Journal de Mathematiques Pures et Appliquees最新文献

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of $R^2$ with uniform magnetic field $\beta >0$ and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy ${\lambda }_1$ and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields $\beta =\beta \left(x\right)$ on a simply connected domain in a Riemannian surface.

In particular: we prove the upper bound ${\lambda }_1<\beta$ for a general plane domain for a constant magnetic field, and the upper bound ${\lambda }_1<{\max }_{x\in \bar{\Omega }}\left|\beta \right(x\left)\right|$ for a variable magnetic field when Ω is simply connected.

For smooth domains, we prove a lower bound of ${\lambda }_1$ depending only on the intensity of the magnetic field β and the rolling radius of the domain.

The estimates on the Riesz mean imply an upper bound for the averages of the first k eigenvalues which is sharp when $k\to \infty$ and consists of the semiclassical limit $\frac{2\pi k}{\left|\Omega \right|}$ plus an oscillating term.

We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which ${\lambda }_1$ is always small.

We propose a coordinate-free approach to study the holomorphic maps between the real hyperquadrics in complex projective spaces. It is based on a notion of orthogonality on the projective spaces induced by the Hermitian structures that define the hyperquadrics. There are various kinds of special linear subspaces associated to this orthogonality which are well respected by the relevant holomorphic maps and we obtain rigidity theorems by analyzing the images of these linear subspaces, together with techniques in projective geometry. Our method allows us to recover and generalize a number of well-known results in the field with simpler arguments.

Consider the nonlinear stability of the Couette flow in the Boussinesq equations with vertical dissipation on $T\times R$. We prove that if the initial perturbations $u_{\mathrm{in}}$ and ${\theta }_{\mathrm{in}}$ to the Couette flow $v_s={(y,0)}^{\top }$ and ${\theta }_s=1$, respectively, satisfy ${\Vert u_{\mathrm{in}}\Vert }_{H^{N+1}}+{\nu }^{-\frac{1}{2}}{\Vert {\theta }_{\mathrm{in}}\Vert }_{H^N}+{\nu }^{-\frac{1}{3}}{\Vert |{\partial }_x{|}^{\frac{1}{3}}\theta \Vert }_{H^N}\ll {\nu }^{\frac{1}{3}}$, $N>7$, then the resulting solution remains close to the Couette flow in $L^2$ at the same order for all time.

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded $C^2$ domains for a large class of prescribed boundary data. This result can be seen as a natural generalization of the classical sharp criterion for solvability of the minimal surface equation by Jenkins-Serrin. In contrast, we also construct a class of prescribed boundary data on just mean convex domains for which the Dirichlet problem in codimension 2 is not solvable. Moreover, we study existence and the uniqueness of minimal graphs by perturbation.

We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian $\mathrm{Gr}(3,7)$ parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from two operations with quadric bundles that might be interesting in themselves.

We construct bounded Poincaré operators for twisted complexes and BGG complexes with a wide class of function classes (e.g., Sobolev spaces) on bounded Lipschitz domains. These operators are derived from the de Rham versions using BGG diagrams and, for vanishing cohomology, satisfy the homotopy identity $dP+Pd=I$ in degrees >0. The operators preserve polynomial classes if the de Rham versions do so. Nontrivial cohomology and the complex property $P\circ P=0$ can be incorporated. We present applications to polynomial exact sequences.

In this paper, we study the nearly critical Lane-Emden equations(⁎)$\{\begin{array}{ccc}& -\Delta u=u^{p-\epsilon }& \text{in}\Omega ,\\ & u>0& \text{in}\Omega ,\\ & u=0& \text{on}\partial \Omega ,\end{array}$ where $\Omega \subset R^N$ with $N\ge 3$, $p=\frac{N+2}{N-2}$ and $\epsilon >0$ is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions.

In general, the solutions of (⁎) may blow-up at multiple points $a_1,\cdots ,a_k$ of Ω as $\epsilon \to 0$. In particular, when Ω is convex, there must be a unique blow-up point (i.e., $k=1$). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎).

A parabolic automorphism of a hyperkähler manifold M is a holomorphic automorphism acting on $H^2\left(M\right)$ by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangian fibration acts on almost all fibers ergodically. The existence of an invariant Lagrangian fibration is automatic for manifolds satisfying the hyperkähler SYZ conjecture; this includes all known examples of hyperkähler manifolds. When there are two parabolic automorphisms preserving two distinct Lagrangian fibrations, it follows that the group they generate acts on M ergodically. Our results generalize those obtained by S. Cantat for K3 surfaces.

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a “projection” of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one.

Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.

In this paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power (${\left({\rho }^{ϵ}\right)}^{\delta }$ with $\delta >1$), for regular solutions to the corresponding Cauchy problem, via introducing one “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the velocity near the vacuum, we establish the uniform energy estimates for the local sound speed in $H^3$ and ${\left({\rho }^{ϵ}\right)}^{\frac{\delta -1}{2}}$ in $H^2$ with respect to the viscosity coefficients for arbitrarily large time under some smallness assumption on the initial density. Second, by making full use of this structure's quasi-symmetric property and the weak smooth effect on solutions, we prove the strong convergence of the regular solutions of the degenerate viscous flow to that of the inviscid flow with vacuum in $H^2$ for arbitrarily large time. It is worth pointing out that the result obtained here seems to be the first one on the global-in-time inviscid limit of solutions with large velocities and vacuum for compressible flow in 3D space without any symmetric assumption.