1,720,982 research outputs found
Currents with coefficients in groups, applications and other problems in Geometric Measure Theory (2014)
Full text linkLectures on elliptic partial differential equations
Full text linkThe book originates from the Elliptic PDE course given by the first author at the Scuola Normale Superiore in recent years. It covers the most classical aspects of the theory of Elliptic Partial Differential Equations and Calculus of Variations, including also more recent developments on partial regularity for systems and the theory of viscosity solutions
An optimal irrigation network with infinitely many branching points
Full text linkThe Gilbert-Steiner problem is a mass transportation problem, where the cost of the transportation depends on the network used to move the mass and it is proportional to a certain power of the "flow". In this paper, we introduce a new formulation of the problem, which turns it into the minimization of a convex functional in a class of currents with coefficients in a group. This framework allows us to define calibrations. We apply this technique to prove the optimality of a certain irrigation network in the separable Hilbert space l2, having countably many branching points and a continuous amount of endpoints
An elementary proof of the rank-one theorem for BV functions
Full text linkWe provide a simple proof of a result, due to G. Alberti, concerning a rank-one property for the singular part of the derivative of vector-valued functions of bounded variation
On some geometric properties of currents and Frobenius theorem
Full text linkIn this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in [1]
Rank-one theorem and subgraphs of BV functions in Carnot groups
Full text linkWe prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups H^n for n ≥ 2. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph
Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks
No full textWe investigate the relation between energy minimizing maps valued into
spheres having topological singularities at given points and optimal networks
connecting them (e.g. Steiner trees, Gilbert-Steiner irrigation networks). We
show the equivalence of the corresponding variational problems, interpreting in
particular the branched optimal transport problem as a homological Plateau
problem for rectifiable currents with values in a suitable normed group. This
generalizes the pioneering work by Brezis, Coron and Lieb [10].Comment: 19 pages, 2 figure
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