Communications on Pure and Applied Mathematics最新文献

We study the obstacle problem for parabolic operators of the type $��{\partial }_t�+�L�$, where L is an elliptic integro-differential operator of order 2s, such as $�{�(�-�\Delta �)�}^s$, in the supercritical regime $��s�\in �(�0�,�\frac{1}{2}�)�$. The best result in this context was due to Caffarelli and Figalli, who established the $�C_x^{�1�,�s�}$ regularity of solutions for the case $��L�=�{�(�-�\Delta �)�}^s�$, the same regularity as in the elliptic setting.

Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C^{1, 1} in space and time, and that this is optimal. We also deduce the $�C^{�1�}$

Consider the normalized adjacency matrices of random d-regular graphs on N vertices with fixed degree $��d�⩾�3�$. We prove that, with probability $��1�-�N^{�-�1�+�\epsilon �}�$ for any $��\epsilon �>�0�$, the following two properties hold as $��N�\to �\infty �$ provided that $��d�⩾�3�$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N, that is, $��{\lambda }_2�,��\left|�{\lambda }_N�\right|��⩽�2�+�N^{�-�c�}�$. (ii) All eigenvectors of random d-regular graphs are completely delocalized.

Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l.

We are grateful to Peter Thompson for pointing out an error in [1, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that $�\hat{\theta }$ is a vector of constants. However, some of the components of $�\hat{\theta }$ could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with $�\hat{\theta }$ involving states) later in [1, Proposition 3.4].

We give a more explicit version of the statement and provide a correct proof. The desired statement will be deduced from the following:

The following corollary is equivalent to [1, Lemma 3.5, p. 1848] but explicitly highlights that some of the entries of $�\hat{\theta }$ may be initial conditions, not only system parameters.

We show that for an area minimizing m-dimensional integral current T of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most $��m�-�2�$. This provides a strengthening of the existing $��(�m�-�2�)�$-dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate T along blow-up scales.

Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schön assumptions. As the second application, we prove an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013.

We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse-supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.

Motivated by the physics literature on “photonic doping” of scatterers made from “epsilon-near-zero” (ENZ) materials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region $��\Omega �\times �R�$ is affected by the presence of a “dopant” $��D�\subset �\Omega �$ in which the dielectric permittivity is not near zero. Mathematically, this reduces to analysis of a 2D Helmholtz equation $��\mathrm{div}���(�a��\left(�x�\right)��\nabla �u�)��+�k^2�u�=�f�$ with a piecewise-constant, complex valued coefficient a that is nearly infinite (say $��a�=�\frac{1}{\delta }�$ with $��\delta �\approx �0�$) in $��\Omega �\setminus �\overline{D}�$. We show (under suitable hypotheses) that the solution u depends analytically on δ near 0, and we give a simple PDE characterization of the terms in its Taylor expansion. For the appli

This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self-interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal-to-noise model of soft spins in an anisotropic well, for which we prove a negative-second-moment threshold distinguishing positive from zero complexity. A universal near-critical behavior appears within this phase portrait, namely quadratic near-critical vanishing of the complexity of total critical points, and cubic near-critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, that is, beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non-invariant random matrices from our companion paper (Ben Arous, Bourgade, McKenna 2022), and the atypical convexity and integrability of the limiting variational problems.

We use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers $��N_j��\left(�g�\right)��$ of 4-valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus g, we calculate the leading term in the asymptotics of $��N_j��\left(�g�\right)��$ as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David. These phase transitions are of the following four types: (a) one-cut to two-cut through the splitting of the cut at the origin, (b) two-cut to three-cut through the birth of a new cut at the origin, (c) one-cut to three-cut through the splitting of the cut at two symmetric points, and (d) one-cut to three-cut through the birth of two symmetric cuts.