We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas–Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas–Ni–Nirenberg type.
In this paper, we develop a direct for the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre, we work directly on the non-local operator. Using the integral defining the fractional Laplacian, by an elementary approach, we first obtain the key ingredients needed in the either in a bounded domain or in the whole space, such as , , and . Then, using simple examples, semi-linear equations involving the fractional Laplacian, we illustrate how this new can be employed to obtain symmetry and non-existence of positive solutions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities.
Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.
Let be a classical pseudodifferential operator of order on an -dimensional manifold . For the truncation to a smooth subset there is a well-known theory of boundary value problems when has the transmission property (preserves ) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian with , are not covered. They have instead the -transmission property defined in Hörmander's books, mapping into . In an unpublished lecture note from 1965, Hörmander described an -solvability theory for -transmission operators, departing from Vishik and Eskin's results. We here develop the theory in Sobolev spaces ( ) in a modern setting. It leads to not only Fredholm solvability statements but also regularity results in full scales of Sobolev spaces ( ). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Hölder spaces, which radically improve recent regularity results for fractional Laplacians.
Let P be a classical pseudodifferential operator of order m epsilon C on an n-dimensional C-infinity manifold Omega(1). For the truncation P-Omega to a smooth subset Omega there is a well known theory of boundary value problems when P-Omega has the transmission property (preserves C-infinity ((Omega) over bar)) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian (-Delta)(mu) with mu is not an element of Z, are not covered. They have instead the mu-transmission property defined in Hormander's books, mapping x(n)(mu) C-infinity ((Omega) over bar) into C-infinity((Omega) over bar). In an unpublished lecture note from 1965, Hormander described an L-2-solvability theory for mu-transmission operators, departing from Vishik and Eskin's results. We here develop the theory in Lp Sobolev spaces (1 infinity). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Holder spaces, which radically improve recent regularity results for fractional Laplacians. (C) 2014 Elsevier Inc. All rights reserved.
We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well-known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of ‘spiked’ random matrix models about which such predictions (called the ) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed.
For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.
We introduce a new category , which we call the , obtained as a quotient of the bounded derived category of the module category of a finite-dimensional hereditary algebra over a field. We show that, in the simply laced Dynkin case, can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky. Using approximation theory, we investigate the tilting theory of , showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.
Let be the upper half Euclidean space and let be any real number between 0 and 2. Consider the following Dirichlet problem involving the fractional Laplacian: Instead of using the conventional extension method of Caffarelli and Silvestre , we employ a new and direct approach by studying an equivalent integral equation Applying the , we prove the non-existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions.
We consider sequences of graphs and define various notions of convergence related to these sequences: “left convergence” defined in terms of the densities of homomorphisms from small graphs into ; “right convergence” defined in terms of the densities of homomorphisms from into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs , and for graphs with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemerédi partitions, sampling and testing of large graphs.
A multi(sub)linear maximal operator that acts on the product of Lebesgue spaces and is smaller than the -fold product of the Hardy–Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón–Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón–Zygmund operators with functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.
We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.
Consider Hermitian or symmetric random matrices with independent entries, where the distribution of the matrix element is given by the probability measure with zero expectation and with variance . We assume that the variances satisfy the normalization condition for all and that there is a positive constant such that . We further assume that the probability distributions have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order where is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If denotes the of the -th eigenvalue under the semicircle law ordered in increasing order, then the -th eigenvalue is close to in the sense that for some positive constants , for large enough. (2) The proof of (Dyson, 1962 ) which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order up to logarithmic corrections. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large limit provided that the second moments of the two ensembles are identical.
We consider BPS states in a large class of , field theories, obtained by reducing six-dimensional superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich–Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in Gaiotto et al. . It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.
In this paper we provide a novel strategy to prove the validity of Hartreeʼs theory for the ground state energy of bosonic quantum systems in the mean-field regime. For the known case of trapped Bose gases, this can be shown using the strong quantum de Finetti theorem, which gives the structure of infinite hierarchies of -particles density matrices. Here we deal with the case where some particles are allowed to escape to infinity, leading to a lack of compactness. Our approach is based on two ingredients: (1) a weak version of the quantum de Finetti theorem, and (2) geometric techniques for many-body systems. Our strategy does not rely on any special property of the interaction between the particles. In particular, our results cover those of Benguria–Lieb and Lieb–Yau for, respectively, bosonic atoms and boson stars.
By examining asymptotic behavior of certain infinite basic ( -) hypergeometric sums at roots of unity (that is, at a ‘ -microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a -analogue of Ramanujan's formula of the two supercongruences valid for all primes , where denotes the truncation of the infinite sum at the -th place and stands for the quadratic character modulo 3.
The classical Minkowski problem leads to the Minkowski problem and now to the Orlicz Minkowski problem. Existence is demonstrated for the even Orlicz Minkowski problem. A byproduct is a new approach to the solution of the classical Minkowski problem.
The logarithmic Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-volume measure of a convex body. This problem was solved recently by Böröczky, Lutwak, Yang and Zhang for even measures (Böröczky et al. (2013) ). This paper solves the case of discrete measures whose supports are in general position.
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to compare the combinatorial structure directly to the independent discrete process, without renormalizing. The quality of approximation can often be conveniently quantified in terms of total variation distance, for functionals which observe part, but not all, of the combinatorial and independent processes. Among the examples are combinatorial assemblies (e.g., permutations, random mapping functions, and partitions of a set), multisets (e.g., polynomials over a finite field, mapping patterns and partitions of an integer), and selections (e.g., partitions of an integer into distinct parts, and square-free polynomials over finite fields). We consider issues common to all the above examples, including equalities and upper bounds for total variation distances, existence of limiting processes, heuristics for good approximations, the relation to standard generating functions, moment formulas and recursions for computing densities, refinement to the process which counts the number of parts of each possible type, the effect of further conditioning on events of moderate probability, large deviation theory and nonuniform measures on combinatorial objects, and the possibility of getting useful results by overpowering the conditioning.Copyright 1994, 1999 Academic Press, Inc,
We use a new interpolative approach to study the classical polynomial Bohnenblust-Hille inequality. This allows us to greatly improve the best known bounds of this inequality since we actually show that it is subexponential. With this new estimation at hand, we are also able to prove that the Bohr radius of the polydisk D-n behaves, asymptotically, as root log n/n. This, in particular, solves several questions from . (C) 2014 Elsevier Inc. All rights reserved.