengthened form of hanna neumann s conjecture on the best possible upper bound for the rank of the intersection of two subgroups of a free group is equivalent to a conjecture on the best possible upper bound for the number of edges in a bipartite graph with a certain weak symmetry condition we illustrate the usefulness of this equivalence by deriving relatively easily certain previously known results

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

towards the hanna neumann conjecture using dicks method the hanna neumann conjecture states that the intersection of two nontrivial subgroups of rankk 1 andl 1 of a free group has rank at mostkl 1 in a recent paper 3 w dicks proved that a strengthened form of this conjecture is equivalent to his amalgamated graph conjecture he used this equivalence to reprove all known upper bounds on the rank of the intersection we use his method to improve these bounds in particular we prove an upper bound of 2kl k l 1 for the rank of the intersection above k l 2 improving the earlier 2kl min k l bound of 1

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

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

nonconvex variational problems related to a hyperbolic equation we first prove a new lyapunov type theorem which will yield existence of solutions to nonconvex minimum problems involving some hyperbolic equations on rectangular domains with darboux boundary conditions some problems with obstacle and bang bang results are also considered

a generalization of the cellina colombo theorem for a class of non convex variational problems abstract we state a condition under which the integral functional i x t 0 l t x t x t dt attains a minimum under the assumption that x l t x x is concave

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

sobolev trace inequalities of order four we establish sharp trace sobolev inequalities of order four on euclidean d balls for d 4 when d 4 our inequality generalizes the classical second order lebedev milin inequality on euclidean 2 balls our method relies on the use of scattering theory on hyperbolic d balls as an application we characterize the extremal metric of the main term in the log determinant formula corresponding to the conformal laplacian coupled with the boundary robin operator on euclidean 4 balls which surprisingly is not the flat metric on the ball

on the kurosh rank of the intersection of subgroups in free products of groups abstract the kurosh rank r k h of a subgroup h of a free product i g of groups g i is defined accordingly to the classic kurosh subgroup theorem as the number of free factors of h we prove that if h 1 h 2 are subgroups of i g and h 1 h 2 have finite kurosh rank then r k h 1 h 2 2 q q 2 r k h 1 r k h 2 6 r k h 1 r k h 2 where r k h max r k h 1 0 q is the minimum of orders 2 of finite subgroups of groups g i q if there are no such subgroups and q q 2 1 if q in particular if the factors g i are torsion free groups then r k h 1 h 2 2 r k h 1 r k h 2

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

existence uniqueness and qualitative properties of minima to radially symmetric non coercive non convex variational problems we are concerned with the problem of existence uniqueness and qualitative properties of solutions to the radially symmetric variational problem min u in wuu b r int b r pq f pt mod x mod nabla u x h 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 h colon 0 r times r to r is a convex function of the second variable this kind of problems with non convex lagrangians with respect to nabla u arise in various fields of applied sciences such as optimal design and nonlinear elasticity

heat kernel expansions ambient metrics and conformal invariants abstract the conformal powers of the laplacian of a riemannian metric which are known as the gjms operators admit a combinatorial description in terms of the taylor coefficients of a natural second order one parameter family h r g of self adjoint elliptic differential operators h r g is a non laplace type perturbation of the conformal laplacian p 2 g h 0 g it is defined in terms of the metric g and covariant derivatives of the curvature of g we study the heat kernel coefficients a 2 k r g of h r g on closed manifolds we prove general structural results for the heat kernel coefficients a 2 k r g and derive explicit formulas for a 0 r g and a 2 r g in terms of renormalized volume coefficients the taylor coefficients of a 2 k r g as functions of r interpolate between the renormalized volume coefficients of a metric g k 0 and the heat kernel coefficients of the conformal laplacian of g r 0 although h r g is not conformally covariant there is a beautiful formula for the conformal variation of the trace of its heat kernel its proof rests on the combinatorial relations between taylor coefficients of h r g and gjms operators as a consequence we give a heat equation proof of the conformal transformation law of the integrated renormalized volume coefficients by refining these arguments we also give a heat equation proof of the conformal transformation law of the renormalized volume coefficients itself the taylor coefficients of a 2 r define a sequence of higher order riemannian curvature functionals with extremal properties at einstein metrics which are analogous to those of integrated renormalized volume coefficients among the various additional results the reader finds a polyakov type formula for the renormalized volume of a poincare einstein metric in terms of q curvature of its conformal infinity and additional holographic terms the present study of relations between spectral theoretic heat kernel coefficients and geometric quantities like renormalized volume coefficients is stimulated by the holographic perspective of the ads cft duality proposing relations between functional determinants and renormalized volumes

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

on fractional gjms operators we describe a new interpretation of the fractional gjms operators as generalized dirichlet to neumann operators associated to weighted gjms operators on naturally associated smooth metric measure spaces this gives a geometric interpretation of the caffarelli silvestre extension for when 0 1 and both a geometric interpretation and a curved analogue of the higher order extension found by r yang for when 1 we give three applications of this correspondence first we exhibit some energy identities for the fractional gjms operators in terms of energies in the compactified poincare einstein manifold including an interpretation as a renormalized energy second for 1 2 we show that if the scalar curvature and the fractional q curvature q2 of the boundary are nonnegative then the fractional gjms operator p2 is nonnegative third by assuming additionally that q2 is not identically zero we show that p2 satisfies a strong maximum principle 2014 wiley periodicals inc

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