TY - JOUR
T1 - Interior continuity of two-dimensional weakly stationary-harmonic multiple-valued functions
AU - Lin, Chun Chi
N1 - Funding Information:
Acknowledgements The author started the project on stationary-harmonic multiple-valued functions in [10]. He would like to thank his thesis advisor Bob Hardt for his warm encouragement and many helpful discussions at Rice University. During the preparation of this manuscript, the author also received partial support from the research grant of the National Science Council of Taiwan (NSC-100-2918-I-003-009), National Center for Theoretical Sciences in Taiwan and the Max-Planck-Institute for Mathematics in the Sciences in Leipzig. The author would like to acknowledge Professor Dr. Luckhaus, Professor Dr. Otto, Professor Dr. Stevens for their hospitality; and Dr. Spadaro for sharing opinions on the subject of Almgren’s multiple-valued functions during his visit in Leipzig.
PY - 2014/7
Y1 - 2014/7
N2 - In his big regularity paper, Almgren has proven the regularity theorem for mass-minimizing integral currents. One key step in his paper is to derive the regularity of Dirichlet-minimizing QQ (ℝn)-valued functions in the Sobolev space γ2(Ω, QQ (ℝn)), where the domain Ω is open in ℝm . In this article, we introduce the class of weakly stationary-harmonic Q Q (ℝn)-valued functions. These functions are the critical points of Dirichlet's integral under smooth domain-variations and range-variations. We prove that if Ω is a two-dimensional domain in ℝ2 and f∈γ2 (Ω,QQ(ℝ n)) is weakly stationary-harmonic, then f is continuous in the interior of the domain Ω.
AB - In his big regularity paper, Almgren has proven the regularity theorem for mass-minimizing integral currents. One key step in his paper is to derive the regularity of Dirichlet-minimizing QQ (ℝn)-valued functions in the Sobolev space γ2(Ω, QQ (ℝn)), where the domain Ω is open in ℝm . In this article, we introduce the class of weakly stationary-harmonic Q Q (ℝn)-valued functions. These functions are the critical points of Dirichlet's integral under smooth domain-variations and range-variations. We prove that if Ω is a two-dimensional domain in ℝ2 and f∈γ2 (Ω,QQ(ℝ n)) is weakly stationary-harmonic, then f is continuous in the interior of the domain Ω.
KW - Almgren's big regularity paper
KW - Interior continuity
KW - Multiple-valued functions
KW - Weakly stationary-harmonic
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U2 - 10.1007/s12220-012-9385-2
DO - 10.1007/s12220-012-9385-2
M3 - Article
AN - SCOPUS:84904381482
SN - 1050-6926
VL - 24
SP - 1547
EP - 1582
JO - Journal of Geometric Analysis
JF - Journal of Geometric Analysis
IS - 3
ER -