Milan Journal of Mathematics最新文献

Motivated by a conjecture of De Giorgi, we consider the Almgren-Taylor-Wang scheme for mean curvature flow, where the volume penalization is replaced by a term of the form $\begin{array}{c}{\int }_{E\Delta F}f\left(\frac{{\text{d}}_F}{\tau }\right)dx\end{array}$ for $f$ ranging in a large class of strictly increasing continuous functions, where $E\Delta F=(E\cup F)E\cap F$ is the symmetric difference between sets E and F, and $d_F$ is the distance function from $\partial F$ . In particular, our analysis covers the case $\begin{array}{c}f\left(r\right)=r^{\alpha },r\ge 0,\alpha >0,\end{array}$ considered by De Giorgi. We show that the generalized minimizing movement scheme converges to the geometric evolution equation $\begin{array}{c}f\left(v\right)=-\kappa \text{on}\partial E\left(t\right),\end{array}$ where $\left\{E\right(t\left)\right\}$ are evolving subsets of $R^n,$ $v$ is the normal velocity of $\partial E\left(t\right),$ and $\kappa$ is the mean curvature of $\partial E\left(t\right)$ . We extend our analysis to the anisotropic setting, and in the presence of a driving force. We also show that minimizing movements coincide with the smooth classical solution as long as the latter exists. Finally, we prove that in the absence of forcing, mean convexity and convexity are preserved by the weak flow.

In this paper, we prove a Serrin-type result for an elliptic system of equations, overdetermined with both Dirichlet and generalized Neumann conditions. With this tool, we characterize the critical shapes of some domain functionals under volume constraints.

We consider the prototypical example of the (2times 2) liquid chromatography system and characterize the set of initial data leading to a given attainable profile at (t=T). For profiles that are not attainable at time T, we study a non-smooth optimization problem: recovering the initial data that lead as close as possible to the target in the (L^2)-norm. We then study the system on a bounded domain and use a boundary control to steer its dynamics to a given trajectory. Finally, we implement a suitable finite volumes scheme to illustrate these results and show its numerical convergence. Minor modifications of our arguments apply to the Keyfitz–Kranzer system.

This note is devoted to a discussion of the potential links and differences between three topics: regularization by noise, convex integration, spontaneous stochasticity. All of them deal with the effect on large scales of a small-scale perturbation of fluid dynamic equations. The effects sometimes have something in common, like convex integration and spontaneous stochasticity, sometimes they look the opposite, as in regularization by noise. We are not aware of rigorous links or precise explanations of the differences, and hope to drive new research with this comparative examination.

We study the geometry and partial differential equations arising from the consideration of group-determinants, and representation theory. The simplest and most striking such example is undoubtedly that of the Humbert operator, associated with the cyclic group (mathbb Z/3mathbb Z), (displaystyle Delta _3=dfrac{partial ^3}{partial x^3}+dfrac{partial ^3}{partial y^3}+dfrac{partial ^3}{partial z^3}-3dfrac{partial ^3}{partial xpartial ypartial z}). This operator appears as a natural extension of the Laplacian in dimension 2. Another originality of our work is to show that the spectral theory of operators associated with Frobenius determinants is closely linked to finite Fourier transform theory.

We show that the formalism of the six Grothendieck operations holds in the sub-category of definably locally closed definable subsets equipped with the o-minimal site in o-minimal expansions of ordered groups.

Let (Gsubset textrm{GL}_n(k)) be a finite subgroup and (k[x_1,dots , x_n]^Gsubset k[x_1,dots , x_n]) its ring of invariants. We show that, in many cases, the automorphism group of (k[x_1,dots , x_n]^G) is (k^times ).

We consider a slightly subcritical elliptic system with Dirichlet boundary conditions and a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use the dual method on Orlicz spaces, showing that our problem possesses a mountain pass type solution.

In this paper, we deal with models with Born-Infeld (or relativistic) type diffusion and monostable reaction, investigating the effect of the introduction of a convection term on the limit shape of the critical front profile for vanishing diffusion. We first provide an estimate of the critical speed and then, through a careful analysis of an equivalent first-order problem, we show that different convection terms may lead either to a complete sharpening of the limit profile or to its complete regularization, presenting some related numerical simulations.

In this paper we deal with a reaction–diffusion equation in a bounded interval of the real line with a nonlinear diffusion of Perona–Malik’s type and a balanced bistable reaction term. Under very general assumptions, we study the persistence of layered solutions, showing that it strongly depends on the behavior of the reaction term close to the stable equilibria (pm 1), described by a parameter (theta >1). If (theta in (1,2)), we prove existence of steady states oscillating (and touching) (pm 1), called compactons, while in the case (theta =2) we prove the presence of metastable solutions, namely solutions with a transition layer structure which is maintained for an exponentially long time. Finally, for (theta >2), solutions with an unstable transition layer structure persist only for an algebraically long time.