Annali di Matematica Pura ed Applicata最新文献

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.

In this note, we prove that for every integer (dge 2) which is not a prime power, there exists a finite solvable group G such that (dmid |G|), (pi (G)=pi (d)) and G has no subgroup of order d. We also introduce the CLT-degree of a finite group and answer two questions about it.

We consider a multidimensional scalar problem of the calculus of variations with a nonnegative general Lagrangian depending on the space variable, on a Sobolev function and on its gradient. The main result of the paper is a sufficient condition discarding the Lavrentiev phenomenon between Sobolev and Lipschitz functions, with a prescribed boundary datum. The result unifies most of the known conditions in the literature and does not require that the Lagrangian obey a p-q growth condition or be an N-function.

We study left invariant locally conformally product structures on simply connected Lie groups and give their complete description in the solvable unimodular case. Based on previous classification results, we then obtain the complete list of solvable unimodular Lie algebras up to dimension 5 which carry LCP structures, and study the existence of lattices in the corresponding simply connected Lie groups.

The relative fixity of a digraph (Gamma ) is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of (Gamma ) and the number of vertices of (Gamma ). We characterize the vertex-primitive digraphs whose relative fixity is at least (frac{1}{3}), and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.

The (L_p) chord Minkowski problem based on chord measures and (L_p) chord measures introduced firstly by Lutwak et al. (Comm Pure Appl Math 1–54, 2023) is a very important and meaningful geometric measure problem in the (L_p) Brunn–Minkowski theory. Xi et al. (Adv Nonlinear Stud 23:20220041, 2023) using variational methods gave a measure solution when (p > 1) and (0<p<1) in the symmetric case. Recently, Guo et al. (Math Ann 2023. 10.1007/s00208-023-02721-8) also obtained a measure solution for (0le p<1) by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the (L_p) chord Minkowski problem by replacing p with a fixed continuous function (varphi :(0,infty )rightarrow (0,infty )), and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.

We prove a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems whose prototype is

with parameters (p>1) and (q>0) and (Omega subset {mathbb {R}}^n). In this paper, we are concerned with the ranges (q>1) and (p>frac{n(q+1)}{n+q+1}). A key ingredient in the proof is an intrinsic geometry that takes both the solution u and its spatial gradient Du into account.