A characterization of ℓ₁ double bubbles with general interface interaction

Abstract We investigate the optimal arrangements of two planar sets of given volume which are minimizing the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℓ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {\ell_{1}} double-bubble interaction functional. The latter features a competition between the minimization of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℓ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {\ell_{1}} perimeters of the two sets and the maximization of their <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℓ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {\ell_{1}} interface. We investigate the problem in its full generality for sets of finite perimeter, by considering the whole range of possible interaction intensities and all relative volumes of the two sets. The main result is the complete classification of minimizers.