"But it's taken us since the geometry of Euclid and Archimedes to prove it." "Double soap bubbles have known what they’re doing all along," says Morgan. The team's solution of the Double Bubble Conjecture was announced in March 2000 before the Undergraduate Mathematics Conference at the Rose-Hulman Institute of Technology, and has been heralded as a "mathematical milestone" by the American Mathematical Society. The National Science Foundation and the Spanish scientific research foundation DirecciÓn General de InvestigaciÓn CientÍfica y TÉcnica supported their work. He and three other mathematicians - Michael Hutchings of the University of California at Berkeley, and Manuel Ritoré and Antonio Ros of Universidad de Granada in Spain - proved that the common double soap bubble is the most efficient structure for containing two separate volumes of air within the least amount of surface area. Throughout his career, Morgan, a professor at Williams College, has studied soap bubbles, films and double bubbles for insight into the interplay of area and volume. Optimal geometry is concerned with geometric figures and their properties, and specifically: what shape or configuration of shapes best meets a situation's given constraints. Further research using techniques from that proof could enhance our understanding of the physical properties of structures ranging in size from the nanoscale to the galactic. It took mathematicians centuries to arrive at the proof, which was announced in 2000. Every time two soap bubbles form a double bubble, they demonstrate the best - or optimal - geometric figure for enclosing two separate volumes of air within the least amount of surface area. And they always meet at angles of 120 degrees."Īlthough the question may sound like a riddle, it involves complex mathematics and science. Soc.Ask Frank Morgan, a leading researcher in optimal geometry, "What happens when one soap bubble likes another soap bubble?" and he'll answer with effervescent enthusiasm. Ziemer, W.P.: A Poincaré type inequality for solutions of elliptic differtential equations. Weinberger, H.F.: Remark on the preceding paper of Serrin. Väisälä, J.: Exhaustions of John domains. Serrin, J.: A symmetry problem in potential theory. Reilly, R.C.: Mean curvature, the Laplacian, and soap bubbles. Reilly, R.C.: Applications of the Hessian operator in a Riemannian manifold. thesis, Università di Firenze, defended on February 2019, preprint arXiv:1902.08584 Poggesi, G.: The Soap Bubble Theorem and Serrin’s problem: quantitative symmetry. Poggesi, G.: Radial symmetry for \(p\)-harmonic functions in exterior and punctured domains. Payne, L., Schaefer, P.W.: Duality theorems un some overdetermined boundary value problems. Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Magnanini, R., Poggesi, G.: Serrin’s problem and Alexandrov’s Soap Bubble Theorem: enhanced stability via integral identities. Magnanini, R., Poggesi, G.: On the stability for Alexandrov’s Soap Bubble theorem. Magnanini, R.: Alexandrov, Serrin, Weinberger, Reilly: symmetry and stability by integral identities. Krummel, B., Maggi, F.: Isoperimetry with upper mean curvature bounds and sharp stability estimates. Ishiwata, M., Magnanini, R., Wadade, H.: A natural approach to the asymptotic mean value propery for the p-Laplacian. Hurri-Syrjänen, R.: An improved Poincaré inequality. Heintze, E., Karcher, H.: A general comparison theorem with applications to volume estimates for submanifolds. 20, 261–299 (2018)įeldman, W.M.: Stability of Serrin’s problem and dynamic stability of a model for contact angle motion. 195, 1333–1345 (2016)Ĭiraolo, G., Vezzoni, L.: A sharp quantitative version of Alexandrov’s theorem via the method of moving planes. 70, 665–716 (2017)Ĭiraolo, G., Magnanini, R., Vespri, V.: Hölder stability for Serrin’s overdetermined problem. 103, 172–176 (1998)Ĭiraolo, G., Maggi, F.: On the shape of compact hypersurfaces with almost constant mean curvature. 60, 633–660 (2011)īoas, H.B., Straube, E.J.: Integral inequalities of Hardy and Poincaré type. 245, 1566–1583 (2008)īrasco, L., Magnanini, R., Salani, P.: The location of the hot spot in a grounded convex conductor. 58, 303–315 (1962)īrandolini, B., Nitsch, C., Salani, P., Trombetti, C.: On the stability of the Serrin problem. 2, 412–416Īlexandrov, A.D.: A characteristic property of spheres. 4, 907–932 (1999)Īlexandrov, A.D.: Uniqueness theorem for surfaces in the large. Aftalion, A., Busca, J., Reichel, W.: Approximate radial symmetry for overdetermined boundary value problems.
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