Communications on Pure and Applied Mathematics最新文献

Understanding the transition mechanism of boundary layer flows is of great significance in physics and engineering, especially due to the current development of supersonic and hypersonic aircraft. In this paper, we construct multiple unstable acoustic modes so-called, Mack modes, which play a crucial role during the early stage of transition in the supersonic boundary layer. To this end, we develop an inner-outer gluing iteration to solve a hyperbolic-elliptic mixed type and singular system.

Let $��\Omega �\subset �R^n�$ be an open set with the same volume as the unit ball $�B$ and let $��{\lambda }_k��\left(�\Omega �\right)��$ be the $�k$-th eigenvalue of the Laplace operator of $�\Omega$ with Dirichlet boundary conditions on $��\partial �\Omega �$. In this work, we answer the following question:

We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin–Li–Yau and Kröger, valid for Riesz exponents $��\gamma �\ge �1�$, extend to certain values , provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of Pólya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.

In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove $�\beta$-Hölder continuity of the integrated density of states for supercritical quasi-periodic Schrödinger operators restricted to the $�\ell$th stratum, for any and $��\ell �\ge �2�$. We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.

This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand – whose precise form derives directly from the theory of perfect plasticity – behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field – the Cauchy stress in the terminology of perfect plasticity – which allows us to define characteristic lines and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study [5], we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.

Dimer models have been the focus of intense research efforts over the last years. Our paper grew out of an effort to develop new methods to study minimizers or the asymptotic height functions of general dimer models and the geometry of their frozen boundaries. We prove a complete classification of the regularity of minimizers and frozen boundaries for all dimer models for a natural class of polygonal domains, much studied in numerical simulations and elsewhere. In particular, we show that the frozen boundaries are always algebraic curves. Our classification also implies that the Pokrovsky-Talapov law holds for all dimer models at a generic point on the frozen boundary and, in addition, shows a very strong local rigidity of dimer models, which can be interpreted as a geometric universality result. Indeed, we prove a converse result, showing that any geometric situation for any dimer model is, in the simply connected case, realized already by the lozenge model. To achieve these goals we develop a new study on the boundary regularity for a class of Monge–Ampère equations in non-strictly convex domains, of independent interest, as well as a new approach to minimality for a general dimer functional. In the context of polygonal domains, we give the first general results for the existence of gas domains for minimizers.

One of the greatest success stories of randomized algorithms in linear algebra has been the development of fast, randomized solvers for highly overdetermined linear least-squares problems. However, none of the existing algorithms is backward stable, preventing them from being deployed as drop-in replacements for existing QR-based solvers. This paper introduces sketch-and-precondition with iterative refinement (SPIR) and FOSSILS, two provably backward stable randomized least-squares solvers. SPIR and FOSSILS combine iterative refinement with a preconditioned iterative method applied to the normal equations and converge at the same rate as existing randomized least-squares solvers. This work offers the promise of incorporating randomized least-squares solvers into existing software libraries while maintaining the same level of accuracy and stability as classical solvers.