scholarly journals Solutions of the (free boundary) Reifenberg Plateau problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Camille Labourie
Keyword(s):  
Free Boundary ◽  
Weak Convergence ◽  
Direct Method ◽  
Weak Limit ◽  
Plateau Problem ◽  
Minimizing Sequence ◽  
Calculus Of Variation ◽  
Link Type ◽  
Weak Limits

AbstractWe solve two variants of the Reifenberg problem for all coefficient groups. We carry out the direct method of the calculus of variation and search a solution as a weak limit of a minimizing sequence. This strategy has been introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi and allowed them to solve the Reifenberg problem. We use an analogous strategy proved in [C. Labourie, Weak limits of quasiminimizing sequences, preprint 2020, https://arxiv.org/abs/2002.08876] which allows to take into account the free boundary. Moreover, we show that the Reifenberg class is closed under weak convergence without restriction on the coefficient group.

Regularity results for solutions to the c-Plateau problem with free boundary having small singular set, and generally in codimension zero

2016 ◽  
Vol 9 (3) ◽  
pp. 259-282 ◽  
Author(s):  
Leobardo Rosales
Keyword(s):  
Free Boundary ◽  
Minimization Problem ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Plateau Problem ◽  
Singular Set ◽  
Codimension Two ◽  
Rectifiable Currents ◽  
Problem With Free Boundary ◽  
Regularity Results

AbstractWe prove two results for the c-Plateau problem, introduced in [17], which is a minimization problem for integer rectifiable currents. First, we prove there is no solution to the c-Plateau problem with free boundary having singular set of finite Hausdorff codimension two measure and with regular part having constant mean curvature. Second, we prove regularity up to Hausdorff codimension seven of the free boundary of top-dimensional solutions to the c-Plateau problem.

Weak limits of sample range

1974 ◽  
Vol 11 (04) ◽  
pp. 836-841 ◽  
Author(s):  
Laurens De Haan
Keyword(s):  
Weak Convergence ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Weak Limits ◽  
Sample Range ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

One-dimensional three-state quantum walks: Weak limits and localization

2016 ◽  
Vol 19 (04) ◽  
pp. 1650025 ◽  
Author(s):  
Chul Ki Ko ◽  
Etsuo Segawa ◽  
Hyun Jae Yoo
Keyword(s):  
Limit Distribution ◽  
Evolution Operator ◽  
Adjoint Operator ◽  
Quantum Walk ◽  
Vacuum Expectation ◽  
Weak Limit ◽  
Quantum Walks ◽  
Continuous Part ◽  
One Dimensional ◽  
Weak Limits

We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.

Weak limits of sample range

1974 ◽  
Vol 11 (4) ◽  
pp. 836-841 ◽  
Author(s):  
Laurens De Haan
Keyword(s):  
Weak Convergence ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Weak Limits ◽  
Sample Range ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

Weak convergence of random walks, conditioned to stay away from small sets

2013 ◽  
Vol 50 (1) ◽  
pp. 122-128
Author(s):  
Zsolt Pajor-Gyulai ◽  
Domokos Szász
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Random Walks ◽  
Continuous Process ◽  
Random Variables ◽  
Weak Limit ◽  
Limit Laws ◽  
Diffusive Limit

Let {Xn}n∈ℕ be a sequence of i.i.d. random variables in ℤd. Let Sk = X1 + … + Xk and Yn(t) be the continuous process on [0, 1] for which Yn(k/n) = Sk/n1/2 for k = 1, … n and which is linearly interpolated elsewhere. The paper gives a generalization of results of ([2]) on the weak limit laws of Yn(t) conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on ℤd: d ≧ 2 is the Brownian motion.

Weak convergence of the simple birth-and-death process

1981 ◽  
Vol 18 (01) ◽  
pp. 245-252
Author(s):  
Constantin Ivan
Keyword(s):  
Markov Process ◽  
Weak Convergence ◽  
Weak Limit ◽  
Death Process ◽  
Birth And Death Process

The existence of a weak limit birth-and-death process on the natural integers for the simple birth-and-death process conditional on non-extinction up to time t as t→∞ is proved. Starting from the latter a new weak limiting procedure yields a diffusion Markov process on the positive infinite semi-axis.

COMMENT ON “WEAK CONVERGENCE TO A MATRIX STOCHASTIC INTEGRAL WITH STABLE PROCESSES”

2011 ◽  
Vol 27 (4) ◽  
pp. 907-911 ◽  
Author(s):  
Vygantas Paulauskas ◽  
Svetlozar T. Rachev ◽  
Frank J. Fabozzi
Keyword(s):  
Weak Convergence ◽  
Stochastic Integral ◽  
Weak Limit ◽  
Limit Behavior ◽  
Infinite Variance ◽  
Stochastic Integrals ◽  
Stable Processes ◽  
Heavy Tailed

In this comment we identify a lacuna in a proof in the paper by M. Caner published in 1997 in Econometric Theory concerning the weak limit behavior of various expressions involving heavy-tailed multivariate vectors and the convergence of stochastic integrals. In a later paper (Caner, 1998) the results for these limit relations are used to formulate tests for cointegration with infinite variance errors.

Sign in / Sign up

Export Citation Format

Share Document