Search

In a recent work [Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 2024], Gastel and Neff introduced an interesting system from a geometrically nonlinear flat cosserat micropolar model and established interior regularity in the critical dimension. Motived by this work, in this article, we establish both interior regularity and sharp $L^p$ regularity for their system in supercritical dimensions., Comment: 21 pages

This little note is devoted to refining the almost optimal regularity results of Breiner and Lamm \cite{Breiner-Lamm-2015} on minimizing and stationary biharmonic maps via the powerful quantitative stratification method introduced by Cheeger and Naber \cite{Cheeger-Naber-2013} and further developed by Naber and Valtorta \cite{Naber-V-2017,Naber-V-2018} for harmonic maps. In particular, we obtain an optimal regularity results for minimizing biharmonic maps.

We study the limit behavior of the solutions to energy-critical complex Ginzburg-Landau equation. We give a rigorous theory of the zero-dispersion limit from energy-critical complex Ginzburg-Landau equation to energy-critical nonlinear heat equation for dimensions 3 and 4 in both the defocusing and focusing cases by energy method. Furthermore, we also show the invisicid limit of energy-critical complex Ginzburg-Landau equation to energy-critical nonlinear Schr\"odinger equation for dimension 4 in the focusing case.

In this short note, we establish a sharp Morrey regularity theory for an even order elliptic system of Rivi\`ere type: \begin{equation*} \Delta^{m}u=\sum_{l=0}^{m-1}\Delta^{l}\left\langle V_{l},du\right\rangle +\sum_{l=0}^{m-2}\Delta^{l}\delta\left(w_{l}du\right)+f\qquad \text{in} B^{2m} \end{equation*} under minimal regularity assumptions on the coefficients functions V^l, w^l and that f belongs to certain Morrey space. This can be regarded as a further extension of the recent L^p-regularity theory obtained by Guo-Xiang-Zheng [15], and generalizes [7, 27] for second and fourth order elliptic systems., Comment: 12 pages

In this short note, we provide a partial extension of Rivi\`ere's convervation law in higher dimensions under certain Lorentz integrability condition for the connection matrix. As an application, we obtain a conservation law for weakly harmonic mappings around regular points in supercritical dimensions., Comment: 6 pages

In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schr\"odinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Sch\"ordinger equation. In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1( \mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schr\"odinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is bounded by a constant depending on the stationary solution $W$., Comment: 35 pages

In this short note, we consider quasiregular local homeomorphisms on uniform domains. We prove that such mappings always can be extended to some boundary points along John curves, which extends the corresponding result of Rajala [Amer. J. Math. 2008]., Comment: 18 pages

It was proposed recently that the fine-grained entropy of the Hawking radiation can be expressed by the semiclassical island formula, which reproduces the unitary Page curve. In this paper, we choose the ``in'' vacuum state and apply the quantum extremal surface construction to study the Page curve for the step-function Vaidya model of evaporating black holes in four dimensions, which is produced by the spherical null shells. Metrics of the three regions of this spacetimes are obtained. In addition, the entanglement islands for the step-function Vaidya model of evaporating black holes at very late times are studied. When cutoff surface $A$ is located in Minkowski region III with $u_A < u_H$ at very late times, we find that the location of the boundary of island $\partial I$ depends on the value of $8M-v_A+v_I$. Specifically, $\partial I$ is inside, at or outside the horizon when $8M-v_A+v_I$ is less than, equal to or larger than zero respectively. Moreover, when cutoff surface $A$ is located in Minkowski region III with $u_A > u_H$ after the black hole evaporates completely, we find that entanglement island still exists and $\partial I$ is located on an equal-time Cauchy surface of the observer $A$ when $r_{(A)}^2\geq64G_N\kappa c $., Comment: 29 pages, 12 figures, improvements to presentation, references added, and new content added in appendix to comment about the stress-energy tensor of the matter field, published version in JHEP

In this survey paper, we give an overview of the conservation law approach in the study of geometric PDEs that models in particular polyharmonic maps., Comment: The proof of Theorem 3.3 is now sketched. Surveys in Geometric Analysis 2022

In this paper we analyse and discuss 2D Jackiw-Teitelboim (JT) gravity coupled to primary fermion fields in asymptotically anti-de Sitter (AdS) spacetimes. We get a particular solution of the massless Dirac field outside the extremal black hole horizon and find the solution for the dilaton in JT gravity. Two dimensional JT gravity spacetime is conformally flat, we calculate the two point correlators of primary fermion fields under the Weyl transformations. The key point of this work is to present a standard technique which is called resolvent rather than CFT methods. We redefine the fields in terms of the conformal factor as the fermion fields, and we use the resolvent technique to derive the renormalized entanglement entropy for massless Dirac fields in JT gravity., Comment: 15 pages, 3 figures; improvements to presentation, new content added in appendix to comment about the correlator trace formula of primary fermion fields, published version in CPC

Quantum sensing provides sensitive tabletop tools to search for exotic spin-dependent interactions beyond the Standard Model, which has attracted great attention in theories and experiments. Here we develop a technique based on quantum Spin Amplifier for Particle PHysIcs REsearch (SAPPHIRE) to resonantly search for exotic interactions, specifically parity-odd spin-spin interactions. The present technique effectively amplifies the pseudomagnetic field generated by exotic interactions by a factor of about 200 while being insensitive to spurious external magnetic fields. Our studies, using such a quantum amplification technique, open the doors to exploring the parity-violation interactions mediated by Z' bosons in the challenging parameter space (force range between 3 mm and 0.1 km) and set the most stringent constraints on Z'-mediated electron-neutron couplings, significantly improving previous limits by up to five orders of magnitude. Moreover, our bounds on Z'-mediated couplings between nucleons reaches into a hitherto unexplored parameter space (force range below 1 m), complementing the existing astrophysical and laboratory studies., Comment: 8 pages, 5 figures

In this note, we refine the local conservation law obtained by Lamm-Rivi\`ere for fourth order systems and de Longueville-Gastel for general even order systems to a global conservation law.

Searches for the axion and axionlike particles may hold the key to unlocking some of the deepest puzzles about our universe, such as dark matter and dark energy. Here we use the recently demonstrated spin-based amplifier to constrain such hypothetical particles within the well-motivated ``axion window'' (1 $\mu$eV-1 meV) through searching for an exotic spin-spin interaction between polarized electron and neutron spins. The key ingredient is the use of hyperpolarized long-lived $^{129}$Xe nuclear spins as an amplifier for the pseudomagnetic field generated by the exotic interaction. Using such a spin sensor, we obtain a direct upper bound on the product of coupling constants $g_p^e g_p^n$. The spin-based amplifier technique can be extended to searches for a wide variety of hypothetical particles beyond the Standard Model., Comment: 6 pages, 4 figures

We establish an optimal $L^p$-regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $n\ge 5$: $$ \Delta^2 u=\Delta(D\cdot\nabla u)+div(E\cdot\nabla u)+(\Delta\Omega+G)\cdot\nabla u +f \qquad \ {\rm{in}}\ B^n, $$ where $\Omega\in W^{1,2}(B^n, so_m)$ is antisymmetric and $f\in L^p(B^n)$, and $D, E, \Omega, G$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $\nabla u$ and $\nabla^2 u$. This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm-Rivi\`ere, Struwe, and Wang. In particular, our results improve Struwe's H\"older regularity theorem to any H\"older exponent $\alpha\in (0,1)$ when $f\equiv 0$, and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of the techniques, we also extend the $L^p$-regularity theory of harmonic maps by Moser to Rivi\`ere-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions, which confirms an expectation by Sharp., Comment: there are changes of title and abstract, and introduction