Communications on Pure and Applied Mathematics最新文献

We prove that round balls of volume $��\le �1�$ uniquely minimize in Gamow's liquid drop model.

We capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well-prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one-dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.

We determine the asymptotics of the block Toeplitz determinants $��det�T_n��\left(�\varphi �\right)��$ as $��n�\to �\infty �$ for $��N�\times �N�$ matrix-valued piecewise continuous functions $�\varphi$ with a finitely many jumps under mild additional conditions. In particular, we prove that

We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in $�R^2$. We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

We construct using variational methods Hamiltonian stationary surfaces with isolated Schoen–Wolfson conical singularities. We obtain these surfaces through a convergence process reminiscent to the Ginzburg–Landau asymptotic analysis in the strongly repulsive regime introduced by Bethuel, Brezis and Hélein. We describe in particular how the prescription of Schoen–Wolfson conical singularities is related to optimal Wente constants.

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $�u$ be a real-valued harmonic function in $�R^n$ with $��u�\left(�0�\right)�=�0�$ and $��n�\ge �3�$. We prove

We construct an entire solution $��U�:�R^2�\to �R^2�$ to the elliptic system

We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish that these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher–Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial.