- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 05:39:21
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=20&iss=4&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
20
2018
4
Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions
Scott
Armstrong
Université Paris-Dauphine, France
Pierre
Cardaliaguet
Université Paris-Dauphine, France
Stochastic homogenization, mean curvature equation, Hamilton–Jacobi equation, error estimate
We study random homogenization of second-order, degenerate and quasilinear Hamilton–Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing geometric motions of level sets as well as a large class of viscous, non-convex Hamilton–Jacobi equations. The main results include the first proof of qualitative stochastic homogenization for such equations. We also present quantitative error estimates which give an algebraic rate of homogenization.
Partial differential equations
Probability theory and stochastic processes
797
864
10.4171/JEMS/777
http://www.ems-ph.org/doi/10.4171/JEMS/777
2
28
2018
Boundedness of moduli of varieties of general type
Christopher
Hacon
University of Utah, Salt Lake City, USA
James
McKernan
UCSD, La Jolla, USA
Chenyang
Xu
Beijing University, China
Moduli, boundedness, general type, minimal model program, abundance
We show that the family of semi log canonical pairs with ample log canonical class and with fixed volume is bounded.
Algebraic geometry
865
901
10.4171/JEMS/778
http://www.ems-ph.org/doi/10.4171/JEMS/778
3
5
2018
Counting designs
Peter
Keevash
University of Oxford, UK
Hypergraph Decomposition, Design Theory
We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed method of Randomised Algebraic Construction: we give a simpler proof of a special case of our result on clique decompositions of hypergraphs, namely triangle decompositions of quasirandom graphs.
Combinatorics
903
927
10.4171/JEMS/779
http://www.ems-ph.org/doi/10.4171/JEMS/779
3
5
2018
Vectorial nonlinear potential theory
Tuomo
Kuusi
University of Oulu, Finland
Giuseppe
Mingione
Università di Parma, Italy
Nonlinear potential theory, regularity, degenerate elliptic systems, measure data
We settle the longstanding problem of establishing pointwise potential estimates for vectorial solutions $u\colon \Omega \to \mathbb R^{N}$ to the non-homogeneous $p$-Laplacean system $$ -\rm{div} (|Du|^{p-2}Du)=\mu \qquad \mbox{in}\ \Omega \subset \mathbb R^{n}\,, $$ where $\mu$ is a $\mathbb R^{N}$-valued measure with finite total mass. In particular, for solutions $u \in W^{1,1}(\mathbb R^{n})$, the global estimates via Riesz and Wolff potentials $$|Du(x_0)|^{p-1} \lesssim \int_{\mathbb R^{n}}\frac{d|\mu|(x)}{|x-x_0|^{n-1}}$$ and $$ |u(x_0)|\lesssim {\bf W}^{\mu}_{1, p}(x_0,\infty) = \int_0^\infty \left(\frac{|\mu|(B_\varrho(x_0))}{\varrho^{n-p}}\right)^{1/(p-1)}\, \frac{d\varrho}{\varrho} $$ respectively, hold at every point $x_0$ such that the corresponding potentials are finite. The estimates allow to give sharp descriptions of fine properties of solutions which are the exact analog of the ones in classical linear potential theory. For instance, sharp characterisation of Lebesgue points of $u$ and $Du$ and optimal regularity criteria for solutions are provided exclusively in terms of potentials.
Partial differential equations
929
1004
10.4171/JEMS/780
http://www.ems-ph.org/doi/10.4171/JEMS/780
2
28
2018
Multiplicative stochastic heat equations on the whole space
Martin
Hairer
University of Warwick, Coventry, UK
Cyril
Labbé
Université Paris-Dauphine, France
Stochastic heat equation, parabolic Anderson model, white noise, weighted spaces, regularity structures
We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains. The two main equations our result covers are the parabolic Anderson model on $\mathbb R^3$, and the KPZ equation on $\mathbb R$ via the Cole–Hopf transform. To perform these constructions, we adapt the theory of regularity structures to the setting of weighted Besov spaces. One particular feature of our construction is that it allows one to start both equations from a Dirac mass at the initial time.
Probability theory and stochastic processes
Partial differential equations
1005
1054
10.4171/JEMS/781
http://www.ems-ph.org/doi/10.4171/JEMS/781
3
15
2018