Harmonic maps between Alexandrov spaces
In this paper, the authors study the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature \(\leq 1\) in the sense of Alexandrov. Let \(X\) be an Alexandrov space of Hausdorff dimension \(n\) with curvature \(\geq k\) and an \(n\)-dimensional Hausdorff measure \(\mu\), \(\Omega \subset X\) be a bounded domain, and let \(Y\) be a complete geodesic space with curvature \(\leq 1\) in the sense of Alexandrov. Given \(\epsilon >0\) and a Borel measurable function \(u: \Omega \to Y\), the approximating energy functional \(E^u_\epsilon\) of \(u\) is defined as follows: For each compactly supported continuous function \(\varphi \in C_c(\Omega)\), \[ E^u_\epsilon(\varphi) := C_n \int_\Omega \varphi(x) d \mu(x) \int_{B_\epsilon(x)\cap \Omega} \frac{d^2(u(x), u(y))}{\epsilon^{n+2}}d \mu(y), \] where \(C_n\) is a normalized constant. And the energy functional of \(u\) is defined by \[ E^u(\varphi) = \limsup_{\epsilon \to 0} E^u_{\epsilon}(\varphi),\quad \forall \varphi \in C_c(\Omega). \] We say that \(u \in W^{1, 2}(\Omega, Y)\) if \(u \in L^2(\Omega)\) and it has finite energy \[ \sup_{\varphi\in C_c(\Omega), 0 \leq \varphi \leq 1} E^u(\varphi) <\infty. \] For a ball \(B_\rho(q) \subset Y\) with \(\rho< \pi/2\) and \(\varphi \in W^{1,2}(\Omega, Y)\) with \(\varphi(\Omega) \subset B_\rho(q)\), let \[ W^{1,2}_\varphi (\Omega, B_\rho(q)):= \{v \in W^{1,2}(\Omega, Y)\,:\, d(v, \varphi) \in W^{1,2}_0, v(\Omega) \subset B_\rho(q)\,\}. \] The first result proved by authors in this paper is that \(W^{1,2}_\varphi (\Omega, B_\rho(q))\) has a unique element \(u\) of least energy. It is called the harmonic map on \(\Omega\) which agrees \(\varphi\) on \(\partial \Omega\). The second result is the regularity of harmonic maps given in the above. Assume both \(\Omega\) and \(X\setminus \overline{\Omega}\) satisfy the measure density condition, i.e., there exists a constant \(C >0\) such that \(\mu(B_r(x) \cap \Omega) \geq C \mu(B_r(x))\) and \(\mu(B_r(x) \cap X\setminus \overline{\Omega}) \geq C \mu(B_r(x))\) for \(x\in \overline{\Omega}\) and for all \(0<r <\min\{1, \text{Diam}(\Omega)\}\). Suppose that \(w \in W^{1,2}(X, Y)\) is Hölder continuous on \(\overline {\Omega}\) and the image \(w(X)\) is contained in a geodesic ball \(B_\rho(q)\subset Y\) with radius \(\rho < \pi/2\). The authors show that if \(u\) is the harmonic map on \(\Omega\) which agrees \(w\) on \(\partial \Omega\), then \(u\) is Hölder continuous on \(\overline{\Omega}\).
scientific article; zbMATH DE number 6756126
scientific article