Related papers: Symmetric double bubbles in the Grushin plane
We study the double bubble problem where the perimeter is taken with respect to the hexagonal norm, i.e. the norm whose unit circle in $\mathbb{R}^2$ is the regular hexagon. We provide an elementary proof for the existence of minimizing…
We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of…
The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of…
In 1993 Foisy et al. proved that the optimal Euclidean planar double bubble---the least-perimeter way to enclose and separate two given areas---is three circular arcs meeting at 120 degrees. We consider the plane with density $r^p$, joining…
The multi-bubble isoperimetric conjecture in $n$-dimensional Euclidean and spherical spaces from the 1990's asserts that standard bubbles uniquely minimize total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q \leq…
We study the double bubble problem with perimeter taken with respect to the $\ell_1$ norm on $\mathbb{R}^2$. We give an elementary proof for the existence of minimizing sets for any volume ratio parameter $0<\alpha\le1$ by direct comparison…
It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations.…
We investigate the optimal arrangements of two planar sets of given volume which are minimizing the $\ell_1$ double-bubble interaction functional. The latter features a competition between the minimization of the $\ell_1$ perimeters of the…
We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the…
We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of…
We characterize the unique minimizer of the three-dimensional double-bubble problem with respect to the $\ell_1$-norm for volume ratios between $1/2$ and $2$.
We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analogue of…
We explore the collision of two cylindrical bubbles in classical general relativity with a scalar field stress-energy tensor. Inside each bubble the field rests at a local minimum of the potential with non-negative energy density. Outside…
In this paper we solve several reverse isoperimetric problems in the class of $\lambda$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $\lambda > 0$. We give an affirmative answer…
We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of…
We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned…
In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to…
We use a new approach that we call unification to prove that standard weighted double bubbles in $n$-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for…
We prove ``half-space" intersection properties in three settings: the hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian space. For instance, any two embedded minimal hypersurfaces in the sphere must intersect…
This note presents two nontrivial, rotational equilibrium solutions to the spatial uniform gas pressure (isobaric) approximate model of Prosperetti in the inviscid case. Building on Gavrilov's work [GAFA 2019], we first establish the…