patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

a partial differential equation for the rank one convex envelope a partial differential equation pde for the rank one convex envelope is introduced the existence and uniqueness of viscosity solutions to the pde is established elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the pde is proven computational results are presented and laminates are computed from the envelopes results include the kohn strang example the classical four gradient example and an example with eight gradients which produces nontrivial laminates

removable sets in elliptic equations with musielak orlicz growth abstract in this study in terms of the intrinsic hausdorff measures we characterize the size of removable sets for holder continuous solutions to elliptic equations with musielak orlicz growth in the general case we provide a result on a new scale that is more relevant and that captures the classical results as special cases the results slightly refine the known ones provided for problems stated in the variable exponent and double phase spaces and they essentially improve the one known in the orlicz case

calderon zygmund estimates for generalized double phase problems abstract we prove calderon zygmund type estimates for distributional solutions to non uniformly elliptic equations of generalized double phase type in divergence form in particular we provide sharp conditions on the nonlinear operators to establish the calderon zygmund type estimates

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

growth conditions and regularity for weak solutions to nonlinear elliptic pdes abstract we describe some aspects of the process approach to interior regularity of weak solutions to a class of nonlinear elliptic equations in divergence form as well as of minimizers of integrals of the calculus of variations in particular with respect to the growth conditions we emphasize some differences between solving equations and minimizing energy integrals and also between x dependence or x independence

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

boundary regularity of minimizers of double phase functionals abstract in this paper we treat the functional of double phase with variable exponents d u g p x a x d u g q x d x where a x is a non negative holder continuous function with 0 1 p x and q x holder continuous functions with 1 p x q x p x and g i j g x i j 1 2 for a continuous positive definite matrix valued function g g we prove that the minimizer of the above functional with suitable dirichlet boundary condition is holder continuous up to the boundary when g is holder continuous we see also that du is locally holder continuous

global gradient estimates for the borderline case of double phase problems with bmo coefficients in nonsmooth domains abstract we consider a double phase problem with bmo coefficient in divergence form on a bounded nonsmooth domain the problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position which describes a feature of strongly anisotropic materials we obtain the global calderon zygmund type estimates for the distributional solution in the case that the associated nonlinearity has a small bmo and the boundary of the domain is sufficiently flat in the reifenberg sense

isometry invariant solutions to a critical problem on non compact riemannian manifolds abstract we analyse an elliptic equation with critical growth set on a d dimensional d 3 hadamard manifold m g by adopting a variational perspective we prove the existence of non zero non negative solutions invariant under the action of a specific family of isometries our result remains valid when the original nonlinearity is singularly perturbed preserving the same variational approach but considering other groups of isometries we finally show that when m r d d 3 and the nonlinearity is odd there exist at least 1 d d 3 2 pairs of sign changing solutions

powers of large random unitary matrices and toeplitz determinants we study the limiting behavior of tr u k n where u is an n x n random unitary matrix and k n is a natural number that may vary with n in an arbitrary way our analysis is based on the connection with toeplitz determinants the central observation of this paper is a strong szego limit theorem for toeplitz determinants associated to symbols depending on n in a particular way as a consequence of this result we find that for each fixed m n the random variables tru k j n min k j n n j 1 m converge to independent standard complex normals

partial holder regularity for elliptic systems with non standard growth abstract we study partial holder regularity for elliptic systems with non standard growth we consider general systems with orlicz growth and discontinuous coefficient factor for which we prove that their weak solutions are partially holder continuous for any holder exponents in addition we also obtain a similar result for systems with double phase growth

