In this paper we prove that every weak and strong local minimizer $u\in W^{1,2}(\Omega ,{ℝ}^3)$ of the functional $I\left(u\right)={\int }_{\Omega }{\left|Du\right|}^2+f\left(\mathrm{Adj}Du\right)+g\left(\det Du\right),$ where $u:\Omega \subset {ℝ}^3\to {ℝ}^3$, f grows like ${\left|\mathrm{Adj}Du\right|}^p$, g grows like ${\left|\det Du\right|}^q$ and 1<q<p<2, is $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of Ω such that $\mathrm{𝑚𝑒𝑎𝑠}(\Omega \setminus {\Omega }_0)=0$. Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers. ...

A pneumatic tyre in rotating motion with a constant angular velocity $\Omega$ is assimilated to a torus whose generating circle has a radius $R$. The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$. When $(\Omega R/C_0)\ll 1$ and $(a/R)\ll 1$ (where $C_0$ is the velocity of the sound), we show that at the rapid time scale $R/C_0$, the air motion within a torus periodically excited on its surface generates an acoustic wave $h$. A study of this acoustic wave is conducted and shows that the mode associated to...

We study the unsaturated flow of an incompressible liquid carrying a bacterial population through a porous medium contaminated with some pollutant. The biomass grows feeding on the pollutant and affecting at the same time all the physics of the flow. We formulate a mathematical model in a one-dimensional setting and we prove an existence theorem for it. The so-called fluid media scaling approach, often used in the literature, is discussed and its limitations are pointed out on the basis of a specific...

Plant growth occurs due to cell proliferation in the meristem. We model the case of apical meristem specific for branch growth and the case of basal meristem specific for bulbous plants and grass. In the case of apical growth, our model allows us to describe the variety of plant forms and lifetimes, endogenous rhythms and apical domination. In the case of basal growth, the spatial structure, which corresponds to the appearance of leaves, results...

Two-scale convergence is a powerful mathematical tool in periodic homogenization developed for modelling media with periodic structure. The contribution deals with the classical definition, its problems, the ``dual'' definition based on the so-called periodic unfolding. Since in the case of domains with boundary the unfolding operator introduced by D. Cioranescu, A. Damlamian, G. Griso does not satisfy the crucial integral preserving property, the contribution proposes a modified unfolding operator...

We consider an Hamilton-Jacobi equation of the form$$H(x,Du)=0x\in \Omega \subset {ℝ}^N,\left(1\right)$$where $H(x,p)$ is assumed Borel measurable and quasi-convex in $p$. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

We consider an Hamilton-Jacobi equation of the form

 $$H(x,Du)=0x\in \Omega \subset {ℝ}^N,\left(1\right)$$ 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also...

We present some monotonicity and symmetry results for positive solutions of the equation $-\text{div}\left({\left|Du\right|}^{p-2}Du\right)=f\left(u\right)$ satisfying an homogeneous Dirichlet boundary condition in a bounded domain $\mathrm{\Omega }$. We assume 1 < p < 2 and $f$ locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if $\mathrm{\Omega }$ is a ball then the solutions are radially symmetric and strictly radially decreasing.

In a recent paper [Forum Math., 2008] the authors established some global, up to the boundary of a domain Ω ⊂ ℝⁿ, continuity and Morrey regularity results for almost minimizers of functionals of the form $u\mapsto {\int }_{\Omega }g(x,u\left(x\right),\nabla u\left(x\right))dx$. The main assumptions for these results are that g is asymptotically convex and that it satisfies some growth conditions. In this article, we present a specialized but significant version of this general result. The primary purpose of this paper is provide several applications of this simplified...

In this Note we consider the following problem $$\left\{\begin{array}{cc}-\mathrm{△}u=N\left(N-2\right)u^{p_{ϵ}}-\lambda u\hfill & \text{in }\mathrm{\Omega }\hfill \\ u>0\hfill & \text{in }\mathrm{\Omega }\hfill \\ u=0\hfill & \text{on }\partial \mathrm{\Omega }.\hfill \end{array}\right.$$ where $\mathrm{\Omega }$ is a bounded smooth starshaped domain in ${\mathbb{R}}^N$, $N\ge 3$, $p_{ϵ}=\frac{N+2}{N-2}-ϵ$, $ϵ>0$, and $\lambda \ge 0$. We prove that if $u_{ϵ}$ is a solution of Morse index $m>0$ than $u_{ϵ}$ cannot have more than $m$ maximum points in $\mathrm{\Omega }$ for $ϵ$ sufficiently small. Moreover if $\mathrm{\Omega }$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $ϵ$ sufficiently small.