Movement of centers with respect to various potentials
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- by Shigehiro Sakata PDF
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Abstract:
We investigate a potential with a radially symmetric and strictly decreasing kernel depending on a parameter. We regard the potential as a function defined on the upper half-space $\mathbb {R}^m \times (0,+\infty )$ and study some geometric properties of its spatial maximizer. To be precise, we give some sufficient conditions for the uniqueness of a maximizer of the potential and study the asymptotic behavior of the set of maximizers.
Using these results, we imply geometric properties of some specific potentials. In particular, we consider applications for the solution of the Cauchy problem for the heat equation, the Poisson integral (including a solid angle) and $r^{\alpha -m}$-potentials.
References
- Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
- Lorenzo Brasco and Rolando Magnanini, The heart of a convex body, Geometric properties for parabolic and elliptic PDE’s, Springer INdAM Ser., vol. 2, Springer, Milan, 2013, pp. 49–66. MR 3050226, DOI 10.1007/978-88-470-2841-8_{4}
- Lorenzo Brasco, Rolando Magnanini, and Paolo Salani, The location of the hot spot in a grounded convex conductor, Indiana Univ. Math. J. 60 (2011), no. 2, 633–659. MR 2963787, DOI 10.1512/iumj.2011.60.4578
- Isaac Chavel and Leon Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math. 55 (1990), 271–286. MR 1094719, DOI 10.1007/BF02789205
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879, DOI 10.1007/BF01221125
- Shuichi Jimbo and Shigeru Sakaguchi, Movement of hot spots over unbounded domains in $\textbf {R}^N$, J. Math. Anal. Appl. 182 (1994), no. 3, 810–835. MR 1272155, DOI 10.1006/jmaa.1994.1123
- Lester L. Helms, Potential theory, Universitext, Springer-Verlag London, Ltd., London, 2009. MR 2526019, DOI 10.1007/978-1-84882-319-8
- Rolando Magnanini and Shigeru Sakaguchi, On heat conductors with a stationary hot spot, Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 1–23. MR 2044330, DOI 10.1007/s10231-003-0077-1
- Frank Morgan, A round ball uniquely minimizes gravitational potential energy, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2733–2735. MR 2146221, DOI 10.1090/S0002-9939-05-08070-6
- Maria Moszyńska, Looking for selectors of star bodies, Geom. Dedicata 83 (2000), no. 1-3, 131–147. Special issue dedicated to Helmut R. Salzmann on the occasion of his 70th birthday. MR 1800016, DOI 10.1023/A:1005208712952
- Jun O’Hara, Renormalization of potentials and generalized centers, Adv. in Appl. Math. 48 (2012), no. 2, 365–392. MR 2873883, DOI 10.1016/j.aam.2011.09.003
- Shigehiro Sakata, Extremal problems for the central projection, J. Geom. 103 (2012), no. 1, 125–129. MR 2944554, DOI 10.1007/s00022-012-0113-7
- James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
- Katsuyuki Shibata, Where should a streetlight be placed in a triangle-shaped park? Elementary integro-differential geometric optics, http://www1.rsp.fukuoka-u.ac.jp/kototoi/ shibataaleph-sjs.pdf
Additional Information
- Shigehiro Sakata
- Affiliation: Global Education Center, Waseda University, 1-104 Totsuka-machi, Shinjuku-ku, Tokyo 169-8050, Japan
- Email: s.sakata@aoni.waseda.jp
- Received by editor(s): May 21, 2012
- Received by editor(s) in revised form: March 14, 2013
- Published electronically: August 18, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 8347-8381
- MSC (2010): Primary 31C12, 35K05, 35J05; Secondary 35B38, 51M16, 51M25, 52A40
- DOI: https://doi.org/10.1090/tran/6138
- MathSciNet review: 3403058