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.

We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are deffned on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the discrete ropelength for n → ∞, regarding the topology induced by the Sobolev norm ‖ · ‖ W1,∞(S1,ℝd). This result directly implies the convergence of almost minimizers of the discrete energies...

An axisymmetric second order elliptic problem with mixed boundary conditions is considered. The shape of the domain has to be found so as to minimize a cost functional, which is given in terms of the cogradient of the solution. A new dual finite element method is used for approximate solutions. The existence of an optimal domain is proven and a convergence analysis presented.

In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constant and prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higher dimensions (cf. [15]). 
Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if the n-dimensional problem always reduces to the...

The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable...