ALBERT

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Chang-Yu Guo, Chang-Lin Xiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉M〈/mi〉〈/math〉 be a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-smooth Riemannian manifold with boundary and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉 a complete 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-smooth Riemannian manifold. We show that each stationary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mi〉p〈/mi〉〈/math〉-harmonic mapping 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉u〈/mi〉〈mo〉:〈/mo〉〈mi〉M〈/mi〉〈mo〉→〈/mo〉〈mi〉N〈/mi〉〈/mrow〉〈/math〉, whose image lies in a compact subset of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉, is locally 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 for some 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, provided that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉 is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mi〉p〈/mi〉〈/math〉-harmonic mappings with image being contained in a regular geodesic ball. Moreover, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉M〈/mi〉〈/math〉 has non-negative Ricci curvature and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉 is simply connected with non-positive sectional curvature, we deduce a gradient estimate for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉-smooth weakly 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mi〉p〈/mi〉〈/math〉-harmonic mappings from which follows a Liouville-type theorem in the same setting.〈/p〉〈/div〉