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Creators/Authors contains: "Hirsch, Jonas"

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  1. ; ; ; ; (, Nonlinear Analysis)
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  2. ; (, Archive for Rational Mechanics and Analysis)
    Abstract We prove that 2-dimensionalQ-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions. 
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  3. ; ; ; (, Memoirs of the American Mathematical Society)
    Let Σ<#comment/> \Sigma be a smooth Riemannian manifold, Γ<#comment/> ⊂<#comment/> Σ<#comment/> \Gamma \subset \Sigma a smooth closed oriented submanifold of codimension higher than 2 2 and T T an integral area-minimizing current in Σ<#comment/> \Sigma which bounds Γ<#comment/> \Gamma . We prove that the set of regular points of T T at the boundary is dense in Γ<#comment/> \Gamma . Prior to our theorem the existence of any regular point was not known, except for some special choice of Σ<#comment/> \Sigma and Γ<#comment/> \Gamma . As a corollary of our theorem we answer to a question in Almgren’sAlmgren’s big regularity paperfrom 2000 showing that, if Γ<#comment/> \Gamma is connected, then T T has at least one point p p of multiplicity 1 2 \frac {1}{2} , namely there is a neighborhood of the point p p where T T is a classical submanifold with boundary Γ<#comment/> \Gamma ; we generalize Almgren’s connectivity theorem showing that the support of T T is always connected if Γ<#comment/> \Gamma is connected; we conclude a structural result on T T when Γ<#comment/> \Gamma consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when Σ<#comment/> = R m + 1 \Sigma = \mathbb R^{m+1} and T T is m m -dimensional. 
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  4. ; ; (, Journal of Differential Geometry)
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  5. ; ; ; (, Communications on Pure and Applied Mathematics)
    Abstract We establish a theory ofQ‐valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currentsmod(p)whenp = 2Q, and to establish a first general partial regularity theorem for everypin any dimension and codimension . © 2020 The Authors.Communications on Pure and Applied Mathematicspublished by Wiley Periodicals LLC. 
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  6. ; ; ; (, Geometric and Functional Analysis)
    Abstract We establish a first general partial regularity theorem for area minimizing currents$${\mathrm{mod}}(p)$$ mod ( p ) , for everyp, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of anm-dimensional area minimizing current$${\mathrm{mod}}(p)$$ mod ( p ) cannot be larger than$$m-1$$ m - 1 . Additionally, we show that, whenpis odd, the interior singular set is$$(m-1)$$ ( m - 1 ) -rectifiable with locally finite$$(m-1)$$ ( m - 1 ) -dimensional measure. 
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