non negative solutions of the evolution p laplacian equation initial traces and cauchy problem when 1 p 2 from the introduction we continue here the investigation initiated earlier by us trans amer math soc 314 1989 no 1 187 224 concerning the solvability of the cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation 1 ut div du p 2 du 0 in st rn 0 t 0 2 both in terms of results and techniques the main difference stems from the fact that unlike solutions in the case p 2 solutions of 1 are not in general locally bounded

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

asymptotic behaviour of the doubly nonlinear diffusion equation ut δpum on bounded domains abstract we study the homogeneous dirichlet problem for the doubly nonlinear diffusion equation u t p u m where p 1 m 0 posed in a bounded domain in r n with homogeneous boundary conditions and with non negative and integrable initial data in this paper we consider the degenerate case m p 1 1 and the quasilinear case m p 1 1 in the first case we establish the large time behaviour by proving the uniform convergence to a unique asymptotic profile and we also give rates for this convergence the difference in the second case is that the asymptotic profile is unique up to a constant factor that we determine

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

bounds for rankin selberg integrals and quantum unique ergodicity for powerful levels reference epfl article 202508doi 10 1090 s0894 0347 2013 00779 1view record in web of science record created on 2014 10 23 modified on 2017 12 10

equidistribution of cusp forms in the level aspect let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing square free level q infinity we prove that the pushforward of the mass off to the modular curve of level 1 equidistributes with respect to the poincare measure

rankin selberg method for real analytic cusp forms of arbitrary real weight 1 2 indeed rankin and selberg applied their method to the more general situation of holomorphic cusp forms of type f r v where f is of finite index in f r 0 is the weight and v is a compatible multiplier system let f z be such a cusp form f f m and m1 mm be representatives of distinct right cosets then the vector valued function f z clz d l f m l z cruz d rf mmz t means transpose c dl is the second row of m is of type f r e where e is a unitary m dimensional multiplier system so the concept of vector valued automorphic forms for f is intimately related to the investigation of automorphic forms for subgroups of finite index in v cf 13

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

quantum unique ergodicity for half integral weight automorphic forms we investigate the analogue of the quantum unique ergodicity que conjecture for half integral weight automorphic forms assuming the generalized riemann hypothesis grh we establish que for both half integral weight holomorphic hecke cusp forms for 0 4 lying in kohnen s plus subspace and for half integral weight hecke maa cusp forms for 0 4 lying in kohnen s plus subspace by combining the former result along with an argument of rudnick it follows that under grh the zeros of these holomorphic hecke cusp equidistribute with respect to hyperbolic measure on 0 4 h as the weight tends to infinity

local behaviour of solutions to doubly nonlinear parabolic equations we give a relatively simple and transparent proof for harnack s inequality for certain degenerate doubly nonlinear parabolic equations we consider the case where the lebesgue measure is replaced with a doubling borel measure which supports a poincare inequality

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

transformations of self similar solutions for porous medium equations of fractional type abstract we consider four different models of nonlinear diffusion equations involving fractional laplacians and study the existence and properties of classes of self similar solutions such solutions are an important tool in developing the general theory we introduce a number of transformations that allow us to map complete classes of solutions of one equation into those of another one thus providing us with a number of new solutions as well as interesting connections special attention is paid to the property of finite propagation

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

the nonlocal porous medium equation barenblatt profiles and other weak solutions a degenerate nonlinear nonlocal evolution equation is considered it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal we show the existence of sign changing weak solutions to the corresponding cauchy problem moreover we construct explicit compactly supported self similar solutions which generalize barenblatt profiles the well known solutions of the classical porous medium equation

the influence of fractional diffusion in fisher kpp equations we study the fisher kpp equation where the laplacian is replaced by the generator of a feller semigroup with power decaying kernel an important example being the fractional laplacian in contrast with the case of the standard laplacian where the stable state invades the unstable one at constant speed we prove that with fractional diffusion generated for instance by a stable levy process the front position is exponential in time our results provide a mathematically rigorous justification of numerous heuristics about this model

on quadratic families of cm elliptic curves given a cm elliptic curve with weierstrass equation y 2 f x and a positive definite binary quadratic form q v v we show that there are infinitely many reduced integer pairs u v such that the twisted elliptic curve q u v y 2 f x has analytic rank and consequently mordell weil rank one in fact it follows that the number of such pairs with u v x is at least x 2 e for any e 0

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

weak subconvexity for central values of l functions we describe a general method to obtain weak subconvexity bounds for many classes of l functions we give several examples of our bound and our work has applications to a conjecture of rudnick and sarnak for the mass equidistribution of hecke eigenforms

sieving for mass equidistribution we approach the holomorphic analogue to the quantum unique ergodicity conjecture through an application of the large sieve we deal with shifted convolution sums as in ho109 with various simplifications in our analysis due to the knowledge of the ramanujan petersson conjecture in this holomorphic case

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

on the nonlinear diffusion equation of kolmogorov petrovsky and piscounov abstract recently bramson proved a theorem that classifies the initial data under which solutions of the k p p equation converge to the appropriate travelling waves in this paper a simplified proof is given by using maximal principles instead of his brownian motion approach the regularity condition on the forcing term is also weakened

a fisher kolmogorov equation with finite speed of propagation abstract in this paper we study a fisher kolmogorov type equation with a flux limited diffusion term and we prove the existence and uniqueness of finite speed moving fronts and the existence of some explicit solutions in a particular regime of the equation

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

on solutions of some doubly nonlinear degenerate parabolic equations with absorption abstract the existence uniqueness regularity and behavior of solutions to the initial boundary value problem for the equation u t i 1 n x i u x i p 2 u x i u 0 with zero dirichlet boundary condition and with nonnegative initial condition are studied in a bounded domain in r n where 0 and p 1

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

exponential convergence towards stationary states for the 1d porous medium equation with fractional pressure abstract we analyse the asymptotic behaviour of solutions to the one dimensional fractional version of the porous medium equation introduced by caffarelli and vazquez 1 2 where the pressure is obtained as a riesz potential associated with the density we take advantage of the displacement convexity of the riesz potential in one dimension to show a functional inequality involving the entropy entropy dissipation and the euclidean transport distance an argument by approximation shows that this functional inequality is enough to deduce the exponential convergence of solutions in self similar variables to the unique steady states

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

mass equidistribution for hecke eigenforms we prove a conjecture of rudnick and sarnak on the mass equidistribution of hecke eigenforms this builds upon independent work of the authors

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions