Journal de Mathematiques Pures et Appliquees最新文献

We set up a one-parameter family of inequalities that contains both the Hardy inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg inequalities (when the parameter is optimal). Moreover, we study these results with the exact remainders to provide direct understandings to the sharp constants, as well as the existence and non-existence of the optimizers of the Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an application of our identities, we establish some sharp versions with optimal constants and theirs attainability of the stability of the Heisenberg Uncertainty Principle and several stability results of the Caffarelli-Kohn-Nirenberg inequalities (see Theorem 1.1). In particular, for the stability of the Heisenberg uncertainty principle, we introduced a new deficit function ${\delta }_1\left(u\right)$ and established the best constant for the stability inequality (see Theorem 1.2) which also leads to the stability inequality with sharp constant with respect to the deficit function ${\delta }_2\left(u\right)$ and improved the known stability result for ${\delta }_2\left(u\right)$ in the literature. (see Theorem 1.3). A sharp stability inequality for the non-scale invariant Heisenberg uncertainty principle with optimal constant is also obtained in Theorem 1.4.

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by $\kappa >0$ whose solution converges to that of the MHD equations as $\kappa \to 0$. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.

We establish local regularity theory for parabolic systems of Uhlenbeck type with φ-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local Hölder continuity of the gradients, providing suitable assumptions on the growth function φ. Our approach, being independent of the degeneracy of the system, allows for a unified treatment of both the degenerate and the singular case.

In this paper, we establish the locally diffeomorphic property of the solution to McKean-Vlasov stochastic differential equations defined in the Euclidean space. Our approach is built upon the insightful ideas put forth by Kunita. We observe that although the coefficients satisfy the global Lipschitz condition and some suitable regularity condition, the solution in general does not satisfy the globally homeomorphic property at any time except the initial time, which sets McKean-Vlasov stochastic differential equations apart significantly from classical stochastic differential equations. Finally, we provide an example to complement our results.

In this paper, we prove the existence of nontrivial contractible domains $\Omega \subset S^d$, $d\ge 2$, such that the overdetermined elliptic problem$\{\begin{array}{cc}-\epsilon {\Delta }_{g}u+u-u^p=0& \text{in Ω, }\\ u>0& \text{in Ω, }\\ u=0& \text{on ∂Ω, }\\ {\partial }_{\nu }u=\text{constant}& \text{on ∂Ω, }\end{array}$ admits a positive solution. Here ${\Delta }_g$ is the Laplace-Beltrami operator in the unit sphere $S^d$ with respect to the canonical round metric g, $\epsilon >0$ is a small real parameter and $1<p<\frac{d+2}{d-2}$ ($p>1$ if $d=2$). These domains are perturbations of $S^d\setminus D$, where D is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.

We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a warped product in a specific way. Moreover, if the manifold is complete, then it either is a weighted analogue of a space form, or it belongs to a particular family of Einstein warped products.

The wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of harmonic and multiscale analysis can be ingeniously exploited to build a signal representation with provable geometric stability properties, such as the Lipschitz continuity to the action of small $C^2$ diffeomorphisms – a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the Hölder regularity scale $C^{\alpha }$, $\alpha >0$. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class $C^{\alpha }$, $\alpha >1$, whereas instability phenomena can occur at lower regularity levels modeled by $C^{\alpha }$, $0\le \alpha <1$. While the analysis at the threshold given by Lipschitz (or even $C^1$) regularity remains beyond reach, we are able to prove a stability bound in that case, up to ε losses.

This paper concerns the homogenization of Schrödinger equations for non-crystalline matter, that is to say the coefficients are given by the composition of stationary functions with stochastic deformations. Two rigorous results of so-called effective mass theorems in solid state physics are obtained: a general abstract result (beyond the classical stationary ergodic setting), and one for quasi-perfect materials (i.e. the disorder in the non-crystalline matter is limited). The former relies on the double-scale limits and the wave function is spanned on the Bloch basis. Therefore, we have extended the Bloch Theory which was restricted until now to crystals (periodic setting). The second result relies on the Perturbation Theory and a special case of stochastic deformations, namely stochastic perturbation of the identity.

Given a continuous Hamiltonian $H:(x,p,u)\mapsto H(x,p,u)$ defined on $T^{⁎}M\times R$, where M is a closed connected manifold, we study viscosity solutions, $u_{\lambda }:M\to R$, of discounted equations:$H(x,d_x{u}_{\lambda },\lambda u_{\lambda }(x\left)\right)=c\text{in }M$ where $\lambda >0$ is called a discount factor and c is the critical value of $H(\cdot ,\cdot ,0)$.

When the Hamiltonian H is convex and superlinear in p and non–decreasing in u, under an additional non–degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions ${\left(u_{\lambda }\right)}_{\lambda >0}$ converges to a specific solution $u_0$ of$H(x,d_x{u}_0,0)=c\text{in }M.$ Our non–degeneracy condition requires H to be increasing (in u) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in u.

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl's asymptotics. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function υ we construct a singular Riemannian structure whose spectrum is discrete and satisfies$N\left(\lambda \right)\sim \frac{{\omega }_n}{{\left(2\pi \right)}^n}{\lambda }^{n/2}\upsilon \left(\lambda \right).$ Examples of slowly varying functions are $\log \lambda$, its iterations ${\log }_k\lambda ={\log }_{k-1}\log \lambda$, any rational function with positive coefficients of ${\log }_k\lambda$, and functions with non-logarithmic growth such as $\exp ({(\log \lambda )}^{{\alpha }_1}\dots {({\log }_k\lambda )}^{{\alpha }_k})$ for ${\alpha }_i\in (0,1)$. A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.