application of optimal transport and the quadratic wasserstein metric to full waveform inversion conventional full waveform inversion fwi using the least squares norm as a misfit function is known to suffer from cycle skipping issues which increases the risk of computing a local rather than the global minimum of the misfit the quadratic wasserstein metric has been proved to have many ideal properties with regards to convexity and insensitivity to noise when the observed and predicted seismic data are considered to be two density functions the quadratic wasserstein metric corresponds to the optimal cost of rearranging one density into the other where the transportation cost is quadratic in distance unlike the least squares norm the quadratic wasserstein metric measures not only amplitude differences but also global phase shifts which helps to avoid cycle skipping issues we propose a new way of using the quadratic wasserstein metric trace by trace in fwi and compare it to the global quadratic wasserstein metric via the solution of the monge ampere equation we incorporate the quadratic wasser

convergent filtered schemes for the monge ampere partial differential equation the theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear partial differential equations such as the elliptic monge ampere equation the approximation theory of barles and souganidis asymptotic anal 4 1991 pp 271 283 requires that numerical schemes be monotone or elliptic in the sense of a m oberman siam j numer anal 44 2006 pp 879 895 but such schemes have limited accuracy in this article we establish a convergence result for filtered schemes which are nearly monotone this allows us to construct finite difference discretizations of arbitrarily high order we demonstrate that the higher accuracy is achieved when solutions are sufficiently smooth in addition the filtered scheme provides a natural detection principle for singularities we employ this framework to construct a formally second order scheme for the monge ampere equation and present computational results on smooth and singular solutions

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

local higher integrability of the gradient of a quasiminimizer under generalized orlicz growth conditions abstract we study local quasiminimizers of the dirichlet energy under generalized growth conditions special cases include standard variable exponent and double phase growths we show that the gradient of a local quasiminimizer has local higher integrability

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

eigenvalue distributions of random unitary matrices let u be an n n random matrix chosen from haar measure on the unitary group for a fixed arc of the unit circle let x be the number of eigenvalues of m which lie in the specified arc we study this random variable as the dimension n grows using the connection between toeplitz matrices and random unitary matrices and show that x e x var x 1 2 is asymptotically normally distributed in addition we show that for several fixed arcs i1 im the corresponding random variables are jointly normal in the large n limit

high powers of random elements of compact lie groups if a random unitary matrix u is raised to a sufficiently high power its eigenvalues are exactly distributed as independent uniform phases we prove this result and apply it to give exact asymptotics of the variance of the number of eigenvalues of u falling in a given arc as the dimension of u tends to infinity the independence result it turns out extends to arbitrary representations of arbitrary compact lie groups we state and prove this more general theorem paying special attention to the compact classical groups and to wreath products this paper is excerpted from the author s doctoral thesis 9

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

holder regularity for nonlocal double phase equations abstract we prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro differential equations whose leading operator switches between two different types of fractional elliptic phases according to the zero set of a modulating coefficient a a the model case is driven by the following nonlocal double phase operator u x u y p 2 u x u y x y n s p d y a x y u x u y q 2 u x u y x y n t q d y where q p and a 0 our results do also apply for inhomogeneous equations for very general classes of measurable kernels by simply assuming the boundedness of the modulating coefficient we are able to prove that the solutions are holder continuous whereas similar sharp results for the classical local case do require a to be holder continuous to our knowledge this is the first regularity result for nonlocal double phase problems

filtered schemes for hamilton jacobi equations we build a simple and general class of finite difference schemes for first order hamilton jacobi hj partial differential equations these filtered schemes are convergent to the unique viscosity solution of the equation the schemes are accurate we implement second third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions indicating how to build higher order ones they are also explicit which means they can be solved using the fast sweeping method the accuracy of the method is validated with computational results for the eikonal equation and other hj equations in one and two dimensions using filtered schemes made from standard centered differences higher order upwinding and eno interpolation

