stationary sets for the wave equation in crystallographic domains let w be a crystallographic group in r n generated by reflections and let be the fundamental domain of w we characterize stationary sets for the wave equation in when the initial data is supported in the interior of the stationary sets are the sets of time invariant zeros of nontrivial solutions that are identically zero at t 0 lie show that for these initial data the n 1 dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group w w w this part comes from a corresponding odd symmetry of the initial data in physical language the result is that if the initial source is localized strictly inside of the crystalline then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one

nodal sets and morse indices of solutions of super linear elliptic pdes in this paper we develop a sturm liouville type theory for the nodal sets and morse indices of solutions of super linear elliptic pdes with dirichlet boundary condition it shows that there are some relationships between analytic properties e g lp norm vanishing order of the nodal point andn 1 dimensional hausdorff measure of the nodal set of the solutions as the functions on the domain and morse indices of the solutions as the critical points of the functionalj u 12 u 2 f x u dxonh10

oscillation criteria for second order matrix differential equations new oscillation criteria are presented for second order matrix differential equation px qx rf x 0 which extend and improve the recent results of li and agarwal for scalar cases and the results of meng wang and zheng for matrix cases

some sphere theorems in linear potential theory in this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities whose equality cases are saturated by domains with spherical sy

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

doubling inequality and nodal sets for solutions of bi laplace equations we investigate the doubling inequality and nodal sets for the solutions of bi laplace equations a polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for laplace eigenfunctions by logunov in addition we derive an implicit upper bound for the nodal sets of solutions we show two types of doubling inequalities for the solutions of bi laplace equations as a consequence the rate of vanishing is given for the solutions

strong unique continuation for higher order elliptic equations with gevrey coefficients article history we address the strong unique continuation problem for higher order elliptic partial differential equations in 2d with gevrey co efficients we provide a quantitative estimate of unique continu ation observability estimate and prove that the solutions satisfy the strong unique continuation property for ranges of the gevrey exponent strictly including non analytic gevrey classes as an ap plication we obtain a new upper bound on the hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients published by elsevier inc

the nodal line of the second eigenfunction of the robin laplacian in r2 can be closed abstract we construct a multiply connected domain in r 2 for which the second eigenfunction of the laplacian with robin boundary conditions has an interior nodal line in the process we adapt a bound of donnelly fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains which does not depend directly on any boundary behaviour this also gives a new proof of the nodal line property of the example in the dirichlet case

nodal sets of thin curved layers abstract this paper is concerned with the location of nodal sets of eigenfunctions of the dirichlet laplacian in thin tubular neighbourhoods of hypersurfaces of the euclidean space of arbitrary dimension in the limit when the radius of the neighbourhood tends to zero it is known that spectral properties of the laplacian are approximated well by an effective schrodinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures by applying techniques of elliptic partial differential equations we strengthen the known perturbation results to get a convergence of eigenfunctions in holder spaces this enables us in particular to conclude that every nodal set has a non empty intersection with the boundary of the tubular neighbourhood

volume estimates on the critical sets of solutions to elliptic pdes in this paper we study solutions to elliptic linear equations l u i aij x ju bi x iu c x u 0 either on n or a riemannian manifold under the assumption that the coefficient functions aij are lipschitz bounded we focus our attention on the critical set c u x u 0 and the singular set c u x u 0 and more importantly on effective versions of these currently with just the lipschitz regularity of the coefficients the strongest results in the literature say that the singular set is n 2 dimensional however at this point it has not even been shown that hn 2 c unless the coefficients are smooth fundamentally this is due to the need of an regularity theorem that requires higher smoothness of the coefficients as the frequency increases we introduce new techniques for estimating the critical and singular set which avoids the need of any such regularity consequently we prove that if the frequency of u is bounded by then we have the estimates hn 2 c u c 2 and hn 2 s u c 2 depending on whether the equation is critical or not more importantly we prove corresponding estimates for the effective critical and singular sets even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature we also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem 2017 wiley periodicals inc

laplace operators with eigenfunctions whose nodal set is a knot we prove that given any knot in a compact 3 manifold m there exists a riemannian metric on m such that there is a complex valued eigenfunction u of the laplacian corresponding to the first nontrivial eigenvalue whose nodal set u 1 0 has a connected component given by higher dimensional analogs of this result will also be considered

oscillation criteria for linear matrix hamiltonian systems abstract in this paper we present oscillation criteria in terms of the coefficient functions for the matrix linear hamiltonian system u a x u b x v v c x u a x v under the hypothesis h a x b x b x 0 and c x c x are n n matrices of real valued continuous functions on the interval i a a these criteria generalize earlier results due to etgen and pawlowski 1976 erbe qingkai kong and shigui ruan 1993 and fanwei meng jizhong wang and zhaowen zheng 1998 for the system p x u q x u 0 or its special cases

