We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.

This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series...

The paper deals with a class of optimal shape design problems for elastic bodies unilaterally supported by a rigid foundation. Cost and constraint functionals defining the problem depend on contact stresses, i.e. their control is of primal interest. To this end, the so-called reciprocal variational formulation of contact problems making it possible to approximate directly the contact stresses is used. The existence and approximation results are established. The sensitivity analysis is carried out....

Four optimal design problems and a weight minimization problem are considered for elastic plates with small bending rigidity, resting on a unilateral elastic foundation, with inner rigid obstacles and a friction condition on a part of the boundary. The state problem is represented by a variational inequality and the design variables influence both the coefficients and the set of admissible state functions. If some input data are allowed to be uncertain a new method of reliable solutions is employed....

We address the numerical minimization of the functional $\mathcal{F}\left(v\right)={\int }_{\mathrm{\Omega }}\left|Dv\right|+{\int }_{\partial \mathrm{\Omega }}\mu vd{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}xvdx$, for $v\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$. We note that $\mathcal{F}$ can be equivalently minimized on the larger, convex, set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and that, on that space, $\mathcal{F}$ may be regularized with a sequence $\mathit{\left\{}\mathcal{F}{}_{ϵ}\right(v)=\int {}_{\mathrm{\Omega }}\sqrt{{ϵ}^2+{\left|Dv\right|}^2}+\int {}_{\partial \mathrm{\Omega }}\mu vd\mathcal{H}{}^{n-1}-\int {}_{\mathrm{\Omega }}xvdx\mathit{\}}{}_{ϵ}$of regular functionals. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$ can be discretized by continuous linear finite elements. The convexity of the functionals in $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful for the numerical minimization of $\mathcal{F}$. We prove the $\mathrm{\Gamma }-L^1\left(\mathrm{\Omega }\right)$-convergence of the discrete functionals to $\mathcal{F}$ and present a few numerical examples.