Journal de Mathematiques Pures et Appliquees最新文献

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $R^n$, for $n\ge 3$. The gradient of solutions may blow up as ε, the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on $S^{n-2}$.

For a complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet of a vector bundle along the submanifold associates the $L^2$-optimal holomorphic extension of it to the ambient manifold. When the vector bundle is given by big tensor powers of a positive line bundle, we give an asymptotic formula for this extension operator.

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

This work deals with a family of Hardy-Sobolev doubly critical system defined in $R^n$. More precisely, we provide a classification of the positive solutions, whose expressions comprise multiplies of solutions of the decoupled scalar equation. Our strategy is based on the symmetry of the solutions, deduced via a suitable version of the moving planes technique for cooperative singular systems, joint with the study of the asymptotic behavior by using the Moser's iteration scheme.

We study Cheeger and p-eigenvalue partition problems depending on a given evaluation function Φ for $p\in [1,\infty )$. We prove existence and regularity of minima, relations between the problems, convergence, and stability with respect to p and to Φ.

In this article, we analyze the stability of the parallel surface problem for semilinear equations driven by the fractional Laplacian. We prove a quantitative stability result that goes beyond the one previously obtained in [15].

Moreover, we discuss in detail several techniques and challenges in obtaining the optimal exponent in this stability result. In particular, this includes an upper bound on the exponent via an explicit computation involving a family of ellipsoids. We also sharply investigate a technique that was proposed in [14] to obtain the optimal stability exponent in the quantitative estimate for the nonlocal Alexandrov's soap bubble theorem, obtaining accurate estimates to be compared with a new, explicit example.

Following a model originally considered by Kac and Luttinger, we study interacting many-particle systems in a random background. The background consists of hard spherical obstacles with fixed radius and that are distributed via a Poisson point process with constant intensity on $R^d$, $2\le d\in N$. Interactions among the (bosonic) particles are described through repulsive pair potentials of mean-field type. As a main result, we prove (complete) Bose–Einstein condensation (BEC) in the thermodynamic limit and into the minimizer of a Hartree-type functional, in probability or with probability almost one depending on the strength of the interaction. As an important ingredient, we use very recent results obtained by Alain-Sol Sznitman regarding the spectral gap of the Dirichlet Laplacian in a Poissonian cloud of hard spherical obstacles in large boxes. To the best of our knowledge, our paper provides the first proof of BEC for systems of interacting particles in the Kac–Luttinger model, or in fact for some higher-dimensional interacting random continuum model.

We investigate weak solutions to a fluid-structure interaction (FSI) problem between the flow of an incompressible, viscous fluid modeled by the Navier-Stokes equations, and a poroviscoelastic medium modeled by the Biot equations. These systems are coupled nonlinearly across an interface with mass and elastic energy, modeled by a reticular plate equation, which is transparent to fluid flow. We provide a constructive proof of the existence of a weak solution to a regularized problem. Next, a weak-classical consistency result is obtained, showing that the weak solution to the regularized problem converges, as the regularization parameter approaches zero, to a classical solution to the original problem, when such a classical solution exists. While the assumptions in the first step only require the Biot medium to be poroelastic, the second step requires additional regularity, namely, that the Biot medium is poroviscoelastic. This is the first weak solution existence result for an FSI problem with nonlinear coupling involving a Biot model for poro(visco)elastic media.

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed.

We provide full characterization of the Schatten properties of $[M_b,T]$, the commutator of Calderón–Zygmund singular integral T with symbol b $\left(M_{b}f\right(x):=b(x\left)f\right(x\left)\right)$ on stratified Lie groups $G$. We show that, when p is larger than the homogeneous dimension $Q$ of $G$, the Schatten $L_p$ norm of the commutator is equivalent to the Besov semi-norm $B_p^{Q/p}$ of the function b; but when $p\le Q$, the commutator belongs to $L_p$ if and only if b is a constant. For the endpoint case at the critical index $p=Q$, we further show that the Schatten $L_{Q,\infty }$ norm of the commutator is equivalent to the Sobolev norm ${\dot{W}}^{1,Q}$ of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.