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In this paper, we will give a C1,β-regularity result on the boundary for two dimensional sliding almost minimal sets in R3. This effect may apply to the regularity of the soap films at the boundary, and may also lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed by Guy David in the case that the boundary is a 2-dimensional smooth submanifold. [ABSTRACT FROM AUTHOR]

In this paper, we will give a C1,β-regularity result on the boundary for two dimensional sliding almost minimal sets in R3. This effect may apply to the regularity of the soap films at the boundary, and may also lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed by Guy David in the case that the boundary is a 2-dimensional smooth submanifold. [ABSTRACT FROM AUTHOR]

We study global Mumford–Shah minimizers in R N , introduced by Bonnet as blow-up limits of Mumford–Shah minimizers. We prove a new monotonicity formula for the energy of u when the singular set K is contained in an ( N − 1 ) -dimensional cone. We then use this monotonicity to prove that for any reduced global minimizer ( u , K ) in R 3 , if K is contained in a half-plane and touches its edge, then it is the half-plane itself. This partially answers to a question of Guy David. [ABSTRACT FROM AUTHOR]

The purpose of this work is to identify a sharp criterion under which energy solutions of L supercritical Hartree equation scatter. From the view of physical point, the criterion is represented in terms of the corresponding Lagrange functional and the virial quantity. This differs from Duyckaers et al. (Math Res Lett 15(6):1233-1250, ) where the criterion for the nonlinear Schrödinger equation is expressed by the mass-energy product. We shall prove in the Hartree case that these two descriptions are equivalent modulo scaling. Besides, we shall provide a third equivalent description, which is represented in terms of the linear energy. Moreover, the limit of the finite time blow-up solution at blow-up time shall be investigated. [ABSTRACT FROM AUTHOR]

We consider the minimization problem for an average distance functional in the plane, among all compact connected sets of prescribed length. For a minimizing set, the blow-up sequence in the neighborhood of any point is investigated. We show existence of the blow up limits and we characterize them, using the results to get some partial regularity statements. [ABSTRACT FROM AUTHOR]

We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Rivière: we state that every connected component of the interior of the coincidence set has at most N 0 singular points, where N 0 is only dependent on some geometric constants. Moreover, if the component is small enough, then this component has at most two singular points. On the other hand, we prove in a simple case a conjecture of D.G. Schaeffer on the generic regularity of the free boundary: for a family of obstacle problems in two dimensions continuously indexed by a parameter λ, the free boundary of the solution uλ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points. [ABSTRACT FROM AUTHOR]