Workshop Title:

Some Themes in Calculus of Variations

Date:

March 5th, 2023 (GMT-5)

Organizer:

Department of Mathematics, Purdue University

Keywords:

• Isoperimetric inequalities
• functional inequalities
• minimization

Workshop Chair:

Personal Bio:

Emanuel Indrei is an Assistant Professor of Mathematics at Purdue University. He received his Ph.D. in Mathematics from the University of Texas at Austin in 2013 under the direction of Alessio Figalli. The thesis was selected for the Frank Gerth III Dissertation Award. He was a 2012 NSF EAPSI Fellow, a Postdoctoral Fellow at the Australian National University, a Huneke Postdoctoral Scholar at the Mathematical Sciences Research Institute in Berkeley, and a PIRE Postdoctoral Associate at Carnegie Mellon University.

Workshop Committee Members:

Dr. Victor Lie, Purdue University

Workshop Description:

Background:

The historical significance of the isoperimetric problem is prevalent in several cultures. A version of the relative isoperimetric problem-known as Dido's problem-was mentioned by the ancient Roman poet Virgil in his Aeneid:

"They came to this spot, where to-day you can behold the mighty Battlements and the rising citadel of New Carthage, And purchased a site, which was named 'Bull's Hide' after the bargain by which they should get as much land as they could enclose with a bull's hide."

According to legend, Dido cut the oxhide into very thin strips and enclosed the largest possible area by laying the tied-together strips in a semicircle against the coast. Indeed, under a fixed perimeter constraint, the semi-circle encloses the maximal area inside a half-space.

Goal/Rationale:

According to thermodynamics, the equilibrium shape of a small drop of water or a small crystal minimizes the free energy under a mass constraint. The phenomenon was independently discovered by W. Gibbs in 1878 and P. Curie in 1885. Assuming the gravitational effect is negligible, the energy minimization is the surface area minimization and the solution is the Wulff shape. In a gravitational field, the equilibrium shape for liquids was studied by P.S. Laplace in the early 1800s. One of the most difficult problems is to investigate convexity of minimizers in a background generated by a convex potential. A recent advance involved techniques in geometric measure theory. The uniqueness of minimizers has recently been addressed with a quantitative isoperimetric inequality.

Scope and Information for participants

The equilibrium shape of a small drop of water or a small crystal minimizes the free energy under a mass constraint. In the free energy, a potential energy and surface energy appear. The surface energy strongly influences candidates for minimizers to be convex assuming the mass is small. Nevertheless, if the mass is large, the potential energy is more dominant and strongly influences the least-energy state: this competition complicates some of the most elementary questions; for instance, convexity for all masses. The scope of the workshop is to outline the historical significance of the isoperimetric problem and emphasize the modern research in pure mathematics underscoring an unanticipated obstruction to answer the many elementary questions.

Highlights:

The calculus of variations is concerned with properties of solution(s) to various minimization problems modeling physical phenomena, e.g. in the formation of a soap bubble, the molecules in the soap film align themselves to form a least-energy structure, and the result is a sphere. Mathematically, this phenomenon is encoded in the isoperimetric inequality. In the workshop, the history and modern context were stated. The equilibrium shape of a small drop of water or a small crystal minimizes the free energy under a mass constraint. In the free energy, a potential energy and surface energy appear.

The surface energy strongly influences candidates for minimizers to be convex assuming the mass is small. Nevertheless, if the mass is large, the potential energy is more dominant and strongly influences the least-energy state: this competition complicates some of the most elementary questions; for instance, convexity for all masses. The workshop included a part on the historical significance of the isoperimetric problem and emphasized the modern research in pure mathematics underscoring an unanticipated obstruction.

There also was a demographic question and around 33.33% of the respondents identified as minorities. In addition, the participants were also given a form with information on scholarships/fellowships and graduate school funding.

Access to Workshop:

CONF-CIAP 2023 Workshop: Some Themes in Calculus of Variations

Venue:

Purdue University, 500 Oval Dr, West Lafayette

VISA:

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