In this paper, we compare a biomechanics empirical model of the heart fibrous structure to two models obtained by a non-periodic homogenization process. To this end, the two homogenized models are simplified using the small amplitude homogenization procedure of Tartar, both in conduction and in elasticity. A new small amplitude homogenization expansion formula for a mixture of anisotropic elastic materials is also derived and allows us to obtain a third simplified model.

We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof...

We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature of a...

We prove that within the frame of smoothed prolongations, rapid coarsening between first two levels can be compensated by massive prolongation smoothing and pre- and post-smoothing derived from the prolongator smoother.

We discuss regularity results concerning local minimizers $u:{ℝ}^n\supset \Omega \to {ℝ}^n$ of variational integrals like $$\begin{array}{c}\hfill {\int }_{\Omega }\{F(·,\epsilon \left(w\right))-f·w\}dx\end{array}$$ defined on energy classes of solenoidal fields. For the potential $F$ we assume a $(p,q)$-elliptic growth condition. In the situation without $x$-dependence it is known that minimizers are of class $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of $\Omega$ with full measure if $q<p\frac{n+2}{n}$ (for $n=2$ we have ${\Omega }_0=\Omega$). In this article we extend this to the case of nonautonomous integrands. Of course our result extends to weak solutions of the corresponding nonlinear...

We prove local embeddings of Sobolev and Morrey type for Dirichlet forms on spaces of homogeneous type. Our results apply to some general classes of selfadjoint subelliptic operators as well as to Dirichlet operators on certain self-similar fractals, like the Sierpinski gasket. We also define intrinsic BV spaces and perimeters and prove related isoperimetric inequalities.

We investigate the following quasilinear and singular problem,$$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\&gt;0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$where $\Omega$ is an open bounded domain with smooth boundary, $1\&lt;p\&lt;\infty$, $p-1\&lt;q\le p^{*}-1$, $\lambda \&gt;0$, and $0\&lt;\delta \&lt;1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\&lt;p\&lt;N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\&gt;N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}\left(\Omega \right)$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions...

In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation $$-{\Delta }_{N}u+b{\left|u\right|}^{N-2}u-{\Delta }_N\left(u^2\right)u=h\left(u\right),x\in {ℝ}^N,$$ where ${\Delta }_N$ is the $N$-Laplacian operator, $h\left(u\right)$ is continuous and behaves as $exp\left(\alpha \right|u{|}^{N/(N-1)})$ when $\left|u\right|\to \infty$. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u\left(x\right)\in W^{1,N}\left({ℝ}^N\right)$ with $u\left(x\right)\to 0$ as $\left|x\right|\to \infty$ is established.

In this paper we prove that some classes of semilinear elliptic problems, formulated in very general terms by using the theory of maximal monotone graphs, admit a finite propagation speed. More concretely we show that if the data of these problems have compact supports, then the same happens to their solutions. These same thechniques will also be applied to some evolution problems. The first results in this direction are due to H. Brézis and to O. Oleinik & A. S. Kalashnikov & C. Yuilin...