A partial regularity theorem for harmonic maps at a free boundary

1989 ◽  
Vol 2 (4) ◽  
pp. 299-343 ◽  
Author(s):  
Frank Duzaar ◽  
Klaus Steffen
Keyword(s):  
Free Boundary ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Regularity Theorem

scholarly journals Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Roger Moser ◽  
James Roberts
Keyword(s):  
Free Boundary ◽  
Distance Function ◽  
Boundary Data ◽  
Partial Regularity ◽  
Harmonic Maps

AbstractWe prove partial regularity of weakly stationary harmonic maps with (partially) free boundary data on manifolds where the domain metric may degenerate or become singular along the free boundary at the rate $$d^\alpha $$ d α for the distance function d from the boundary.

scholarly journals Partial regularity of $p(x)$-harmonic maps

2012 ◽  
Vol 365 (6) ◽  
pp. 3329-3353 ◽  
Author(s):  
Maria Alessandra Ragusa ◽  
Atsushi Tachikawa ◽  
Hiroshi Takabayashi
Keyword(s):  
Partial Regularity ◽  
Harmonic Maps

scholarly journals Regularity of area minimizing currents mod p

2020 ◽  
Vol 30 (5) ◽  
pp. 1224-1336
Author(s):  
Camillo De Lellis ◽  
Jonas Hirsch ◽  
Andrea Marchese ◽  
Salvatore Stuvard
Keyword(s):  
Hausdorff Dimension ◽  
Partial Regularity ◽  
Singular Set ◽  
Regularity Theorem ◽  
Locally Finite ◽  
Dimensional Measure ◽  
Area Minimizing ◽  
Mod P

AbstractWe establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ mod ( p ) , for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current $${\mathrm{mod}}(p)$$ mod ( p ) cannot be larger than $$m-1$$ m - 1 . Additionally, we show that, when p is odd, the interior singular set is $$(m-1)$$ ( m - 1 ) -rectifiable with locally finite $$(m-1)$$ ( m - 1 ) -dimensional measure.

scholarly journals The free-boundary Brakke flow

2020 ◽  
Vol 2020 (758) ◽  
pp. 95-137 ◽  
Author(s):  
Nick Edelen
Keyword(s):  
Free Boundary ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Monotonicity Formula ◽  
Local Regularity ◽  
Regularity Theorem ◽  
Elliptic Regularization ◽  
Barrier Surface ◽  
Boundary Mean Curvature

AbstractWe develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must “pop” upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White [23] to the free-boundary setting. Using Ilmanen’s elliptic regularization procedure [10], we prove existence of free-boundary Brakke flows.

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