On the line with density |x|^p, we prove that the best single bubble is an interval with endpoint at the origin and that the best double bubble is two adjacent intervals that meet at the origin.

Author Bio

This paper is the work of an Institute of Advanced Research winter camp course in Shanghai for high school students, under the direction of Frank Morgan.

Juiyu Huang from Nanjing International School likes mathematics, natural sciences, and badminton.

Xinkai Qian from Vermont International Academy, CISK, loves science. In his school’s “RC pilots” club, he builds model planes and flies them. He really enjoyed doing mathematical research in a group and hopes to do more in the future.

Yiheng Pan, a senior at Hangzhou Foreign Language School, has the spirit of exploration and a passion for geometry.

Mulei Xu, a grade 12 student from Nanjing Experimental School, loves geometry, algebra, and astrophysics.

Lu Yang, a junior at Sun Yat-sen University, served as a teaching assistant. She wants to use statistics to predict the future.

Junfei Zhou from Chongqing Tsinghua High School loves mathematics, likes physics, and has a certain interest in chemistry and biology.