Related papers: Multi-Bubble Isoperimetric Problems
In this report we discuss and propose a correction to a convergence and stability issue occurring in the work of Da et al.[2015], in which they proposed a numerical model to simulate soap bubbles.
We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization.
Soap bubbles are thin liquid films enclosing a fixed volume of air. Since the surface tension is typically assumed to be the only responsible for conforming the soap bubble shape, the realized bubble surfaces are always minimal area ones.…
We present new quantitative estimates for the radially symmetric configuration concerning Serrin's overdetermined problem for the torsional rigidity, Alexandrov's Soap Bubble Theorem, and other related problems. The new estimates improve on…
The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that…
In the last two centuries and more particularly in the last decades, the geometry of foams has become an important research domain, in mathematics, physics, material sciences and biology. Most of the simplest geometrical observations of…
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are…
We consider Serrin's overdetermined problem for the torsional rigidity and Alexandrov's Soap Bubble Theorem. We present new integral identities, that show a strong analogy between the two problems and help to obtain better (in some cases…
Bubbles and droplets are ubiquitous in many areas of engineering, including microfluidics where they can serve as microreactors for screening of chemical reactions. They are often formed out of a constriction (a microfluidic channel or a…
The distinguished names in the title have to do with influential proofs of the celebrated Soap Bubble Theorem and of radial symmetry in certain overdetermined boundary value problems. We shall give an overeview of those results and indicate…
We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian…
We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized.
The stability of hot expanded nuclear droplets against small bulk and surface oscillations is examined and possible consequences for multifragmentation are discussed.
We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive…
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…
In this note we briefly survey and propose some open problems related to isoparametric theory.
The persistent decay of bubble clusters in coarsening two-dimensional soap froths is measured experimentally as a function of cluster volume fraction. Dramatically stronger decay is observed in comparison to soap froth models and to…
We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f(x) = |x|$. Under these conditions, we find that isoperimetric $3$-bubble and $4$-bubble results satisfy a regular structure. As our regions increase…
We review recent stability and separation results in volume comparison problems and use them to prove several hyper- plane inequalities for intersection and projection bodies.