double shuffle relation for associators it is proved that drinfel 0 d s pentagon equation implies the generalized double shue relation as a corollary an embedding from the grothen

zero sets of solutions to semilinear elliptic systems of first order consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n dimensional manifold assume the operator has the strong unique continuation property we show that the zero set of the solution is contained in a countable union of smooth n 2 dimensional submanifolds hence it is countably n 2 rectifiable and its hausdorff dimension is at most n 2 moreover it has locally finite n 2 dimensional hausdorff measure we show by example that every real number between 0 and n 2 actually occurs as the hausdorff dimension for a suitable choice of operator we also derive results for scalar elliptic equations of second order

on the scaling limits at zeros of solutions of parabolic equations abstract we are concerned with the local scaling limits near zero points of solutions of general parabolic equations we show that at each zero point of finite vanishing order the solution is asymptotic to a homogeneous polynomial solution of the osculating equation with constant coefficients this provides useful information about the local structure of zero sets of solutions in particular we give upper bounds for hausdorff dimensions of zero sets of weak solutions our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

the nodal set of solutions to some elliptic problems sublinear equations and unstable two phase membrane problem abstract we are concerned with the nodal set of solutions to equations of the form u u q 1 u q 1 in b 1 where 0 q 1 2 b 1 b 1 0 is the unit ball in r n n 2 and u max u 0 u max u 0 are the positive and the negative part of u respectively this class includes the unstable two phase membrane problem q 1 as well as sublinear equations for 1 q 2 we prove the following main results a the finiteness of the vanishing order at every point and the complete characterization of the order spectrum b a weak non degeneracy property c regularity of the nodal set of any solution the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having hausdorff dimension at most n 2 locally finite when n 2 d a partial stratification theorem ultimately the main features of the nodal set are strictly related with those of the solutions to linear or superlinear equations with two remarkable differences first of all the admissible vanishing orders can not exceed the critical value 2 2 q at threshold we find a multiplicity of homogeneous solutions yielding the non validity of any estimate of the n 1 dimensional measure of the nodal set of a solution in terms of the vanishing order as a byproduct we also prove the strong unique continuation property for the unstable obstacle problem corresponding to the case 0 the proofs are based on monotonicity formulae for a 2 parameter family of weiss type functionals blow up arguments and the classification of homogeneous solutions

boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions in this work we introduce the use of powerful tools from geometric measure theory gmt to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure omega of a domain omega omega subset mathbb r n and the harmonic measure omega of omega omega int omega c in dimension n ge 3

algebraic differential formulas for the shuffle stuffle and duality relations of iterated integrals abstract in this paper we prove certain algebraic identities which correspond to differentiation of the shuffle relation the stuffle relation and the relations which arise from mobius transformations of iterated integrals these formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line

eigenfunctions with prescribed nodal sets in this paper we consider the problem of prescribing the nodal set of low energy eigenfunctions of the laplacian our main result is that given any separating closed hypersurface in a compact n manifold m there is a riemannian metric on m such that the nodal set of its first nontrivial eigenfunction is we present a number of variations on this result which enable us to show in particular that the first nontrivial eigenfunction can have as many non degenerate critical points as one wishes

uniqueness of global positive solution branches of nonlinear elliptic problems has been studied in many publications the first global description was given in 19 nonetheless there were open questions some of which are answered in this paper we motivate some of these questions now for a suitable class of smooth fimctions f ir ir problem 1 1 has infinitely many solutions for 2 1 this was shown in 15 via the method of suband supersolutions which also gives an order of these solutions 0 0 which is not excluded in 15 there is also a global branch continuum c of solutions 2 u bifurcating from the trivial solution line 2 0 at 2 21 where 1 21 f 0 is the principal eigenvalue of the linearization

interval oscillation criteria for second order nonlinear differential equations with damping new oscillation criteria are established for the nonlinear damped differential equation ry py qf y 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t0 rather than on the whole half line our results are sharper than some previous results and extend and improve the recent interval criteria of kong and huang and can be applied to extreme cases such as t0 q s ds in particular several examples that dwell upon the sharp conditions of our results are also included

oscillation criteria of second order nonlinear matrix differential systems in this paper the second order nonlinear matrix differential systems p t y t f t y t y t 0 t t 0is considered where p t y t and f t y t y t are nxn real continuous matrix functions and p t 0 are positive definite and symmetric one some new oscillation criteria for the systems are established and some known results in the literature are improved and generalized

