The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural...

We prove the existence of positive and of nodal solutions for $-\Delta u={\left|u\right|}^{p-2}u+\mu {\left|u\right|}^{q-2}u$, $u\in {\mathrm{H}}_0^1\left(\Omega \right)$, where $\mu \>0$ and $2\<q\<p=2N(N-2)$, for a class of open subsets $\Omega$ of ${ℝ}^N$ lying between two infinite cylinders.

We prove the existence of positive and of nodal solutions for -Δu = |u|p-2u + µ|u|q-2u, $u\in {\mathrm{H}}_0^1\left(\Omega \right)$, where µ > 0 and 2 < q < p = 2N(N - 2) , for a class of open subsets Ω of ${ℝ}^N$ lying between two infinite cylinders.

In this paper we show some results of multiplicity and existence of sign-changing solutions using a mountain pass theorem in ordered intervals, for a class of quasi-linear elliptic Dirichlet problems. As a by product we construct a special pseudo-gradient vector field and a negative pseudo-gradient flow for the nondifferentiable functional associated to our class of problems.

A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...

A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...

We consider a singularly perturbed elliptic equation with superlinear nonlinearity on an annulus in ${ℝ}^4$, and look for solutions which are invariant under a fixed point free 1-parameter group action. We show that this problem can be reduced to a nonhomogeneous equation on a related annulus in dimension 3. The ground state solutions of this equation are single peak solutions which concentrate near the inner boundary. Transforming back, these solutions produce a family of solutions which concentrate...

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 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 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...