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

global gradient estimates for a borderline case of double phase problems with measure data abstract we obtain a global calderon zygmund estimate for a borderline case of double phase problems with measure data in terms of the 1 fractional maximal function of the measure

on the regularity of the ω minima of φ functionals abstract we focus on some regularity properties of minima of variational integrals with growth and provide an upper bound on the hausdorff dimension of their singular set

campanato morrey spaces for the double phase functionals with variable exponents abstract our aim in this paper is to show that the riesz potential operator i of variable order embeds from variable exponent morrey spaces l p g to campanato morrey spaces in the case x p x x our result extends the recent work of rafeiro and samko 2019 and the authors mizuta et al 2020 we show that i embeds from morrey spaces l g of the double phase functionals x t t p x b x t q x to campanato morrey spaces where p and q satisfy log holder conditions p x q x and b is nonnegative bounded and holder continuous of order 0 1 we also discuss the continuity of riesz potentials i f of functions in l g and show that i embeds from l g to vanishing campanato morrey spaces

mock gaussian behaviour for linear statistics of classical compact groups we consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups we compute their moments and find that the first few moments are gaussian whereas the limiting distribution is not the precise number of gaussian moments depends upon the particular statistic considered

global lipschitz continuity for minima of degenerate problems we consider the problem of minimizing the lagrangian int f nabla u f u among functions on omega subset mathbb r n with given boundary datum varphi we prove lipschitz regularity up to the boundary for solutions of this problem provided omega is convex and varphi satisfies the bounded slope condition the convex function f is required to satisfy a qualified form of uniform convexity only outside a ball and no growth assumptions are made

numerical solution of the optimal transportation problem using the monge ampere equation a numerical method for the solution of the elliptic monge ampere partial differential equation with boundary conditions corresponding to the optimal transportation ot problem is presented a local representation of the ot boundary conditions is combined with a finite difference scheme for the monge ampere equation email protected s method is implemented leading to a fast solver comparable to solving the laplace equation on the same grid several times theoretical justification for the method is given by a convergence proof in the companion paper 4 solutions are computed with densities supported on non convex and disconnected domains computational examples demonstrate robust performance on singular solutions and fast computational times

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

regularity for multi phase variational problems abstract we prove c 1 regularity for local minimizers of the multi phase energy w d w p a x d w q b x d w s d x under sharp assumptions relating the couples p q and p s to the holder exponents of the modulating coefficients a and b respectively

finite element methods for fully nonlinear second order pdes based on a discrete hessian with applications to the monge ampere equation the purpose of this paper is twofold first we modify a method due to lakkis and pryer where the notion of a discrete hessian is introduced to compute fully nonlinear second order pdes the discrete hessian used in our approach is entirely local making the resulting linear system within the newton iteration much easier to solve the second contribution of this paper is to analyze both lakkis and pryer s method and its modification in parallel applied to the two dimensional monge ampere equation in both cases we show the well posedness of the methods as well as derive optimal error estimates numerical experiments are presented which i back up the theoretical findings and ii indicate that the methods are able to capture weak viscosity solutions

double phase image restoration abstract in this paper we explore the potential of the double phase functional in an image processing context to this end we study minimizers of the double phase energy for functions with bounded variation and show that this energy can be obtained by convergence or relaxation of regularized functionals a central tool is a capped fractional maximal function of the derivative of bv functions

toeplitz and wiener hopf determinants with piecewise continuous symbols abstract the asymptotics for determinants of toeplitz and wiener hopf operators with piecewise continuous symbols are obtained in this paper if w is the wiener hopf operator defined on l2 0 with piecewise continuous symbol having a finite number of discontinuities at r then under appropriate conditions it is shown that det w g r2k where g exp log 0 r 1 2 log r r and k is a completely determined constant an analogous result is obtained for toeplitz operators the main point of the paper is to obtain a result in the wiener hopf case since the toeplitz case had been treated earlier in the toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known fortunately in the toeplitz case such asymptotics were known the difficulty in the wiener hopf case is that there was not a single singular case where the determinant was explicitly known this problem was overcome by using the fact that wiener hopf determinants when discretized become toeplitz determinants whose entries depend on the size of the matrix no theorem on toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results

hardy sobolev inequalities in the unit ball for double phase functionals abstract we establish hardy sobolev inequalities in the unit ball b for double phase functionals x t t p b x t q where 1 p q b is non negative and radially holder continuous of order 0 1