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

quantitative uniqueness of elliptic equations based on a variant of frequency function we improve the vanishing order of solutions for schr o dinger equations which describes quantitative behavior of strong uniqueness continuation property for the first time we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions furthermore strong unique continuation is established for higher order elliptic equations using this variant of frequency function

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

tubular neighborhoods of nodal sets and diophantine approximation we give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the laplacian on a real analytic closed riemannian manifold m as an application we consider the question of approximating points on m by nodal sets and explore analogy with approximation by rational numbers

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

sharp vanishing order of solutions to stationary schrodinger equations on carnot groups of arbitrary step abstract based on a variant of the frequency function approach of almgren 1 under appropriate assumptions we establish an optimal upper bound on the vanishing order of solutions to stationary schrodinger equations associated to sub laplacian on carnot groups of arbitrary step such a bound provides a quantitative form of strong unique continuation and can be thought of as a subelliptic analogue of the recent results obtained by bakri 3 and zhu 27 for the standard laplacian

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

completing a symmetric 2 2 block matrix and its inverse abstract we consider the following completion problems suppose n1 n2 are nonnegative integers such that n 1 n 2 n 0 let a11 a12 a21 b22 be matrices with dimensions n1 n1 n1 n2 n2 n1 and n2 n2 respectively we determine necessary and sufficient conditions so that there exists an n2 n2 matrix a22 such that a a 11 a 21 a 12 a 22 and i a is nonsingular and symmetric and b22 is the lower right block of a partitioning of a 1 ii a is symmetric positive definite and b22 is the lower right block of a partitioning of a 1

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

asymptotics for minimal discrete energy on the sphere we investigate the energy of arrangements of n points on the surface of the unit sphere sd in rd 1 that interact through a power law potential v 1 rs where s 0 and r is euclidean distance with ed s n denoting the minimal energy for such n point arrangements we obtain bounds valid for all n for ed s n in the cases when 0 s d and 2 d s for s d we determine the precise asymptotic behavior of ed d n as n as a corollary lower bounds are given for the separation of any pair of points in an n point minimal energy configuration when s d 2 for the unit sphere in r3 d 2 we present two conjectures concerning the asymptotic expansion of e2 s n that relate to the zeta function l s for a hexagonal lattice in the plane we prove an asymptotic upper bound that supports the first of these conjectures of related interest we derive an asymptotic formula for the partial sums of l s when 0 s 2 the divergent case

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

nodal length fluctuations for arithmetic random waves using the spectral multiplicities of the standard torus we endow the laplace eigenspaces with gaussian probability measures this induces a notion of random gaussian laplace eigenfunctions on the torus arithmetic random waves we study the distribution of the nodal length of random eigenfunctions for large eigenvalues and our primary result is that the asymptotics for the variance is nonuniversal our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy

a quantitative carleman estimate for second order elliptic operators we prove a carleman estimate for elliptic second order partial differential expressions with lipschitz continuous coefficients the carleman estimate is valid for any complex valued function u w2 2 with support in a punctured ball of arbitrary radius the novelty of this carleman estimate is that we establish an explicit dependence on the lipschitz and ellipticity constants the dimension of the space and the radius of the ball in particular we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and lipschitz constant

on a sharp lemma of cassels and montgomery on manifolds let m g be a d dimensional compact connected riemannian manifold and let m m 0 oo be a complete sequence of orthonormal eigenfunctions of the laplace beltrami operator on m we show that there exists a positive constant c such that for all integers n and x and for all finite sequences of n points in m x j and positive weights a j j 1 n we have a j m x j 2 max c x a j 2 a j 2 in the left hand side the inner sum is for j 1 n and the outer sum is for m 0 x

quantitative uniqueness for elliptic equations at the boundary of c1 dini domains abstract based on a variant of the frequency function approach of almgren we establish an optimal upper bound on the vanishing order of solutions to variable coefficient schrodinger equations at a portion of the boundary of a c 1 dini domain such bound provides a quantitative form of strong unique continuation at the boundary it can be thought of as a boundary analogue of an interior result recently obtained by bakri and zhu for the standard laplacian

sampling and equidistribution theorems for elliptic second order operators lifting of eigenvalues and applications abstract we consider elliptic second order partial differential operators with lipschitz continuous leading order coefficients on finite cubes and the whole euclidean space we prove quantitative sampling and equidistribution theorems for eigenfunctions the estimates are scale free in the sense that for a sequence of growing cubes we obtain uniform estimates these results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum and an uncertainty relation aka spectral inequality for short energy interval spectral projectors several applications including random operators are discussed in the proof we have to overcome several challenges posed by the variable coefficients of the leading term

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

quantitative uniqueness estimates for the general second order elliptic equations abstract in this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials we derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions the lower bounds depend on asymptotic behaviors of magnetic and electric potentials the proof is carried out by the carleman method and bootstrapping arguments

completion of an orthogonal projector and a related inverse mapping problem we study the completion problem for an orthogonal projector a of r n from knowledge of one or several of its columns as well as the related matrix inverse problem of finding an orthogonal projector a with ax y for two given n by k matrices x and y using geometric properties of orthogonal projections and of the singular value decomposition we derive theoretical and numerical solutions for these problems we give algorithms and test their matlab implementations