a remark on a curvature gap for minimal surfaces in the ball we extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional euclidean ball due to l ambrozio and i nunes

counting nodal lines which touch the boundary of an analytic domain we consider the zeros on the boundary of a neumann eigenfunction of a real analytic plane domain we prove that the number of its boundary zeros is o where 2 we also prove that the number of boundary critical points of either a neumann or dirichlet eigenfunction is o it follows that the number of nodal lines of components of the nodal set which touch the boundary is of order this upper bound is of the same order of magnitude as the length of the total nodal line but is the square root of the courant bound on the number of nodal components in the interior more generally the results are proved for piecewise analytic domains

lower bounds for nodal sets of dirichlet and neumann eigenfunctions let phi be a dirichlet or neumann eigenfunction of the laplace beltrami operator on a compact riemannian manifold with boundary we prove lower bounds for the size of the nodal set phi 0

on nodal sets for dirac and laplace operators we prove that the nodal set zero set of a solution of a generalized dirac equation on a riemannian manifold has codimension 2 at least if the underlying manifold is a surface then the nodal set is discrete we obtain a quick proof of the fact that the nodal set of an eigenfunction for the laplace beltrami operator on a riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension we also see that the nodal set of a harmonic differential form on a closed manifold has codimension 2 at least a fact which is not true if the manifold is not closed examples show that all bounds are optimal

oscillation and asymptotics for nonlinear second order differential equations this paper discusses a class of second order nonlinear differential equations by using the generalized riccati technique and the averaging technique new oscillation criteria are obtained for all solutions of the equation to be oscillatory asymptotic behavior for forced equations is also discussed

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

nodal sets of solutions of equations involving magnetic schrodinger operator in three dimensions it is well known that the complexity of the nodal set of a function mainly comes from the singular set on which both the function and the gradient vanish the singular set of a real valued solution of a linear elliptic equation has been well investigated for a complex valued solution of a linear equation involving a magnetic schrodinger operator the structure of the nodal set has not been well investigated yet excepted in the two dimensional case in this paper we extend the arguments of garafalo and lin indiana univ math j 35 245 268 1986 and of han indiana univ math j 43 983 1002 1994 to show that the singular set of such a solution in a three dimensional domain is countably 1 rectifiable the functions considered in this paper include the order parameter in the ginzburg landau theory of superconductivity and the eigenfunctions of the magnetic schrodinger operator

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

critical points of green s functions on complete manifolds we prove that the number of critical points of a li tam green s function on a complete open riemannian surface of finite type admits a topological upper bound given by the first betti number of the surface in higher dimensions we show that there are no topological upper bounds on the number of critical points by constructing for each nonnegative integer n a riemannian manifold diffeomorphic to r n n 3 whose minimal green s function has at least n non degenerate critical points variations on the method of proof of the latter result yield contractible n manifolds whose minimal green s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of r n

iterated integrals on p1 0 1 z and a class of relations among multiple zeta values abstract in this paper we consider iterated integrals on p 1 0 1 z and define a class of q linear relations among them which arises from the differential structure of the iterated integrals with respect to z we then define a new class of q linear relations among the multiple zeta values by taking their limits of z 1 which we call confluence relations i e the relations obtained by the confluence of two punctured points one of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values as a good reason for this we show that confluence relations imply both the regularized double shuffle relations and the duality relations

intersection bounds for nodal sets of planar neumann eigenfunctions with interior analytic curves let r be a bounded piecewise smooth domain and be a neumann or dirichlet eigenfunction with eigenvalue and nodal set n x x 0 let h be an interior c curve consider the intersection number n h h n we first prove that for general piecewise analytic domains and under an appropriate goodness condition on h see theorem 1 1 n h oh as then using theorem 1 1 we prove in theorem 1 2 that the bound in is satisfied in the case of quantum ergodic qe sequences of interior eigenfunctions provided is convex and h has strictly positive geodesic curvature

upper bounds of nodal sets for eigenfunctions of eigenvalue problems the aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains the examples include biharmonic steklov eigenvalue problems buckling eigenvalue problems and champed plate eigenvalue problems the geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions it is done through analytic estimates of morrey nirenberg and carleman estimates

on the geometry of the level sets of bounded static potentials in this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity in the case where the static potential is bounded we introduce new quantities which are proven to be monotone along the level set flow of the potential function we then show how to use these properties to detect the rotational symmetry of the static solutions deriving a number of sharp inequalities among these we prove the validity without any dimensional restriction of the riemannian penrose inequality as well as of a reversed version of it in the class of asymptotically flat static metrics with connected horizon as a consequence of our analysis a simple proof of the classical 3 dimensional black hole uniqueness theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m