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作者：Caffarelli, Luis; Ros-Oton, Xavier; Serra, Joaquim
作者单位：University of Texas System; University of Texas Austin; Universitat Politecnica de Catalunya
摘要：We study the obstacle problem for integro-differential operators of order 2s, with s is an element of(0,1). Our main results establish the optimal C<^>{1,s} regularity of solutions u, and the C<^>{1,\alpha} regularity of the free boundary near regular points. Namely, we prove the following dichotomy at all free boundary points x_0\in\partial\{u=\varphi\}: (i) either u(x)-\varphi(x)=c\,d<^>{1+s}(x)+o(|x-x_0|<^>{1+s+\alpha}) for some c>0, (ii) or u(x)-\varphi(x)=o(|x-x_0|<^>{1+s+\alpha}), where ...
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作者：Botvinnik, Boris; Ebert, Johannes; Randal-Williams, Oscar
作者单位：University of Oregon; University of Munster; University of Cambridge
摘要：We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real K-theory spectrum. Our main results concern the nontriviality of this map. We prove that for 2n >= 6, the natural KO-orientation from the infinite loop space of ...
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作者：He, Hongyu
作者单位：Louisiana State University System; Louisiana State University
摘要：In this paper, we give a proof of the Gan-Gross-Prasad conjecture for the discrete series of U(p, q). There are three themes in this paper: branching laws of a small A(q) (lambda) branching laws of discrete series and inductive construction of discrete series. These themes are linked together by a reciprocity law and the notion of invariant tensor product.
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作者：Armstrong, Scott; Kuusi, Tuomo; Mourrat, Jean-Christophe
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite PSL; Universite Paris-Dauphine; New York University; Aalto University; Centre National de la Recherche Scientifique (CNRS); Ecole Normale Superieure de Lyon (ENS de LYON)
摘要：One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the wea...
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作者：Popa, Mihnea; Schnell, Christian
作者单位：Northwestern University; State University of New York (SUNY) System; Stony Brook University
摘要：We use the theory of Hodge modules to construct Viehweg-Zuo sheaves on base spaces of families with maximal variation and fibers of general type and, more generally, families whose geometric generic fiber has a good minimal model. Combining this with a result of Campana-Pun, we deduce Viehweg's hyperbolicity conjecture in this context, namely the fact that the base spaces of such families are of log general type. This is approached as part of a general problem of identifying what spaces can su...