lower bounds for nodal sets of eigenfunctions we prove lower bounds for the hausdorff measure of nodal sets of eigenfunctions 0 introduction let m be a smooth closed riemannian manifold and the laplace operator a function u is said to be an eigenfunction with eigenvalue if u u 0 1 with our convention on the sign of the eigenvalues are non negative and go to infinity one of the most fundamental questions about eigenfunctions is to understand the sets where they vanish these sets are called nodal sets nodal sets are n 1 dimensional manifolds away from n 2 dimensional singular sets where the gradient also vanishes so it is natural to estimate the n 1 dimensional hausdorff measure hn 1 the main result of this short note is the following lower bound theorem 1 given a closed n dimensional riemannian manifold m there exists c so that hn 1 u 0 c 3 n 4 0 2 in particular theorem 1 gives a uniform lower bound in dimension n 3 in y s t yau conjectured the lower bound c in all dimensions this was proven for surfaces by bruning b and yau independently and for real analytic metrics by donnelly and fefferman in df but remains open in the smooth case the authors were partially supported by nsf grants dms 0606629 dms 0906233 and nsf frg grants dms 0854774 and dms 0853501 778 t h colding w p minicozzi ii the donnelly fefferman argument leads to exponentially decaying lower bounds in the smooth case see also hl in m mangoubi considered eigenfunctions on a ball in a manifold and proved lower bounds for the volume of the subset of the ball where the function is positive and the subset where it is negative combined with the isoperimetric inequality one can get the lower bound c 3 n 2 1 2n for the measure of the nodal set on the entire manifold recently sogge and zelditch sz proved the lower bound c 7 3n 8 their argument is completely different and is based in part on a beautiful new integral formula relating the l1 norm of u on the nodal set and the l1 norm of u on m finally note that some papers use 2 for the eigenvalue i e u 2 u with that convention our bound is c 3 n 2 1 finding good balls throughout m will be a fixed closed manifold with laplace operator and u will be an eigenfunction with eigenvalue and m u 2 1 we will always assume that 1 since our interest is in what happens when goes to infinity the first step is to fix a scale r depending on lemma 1 there exists a 0 so that u has a zero in every ball of radius a 3 this lemma is standard but we will include a proof since it is so short proof if u does not vanish on b r 3 then barta s theorem implies that the lowest dirichlet eigenvalue on this ball is at least let be a function that is identically one on b r 6 and that cuts off linearly to zero on the annulus b r 3 b r 6 it follows that 2 2 36r 2 vol b r 3 vol b r 6 c r2 1 1 where c comes from the bishop gromov volume comparison see p 275 of g and depends only on n the ricci curvature of m and an upper bound for r since this is impossible for r2 c 1 the lemma follows from now on we set r a 1 2 with a given by lemma 1 next we use a standard covering argument to decompose the manifold m into small balls of radius r if b is a ball in m then we write 2b for the ball with the same center as b and twice the radius lemma 2 there exists a collection bi of balls of radius r in m so that m i bi and each point of m is contained in at most cm cm m of the double balls 2bi s proof choose a maximal disjoint collection of balls b r 2 pi it follows immediately from maximality that the double balls bi br pi cover m suppose that p m is contained in balls b2r p1 b2r pk in particular the disjoint balls b r 2 p1 b 2 pk are all contained in b3r p so that k i 1 vol b r 2 pi vol b3r p 1 2

planck scale number of nodal domains for toral eigenfunctions abstract we study the number of nodal domains in balls shrinking slightly above the planck scale for generic toral eigenfunctions we prove that up to the natural scaling the nodal domains count obeys the same asymptotic law as the global number of nodal domains the proof on one hand uses new arithmetic information to refine bourgain s de randomisation technique at planck scale and on the other hand it requires a planck scale version of yau s conjecture which we believe to be of independent interest

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

spectral inequalities for non selfadjoint elliptic operators and application to the null controllability of parabolic systems we consider elliptic operators a on a bounded domain that are compact perturbations of a selfadjoint operator we first recall some spectral properties of such operators localization of the spectrum and resolvent estimates we then derive a spectral inequality that measures the norm of finite sums of root vectors of a through an observation with an exponential cost following the strategy of g lebeau and l robbiano 1995 we deduce the construction of a control for the non selfadjoint parabolic problem partial t u a u b g in particular the l 2 norm of the control that achieves the extinction of the lower modes of a is estimated examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of a

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

optimal asymptotic bounds for spherical designs in this paper we prove the conjecture of korevaar and meyers for each n cdt d there exists a spherical t design in the sphere s d consisting of n points where cd is a constant depending only on d

on the nodal sets of toral eigenfunctions we study the nodal sets of eigenfunctions of the laplacian on the standard d dimensional flat torus the question we address is can a fixed hypersurface lie on the nodal sets of eigenfunctions with ar bitrarily large eigenvalue in dimension two we show that this happens only for segments of closed geodesics in higher dimensions certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues our main result is that this cannot happen for hypersur faces with nonzero gauss kronecker curvature in dimension two the result follows from a uniform lower bound for the l 2 norm of the restriction of eigenfunctions to the curve proved in an earlier paper 1 in high dimensions we currently do not have this bound instead we make use of the real analytic nature of the flat torus to study variations on this bound for restrictions of eigenfunctions to suitable submanifolds in the complex domain in all of our results we need an arithmetic ingredient concerning the cluster structure of lattice points on the sphere we also present an independent proof for the two dimensional case relying on the abc theorem in function fields

bounds for eigenfunctions of the laplacian on compact riemannian manifolds suppose that is an eigenfunction of with eigenvalue 0 it is proved that c1 n 14 2 where n is the dimension of m and c1 depends only upon a bound for the absolute value of the sectional curvature of m and a lower bound for the injectivity radius of m it is then shown that if m admits an isometric circle action and the metric is generic one has exceptional sequences of eigenfunctions satisfying the complementary bounds k c2 n 18k 2

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

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

quadratic forms in random variables theory and applications textbook for a one semester graduate course for students specializing in mathematical statistics or in multivariate analysis or reference for theoretical as well as applied statisticians confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

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