Communications on Pure and Applied Mathematics最新文献

We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices $��D�\ge �3�$; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $�{\dot{H}}^1$-bounded radial solutions to (NLH) in dimensions $��N�\ge �7�$, building upon soliton resolution for such solutions. To our knowledge, this provides the first rigorous classification of bubble tree dynamics within symmetry. We introduce a new approach based on the energy method that does not rely on maximum principle. The key ingredient of the proof is a monotonicity estimate near any bubble tree configurations, which in turn requires a delicate construction of modified multi-bubble profiles also.

We compute the leading order asymptotic of the maximum of the characteristic polynomial for i.i.d. matrices with real or complex entries. In particular, this result is new even for real Ginibre matrices, which was left as an open problem in Lambert et al. Electron. J. Probab. 29 (2024); the complex Ginibre case was covered in Lambert, Comm. Math Phys. 378 (2020). These are the first universality results for the non-Hermitian analog of the first order term of the Fyodorov–Hiary–Keating conjecture. Our methods are based on constructing a coupling to the branching random walk (BRW) via Dyson Brownian motion. In particular, we find a new connection between real i.i.d. matrices and inhomogeneous BRW.

Let $��(�M^n�,�g�)�$ be a complete Riemannian manifold which is not isometric to $�R^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $��G�\subset �(�0�,�\infty �)�$ with density 1 at infinity such that for every $��V�\in �G�$ there is a unique isoperimetric set of volume $�V$ in $�M$; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals $��I_n�\subset ��(�0�,�\infty �)��$ such that isoperimetric sets with volumes <

The problem of deriving a gradient flow structure for the porous medium equation which is thermodynamic, in that it arises from the large deviations of some microscopic particle system is studied. To this end, a rescaled zero-range process with jump rate $��g��\left(�k�\right)��=�k^{\alpha }�,�\alpha �>�1�$ is considered, and its hydrodynamic limit and dynamical large deviations are shown in the presence of both degenerate and unbounded diffusion. The key super-exponential estimate is obtained using pathwise discretised regularity estimates in the spirit of the Aubin–Lions–Simons lemma. This allows to exhibit the porous medium equation as the gradient flow of the entropy in a thermodynamic metric via the energy-dissipation inequality.

For any $��p�\in �(�1�,�\infty �)�$, we construct $�p$-energies and the corresponding $�p$-energy measures on the Sierpiński carpet. A salient feature of our Sobolev space is the self-similarity of energy. An important motivation for the construction of self-similar energy and energy measures is to determine whether or not the Ahlfors regular conformal dimension is attained on the Sierpiński carpet. If the Ahlfors regular conformal dimension is attained, we show that any optimal Ahlfors regular measure attaining the Ahlfors regular conformal dimension must necessarily be a bounded perturbation of the $�p$-energy measure of some function in our Sobolev space, where $�p$ is the Ahlfors regular conformal dimension. Under the attainment of the Ahlfors regular conformal dimension, the $��(�1�,�p�)�$-Newtonian Sobolev space corresponding to any optimal Ahlfors regular metric and measure is shown to coincide with our Sobolev space with comparable norms, where $�p$ is the Ahlfors regular conformal dimension.

In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearized elasticity and an ansatz on the distribution of incompatibilities of the lattice at the interface between two grains. The logarithmic scaling of this formula has been rigorously justified without any ansatz on the geometry of dislocations only recently in an article by Lauteri and Luckhaus. In the present paper, building upon their analysis, we derive a two dimensional sharp interface limiting functional starting from the nonlinear semi-discrete model introduced in Lauteri and Luckhaus: the line tension we obtain via $�\Gamma$-convergence depends on the rotations of the grains and the relative orientations of the interfaces, and for small angle grain boundaries has the Read and Shockley logarithmic scaling.

Perfectly matched layers are extensively used to compute approximate solutions for Maxwell's equations in $�R^{�1�+�3�}$ using a bounded computational domain, usually a rectangular solid. A smaller rectangular domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the boundary of the computational domain, an absorbing boundary condition is imposed that is necessarily imperfect. The method replaces the Maxwell equations by a larger system, and introduces absorption coefficients positive in the layers. Well posedness of the resulting initial boundary value problem is proved here for the first time. The Laplace transform of a resulting Helmholtz system is studied. For positive real values of the transform variable $�\tau$, the Helmholtz system has a unique solution from a variational form that yields limited regularity for rectangular domains. When $�\tau$ is not real the complex variational form loses positivity. We smooth the domain and, in spite of this loss, construct $�H^2$ solutions with uniform $�L^2$ estimates. Using the $�H^2$ regularity, we deduce Maxwell from Helmholtz, then remove the smoothing. The boundary condition at the smoothed boundary must be carefully chosen. A method of Jerison-Kenig-Mitrea is extended to compensate the nonpositivity of the flux.

This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific $�N$-representability problem inherent to DMET.