Communications on Pure and Applied Mathematics最新文献

We prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end, we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with $��k�\text{th}�$-order remainders that satisfy uniform $�C^k$-estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for $��k�=�0�$ known from previous work of the second-named author. For $��k�>�0�$, the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.

We prove the uniqueness of $�L^1$ blow-down limit at infinity for an entire minimizing solution $��u�:�R^2�\to �R^2�$ of a planar Allen–Cahn system with a triple-well potential. Consequently, $�u$ can be approximated by a triple junction map at infinity. The proof exploits a careful analysis of energy upper and lower bounds, ensuring that the diffuse interface remains within a small neighborhood of the approximated triple junction at all scales.

We consider the analysis of singular waveguides separating insulating phases in two-space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one-dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping-accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory.

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 study the problem of algorithmically optimizing the Hamiltonian $�H_N$ of a spherical or Ising mixed $�p$-spin glass. The maximum asymptotic value $�\mathrm{OPT}$ of $��H_N�/�N�$ is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing (AMP) algorithms efficiently optimize $��H_N�/�N�$ up to a value $�\mathrm{ALG}$ given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property (OGP). However, $��\mathrm{ALG}�<�\mathrm{OPT}�$ can also occur, and no efficient algorithm producing an objective value exceeding $�\mathrm{ALG}$ is known. We prove that for mixed even $�p$-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than $�\mathrm{ALG}$ with non-negligible probability.

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.