泛簇上的楊氏函數

Abstract

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In the thesis, we intend to study the convergence of pairs of surfaces and smooth functions thereon. To capture their limit, we study the convergence of pairs of integral varifolds and Young functions (a measure-theoretic model of surfaces with multiplicity and multiple-valued functions) via their associated graph measures on the product space. To take differentiability into account, we develop the notions of weak differentiability and bounded variation of Young functions; moreover, the compactness properties of pairs of integral varifolds and weakly differentiable or BV Young functions are established.To this end, we study the topological vector structures of several test function spaces and introduce the concept of integral indecomposability—a notion of indecomposability tailored to our setting. Moreover, an existence theorem for integral decompositions of integral varifolds is established. The analysis of integral decompositions is carried out for a larger class of rectifiable varifolds, for which a compactness theorem analogous to the one for integral varifolds is obtained.