In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.

Given an open, bounded set $\Omega $ in $\mathbb {R}^N$, we consider the minimization of the anisotropic Cheeger constant $h_K(\Omega )$ with respect to the anisotropy K, under a volume constraint on the associated unit ball. In the planar case, under the assumption that K is a convex, centrally symmetric body, we prove the existence of a minimizer. Moreover, if $\Omega $ is a ball, we show that the optimal anisotropy K is not a ball and that, among all regular polygons, the square provides the minimal value.

We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(\chi,u)$, where $\chi$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.

We consider the problem of finding two free export/import sets $E^+$ and $E^-$ that minimize the total cost of some export/import transportation problem (with export/import taxes $g^\pm $ ), between two densities $f^+$ and $f^-$ , plus penalization terms on $E^+$ and $E^-$ . First, we prove the existence of such optimal sets under some assumptions on $f^\pm $ and $g^\pm $ . Then we study some properties of these sets such as convexity and regularity. In particular, we show that the optimal free export (resp. import) region $E^+$ (resp. $E^-$ ) has a boundary of class $C^2$ as soon as $f^+$ (resp. $f^-$ ) is continuous and $\partial E^+$ (resp. $\partial E^-$ ) is $C^{2,1}$ provided that $f^+$ (resp. $f^-$ ) is Lipschitz.

We study immersed surfaces in ${\mathbb R}^3$ that are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.

We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions.

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

The shapes of vesicles formed by lipid bilayers with phase separation are governed by a bending energy with phase dependent material parameters together with a line energy associated with the phase interfaces. We present a numerical method to approximate solutions to the Euler-Lagrange equations featuring triangulated surfaces, isoparametric quadratic surface finite elements and the phase field approach for the phase separation. Furthermore, the method involves an iterative solution scheme that is based on a relaxation dynamics coupling a geometric evolution equation for the membrane surface with a surface Allen-Cahn equation. Remeshing and grid adaptivity are discussed, and in various simulations the influence of several physical parameters is investigated.

In this paper, operator splitting scheme for dynamic reservoir characterization by binary level set method is employed. For this problem, the absolute permeability of the two-phase porous medium flow can be simulated by the constrained augmented Lagrangian optimization method with well data and seismic time-lapse data. By transforming the constrained optimization problem in an unconstrained one, the saddle point problem can be solved by Uzawas algorithms with operator splitting scheme, which is based on the essence of binary level set method. Both the simple and complicated numerical examples demonstrate that the given algorithms are stable and efficient and the absolute permeability can be satisfactorily recovered.

A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface and do not interact with each other. We consider a generalization of Newton’s minimal resistance problem: given two bounded convex bodies ${{C}_{1}}$ and ${{C}_{2}}$ such that ${{C}_{1}}\,\subset \,{{C}_{2}}\,\subset \,{{\mathbb{R}}^{3}}$ and $\partial {{C}_{1}}\,\cap \,\partial {{C}_{2}}\,=\,\varnothing $, minimize the resistance in the class of connected bodies $B$ such that ${{C}_{1}}\,\subset \,B\,\subset \,{{C}_{2}}$. We prove that the infimum of resistance is zero; that is, there exist “almost perfectly streamlined” bodies.