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Euler's elastica is defined by a critical point of the total squared curvature under the fixed length constraint, and its L p -counterpart is called p-elastica. In this paper we completely classify all p-elasticae in the plane and obtain their explicit formulae as well as optimal regularity. To this end we introduce new types of p-elliptic functions which streamline the whole argument and result. As an application we also classify all closed planar p-elasticae. [ABSTRACT FROM AUTHOR]

We study the isoperimetric problem for anisotropic perimeter measures on R 3 , endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm ϕ on the horizontal distribution. In the case where ϕ is the standard norm in the plane, such isoperimetric problem is the subject of Pansu's conjecture, which is still unsolved. Assuming some regularity on ϕ and on its dual norm ϕ ∗ , we characterize C 2 -smooth isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. The argument is based on a fine study of the characteristic set of ϕ -isoperimetric sets and on establishing a foliation property by sub-Finsler geodesics. When ϕ is a crystalline norm, we show the existence of a partial foliation for constant ϕ -curvature surfaces by sub-Finsler geodesics. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where ϕ is crystalline). [ABSTRACT FROM AUTHOR]