If we consider a small part of the string, we know it has length. A law of nature is that systems tend toward low potential energy and similarly a string would as well. Once again, we must determine the integral being minimized. The problem is determining the function that describes the shape of a string at rest when hung. However, the method I used required the consideration of many variables unlike calculus of variations. This is the equation for a cycloid, the solution to the Brachistochrone problem.Īs previously mentioned, I have approached this problem in a previous post: The Shape of a String. Once again, the -independent Euler equation is applied. This means the integral we wish to minimize is the following. From conservation of energy, we know the speed of the particle is. First let us determine the integral to be solved. Consider the time it takes to roll down a segment. It is simply one case of Euler’s equation. The Brachistochrone problem is the famous problem of finding the optimal track to roll a particle down such that it minimizes the time taken to do so. If we were to roll a ball down a track, what should the track’s shape be such that it minimizes the travel time between two points A and B. As trivial as this may sound, it is not true in other matric spaces and it is important to know exactly how to calculate this minimal distance points in any metric. We have essentially just proved the shortest distance between two points is a line. We know use the -independent Euler equation. What function takes the shortest path between two points and ? Well, the length of the function can be calculated by the following integral which is also the functional we wish to minimize. In fact, the drawn out results from the posts The Shape of a String and The Lagrangian are just two cases of the one equation. Soc.In the first part, we discussed the idea of a functional, what it means, and how to find its extrema using the calculus of variations. However, those equations don’t really capture how amazing and applicable calculus of variations really is so the following will be some examples of this. 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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