Mathematische Annalen最新文献

We study a natural biharmonic analogue of the classical Alt–Caffarelli problem, both under Dirichlet and under Navier boundary conditions. We show existence, basic properties and (C^{1,alpha })-regularity of minimisers. For the Navier problem we also obtain a symmetry result in case that the boundary data are radial. We find this remarkable because the problem under investigation is of higher order. Computing radial minimisers explicitly we find that the obtained regularity is optimal.

Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for the tunneling approximation recently established in Fefferman et al. (SIAM J Math Anal 54: 1105–1130, 2022), Helffer & Kachmar (Pure Appl Anal, 2024), thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

Let ({mathfrak {M}}^alpha ) be the spherical maximal operators of complex order (alpha ) on ({{mathbb {R}}^n}). In this article we show that when (nge 2), suppose

holds for some (alpha ) and (pge 2), then we must have that (textrm{Re},alpha ge max {1/p-(n-1)/2, -(n-1)/p }.) In particular, when (n=2), we prove that ( Vert {mathfrak {M}}^{alpha } f Vert _{L^p({{mathbb {R}}^2})} le CVert f Vert _{L^p({{mathbb {R}}^2})}) if (textrm{Re} ! alpha >max {1/p-1/2, -1/p}), and consequently the range of (alpha ) is sharp in the sense that the estimate fails for (textrm{Re} alpha <max {1/p-1/2, -1/ p}.)

N. García-Fritz and H. Pasten showed that Hilbert’s 10th problem is unsolvable in the ring of integers of number fields of the form (mathbb {Q}(root 3 of {p},sqrt{-q})) for positive proportions of primes p and q. We improve their proportions and extend their results to the case of number fields of the form (mathbb {Q}(root 3 of {p},sqrt{Dq})), where D belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.

We consider a family of singular surface quasi-geostrophic equations

on ([0,infty )times {mathbb {T}}^{2}), where (nu geqslant 0), (gamma in [0,3/2)), (alpha in [0,1/4)) and (xi ) is a space-time white noise. For the first time, we establish the existence of infinitely many non-Gaussian

• probabilistically strong solutions for every initial condition in (C^{eta }), (eta >1/2);

• ergodic stationary solutions.

The result presents a single approach applicable in the subcritical, critical as well as supercritical regime in the sense of Hairer (Invent Math 198(2):269–504, 2014). It also applies in the particular setting (alpha =gamma /2) which formally possesses a Gaussian invariant measure. In our proof, we first introduce a modified Da Prato–Debussche trick which, on the one hand, permits to convert irregularity in time into irregularity in space and, on the other hand, increases the regularity of the linear solution. Second, we develop a convex integration iteration for the corresponding nonlinear equation which yields non-unique non-Gaussian solutions satisfying powerful global-in-time estimates and generating stationary as well as ergodic stationary solutions.

Let R be a quiver Hecke algebra, and let (mathscr {C}_{w,v}) be the category of finite-dimensional graded R-module categorifying a q-deformation of the doubly-invariant algebra (^{N'(w)} {mathbb {C}}[N] ^{N(v)} ). In this paper, we prove that the localization (widetilde{mathscr {C}}_{w,v}) of the category (mathscr {C}_{w,v}) can be obtained as the localization by right braiders arising from determinantial modules. As its application, we show several interesting properties of the localized category (widetilde{mathscr {C}}_{w,v} ) including the right rigidity.

In this note, we introduce a natural analogue of Steinberg’s cross-section in the loop group of a reductive group (textbf{G}). We show this loop Steinberg’s cross-section provides a simple geometric model for the poset (B(textbf{G})) of Frobenius-twisted conjugacy classes (referred to as Newton strata) of the loop group. As an application, we confirms a conjecture by Ivanov on decomposing loop Delgine–Lusztig varieties of Coxeter type. This geometric model also leads to new and direct proofs of several classical results, including the converse to Mazur’s inequality, Chai’s length formula on (B(textbf{G})), and a key combinatorial identity in the study affine Deligne–Lusztig varieties with finite Coxeter parts.

The paper addresses an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729–1756, 2021) on the existence of normalized solutions to Schrödinger equations with potentials and inhomogeneous nonlinearities. We consider the problem

both on ({mathbb R}^N) as well as on domains (rOmega ) where (Omega subset {mathbb R}^N) is a bounded smooth star-shaped domain and (r>0) is large. The exponents satisfy (2<p<2+frac{4}{N}<q<2^*=frac{2N}{N-2}), so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Nonlinear Schrödinger equations with combined power-type nonlinearities have been investigated first by Tao et al. (Commun Partial Differ Equ 32(7-9):1281-1343, 2007). Due to the presence of the potential a by now standard approach based on the Pohozaev identity cannot be used. We develop a robust method to study the existence of normalized solutions of nonlinear Schrödinger equations with potential and find conditions on V so that normalized solutions exist. Our results are new even in the case (beta =0).

We study the perfect conductivity problem with closely spaced perfect conductors embedded in a homogeneous matrix where the current-electric field relation is the power law (J=sigma |E|^{p-2}E). The gradient of solutions may be arbitrarily large as (varepsilon ), the distance between inclusions, approaches to 0. To characterize this singular behavior of the gradient in the narrow region between two inclusions, we capture the leading order term of the gradient. This is the first gradient asymptotics result on the nonlinear perfect conductivity problem.