Boundary regularity ofp-harmonic maps with free and partially constrained boundary conditions

1997 ◽  
Vol 7 (1) ◽  
pp. 83-107
Author(s):  
C. E. Garza-Hume
Keyword(s):  
Boundary Conditions ◽  
Harmonic Maps ◽  
Boundary Regularity

scholarly journals On the boundary regularity of phase-fields for Willmore's energy

2018 ◽  
Vol 149 (04) ◽  
pp. 1017-1035
Author(s):  
Patrick W. Dondl ◽  
Stephan Wojtowytsch
Keyword(s):  
Boundary Conditions ◽  
Partial Regularity ◽  
Boundary Regularity ◽  
Radon Measures ◽  
Practical Applications ◽  
Phase Fields ◽  
Regularity Results ◽  
Willmore Energy ◽  
Uniformly Bounded

AbstractWe demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions.

Boundary regularity for linear and quasilinear variational inequalities

1989 ◽  
Vol 112 (3-4) ◽  
pp. 319-326 ◽  
Author(s):  
Gary M. Lieberman
Keyword(s):  
Boundary Conditions ◽  
Variational Inequalities ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Parabolic Obstacle Problems ◽  
Second Derivatives ◽  
Quasilinear Problems ◽  
Derivatives Of

SynopsisA method of Jensen is extended to show that the second derivatives of the solutions of various linear obstacle problems are bounded under weaker regularity hypotheses on the dataof the problem than were allowed by Jensen. They are, in fact, weak enough that the linear results imply the boundedness of the second derivatives for quasilinear problems as well. Comparisons are made with previously known results, some of which are proved by similar methods. Both Dirichlet and oblique derivative boundary conditions are considered. Corresponding results for parabolic obstacle problems are proved.

scholarly journals Sphere-valued harmonic maps with surface energy and the K13 problem

2019 ◽  
Vol 12 (4) ◽  
pp. 363-392
Author(s):  
Stuart Day ◽  
Arghir Dani Zarnescu
Keyword(s):  
Liquid Crystals ◽  
Surface Energy ◽  
Boundary Conditions ◽  
Critical Point ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Nematic Liquid Crystals ◽  
Energy Functional ◽  
Dirichlet Energy ◽  
Surface Term

AbstractWe consider an energy functional motivated by the celebrated {K_{13}} problem in the Oseen–Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimisers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the global minimisers.

scholarly journals Boundary regularity and the Dirichlet problem for harmonic maps

1983 ◽  
Vol 18 (2) ◽  
pp. 253-268 ◽  
Author(s):  
Richard Schoen ◽  
Karen Uhlenbeck
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Maps ◽  
Boundary Regularity
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