Annali di Matematica Pura ed Applicata最新文献

The Rayleigh Conjecture for the bilaplacian consists in showing that the clamped plate with least principal eigenvalue is the ball. The conjecture has been shown to hold in 1995 by Nadirashvili in dimension 2 and by Ashbaugh and Benguria in dimension 3. Since then, the conjecture remains open in dimension (dge 4). In this paper, we contribute to answer this question, and show that the conjecture is true in any dimension as long as some special condition holds on the principal eigenfunction of an optimal shape. This condition regards the mean value of the eigenfunction, asking it to be in some sense minimal. This main result is based on an order reduction principle allowing to convert the initial fourth order linear problem into a second order affine problem, for which the classic machinery of shape optimization and elliptic theory is available. The order reduction principle turns out to be a general tool. In particular, it is used to derive another sufficient condition for the conjecture to hold, which is a second main result. This condition requires the Laplacian of the optimal eigenfunction to have constant normal derivative on the boundary. Besides our main two results, we detail shape derivation tools allowing to prove simplicity for the principal eigenvalue of an optimal shape and to derive optimality conditions. Finally, because our first result involves the principal eigenfunction of a ball, we are led to compute it explicitly.

A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.

Let (mathcal {S}) denote the class of analytic and univalent (i.e., one-to-one) functions ( f(z)= z+sum _{n=2}^{infty }a_n z^n) in the unit disk (mathbb {D}={zin mathbb {C}:|z|<1}). For (fin mathcal {S}), In 1999, Ma proposed the generalized Zalcman conjecture that

with equality only for the Koebe function (k(z) = z/(1 - z)^2) and its rotations. In the same paper, Ma (J Math Anal Appl 234:328–339, 1999) asked for what positive real values of (lambda ) does the following inequality hold?

Clearly equality holds for the Koebe function (k(z) = z/(1 - z)^2) and its rotations. In this paper, we prove the inequality (0.1) for (lambda =3, n=2, m=3). Further, we provide a geometric condition on extremal function maximizing (0.1) for (lambda =2,n=2, m=3).

We establish a supercritical Trudinger–Moser type inequality for the k-Hessian operator on the space of the k-admissible radially symmetric functions (Phi ^{k}_{0,textrm{rad}}(B)), where B is the unit ball in ({mathbb {R}}^{N}). We also prove the existence of extremal functions for this new supercritical inequality.

We introduce the notion of (mathbb {P})-subnormal subgroup in a finite group, which generalizes the one of subnormal subgroup. We prove an analogous of the well-known Jordan–Hölder Theorem for these subgroups and their related chains of subgroups.

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of almost contact metric manifolds. In this paper we study the new structure of this type, called the weak nearly Sasakian structure. We find conditions that are satisfied by almost contact manifolds and under which the contact distribution is curvature invariant and weak nearly Sasakian manifolds admit two types of totally geodesic foliations. Our main result generalizes the theorem by Cappelletti-Montano and Dileo (Ann Matem Pura Appl 195:897-922, 2016) to the context of weak almost contact geometry.

The theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator (mathcal {E}), (mathcal {E}^2=0), that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product ((mathbb {D})-module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.

The aim of this paper is to study the existence of a finite stopping time for solutions in the form of variational inequality to fluid flows following a power law (or Ostwald–DeWaele law) in dimension (N in {2,3}). We first establish the existence of solutions for generalized Newtonian flows, valid for viscous stress tensors associated with the usual laws such as Ostwald–DeWaele, Carreau–Yasuda, Herschel–Bulkley and Bingham, but also for cases where the viscosity coefficient satisfies a more atypical (logarithmic) form. To demonstrate the existence of such solutions, we proceed by applying a nonlinear Galerkin method with a double regularization on the viscosity coefficient. We then establish the existence of a finite stopping time for threshold fluids or shear-thinning power-law fluids, i.e. formally such that the viscous stress tensor is represented by a p-Laplacian for the symmetrized gradient for (p in [1,2)).

We consider an autonomous, indefinite Lagrangian L admitting an infinitesimal symmetry K whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point p to a flow line (gamma =gamma (t)) of K that does not cross p. By utilizing the invariance of L under the flow of K, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the “arrival time” t, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When L is positively homogeneous of degree 2 in the velocities, the resulting equation establishes a variational principle that extends the Fermat’s principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.

We characterize homogeneous hypersurfaces in complex space forms which arise as critical points of a higher order energy functional. As a consequence, we obtain existence and non-existence results for (mathbb{C}mathbb{P}^n) and (mathbb{C}mathbb{H}^n), respectively. Moreover, we study the stability of biharmonic hypersurfaces and compute the normal index for a large family of solutions.