Improved energy decay estimate for Dir-stationary \(Q\)-valued functions and its applications (Q6592092)

The paper focuses on Dirichlet-stationary Q-valued functions, which are an essential framework in the study of regularity for mass-minimizing currents, initially introduced in Almgren's groundbreaking work. These functions represent a linearized model for exploring the complexities of mass-minimizing integral currents. The author builds upon prior advancements by \textit{C. De Lellis} and \textit{E. N. Spadaro} [\(Q\)-valued functions revisited. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1246.49001)] and addresses the challenging problem of improving energy decay estimates for such functions. Let \(m, n\) be positive integers and \(\Omega \subset \mathbb{R}^m\) be an open domain. \(f \in W^{1,2}(\Omega, \mathcal{A}_Q ( \mathbb{R}^n ))\) is said to be a Dir-stationary \(Q\)-valued function if the following two properties hold:\N\begin{itemize}\N\item[1.] For every \(\phi \in C_c^{\infty}(\Omega , \mathbb{R}^m)\),\N\[\N2\int \sum_i \langle Df_i : Df_i \cdot D\phi \rangle - \int |Df|^2 \mathrm{div} \phi = 0\N\]\N\item[2.] For every \(\psi \in C^{\infty} (\Omega_x \times \mathbb{R}^n , \mathbb{R}^n ) \) s.t. \(\mathrm{supp}(\psi) \subset \Omega' \times \mathbb{R}^n\) for some \(\Omega' \Subset \Omega\), \(|D_u \psi| \le C < \infty\) and \(|\psi| + |D_x \psi| \le C(1+|u|)\),\N\[\N\int \sum_i \langle Df_i : D_x \psi (x, f_i (x) ) \rangle + \int \sum_i \langle Df_i : D_u \psi (x, f_i(x)) \cdot Df_i(x) \rangle = 0\N\]\N\end{itemize}\N\NThe main result of the paper is a refined version of the energy decay estimate for the Dirichlet energy of Dir-stationary Q-valued functions. More precisely is proved that if \(f\) is a Dir-stationary \(Q\)-valued function, for any \(x \in \Omega\) and \(r < \frac{1}{2}\min (\mathrm{dist}(x, \partial \Omega), 1)\), we have\N\[\N\frac{1}{r^{m-2}} \int_{B_r(x)} |Df|^2 \le \frac{C}{\log \frac{1}{r}}.\N\]\Nwhere, \(C\) depends on \(\mathrm{dist}(x, \partial \Omega)\), \(m, Q, ||f||_{W^{1,2}(\Omega, \mathcal{A}_Q ( \mathbb{R}^n ))}\). The previous improved energy decay allows to prove, as a consequence, a Liouville-Type theorem. Namely, that any bounded Dir-stationary \(Q\)-valued function, defined in the whole \(\mathbb{R}^{m}\), has to be a constant function.\N\NAs a further application, it is proved that, for a Dir-stationary \(Q\)-valued function \(f\), any \(x \in \Omega\) is a Lebesgue point and this property limits the set of discontinuities of \(f\) to be a meager (nowhere dense) set.\N\NFinally, it is observed that Dir-stationary \(Q\)-valued functions naturally reside in certain generalized Campanato-Morrey spaces, suggesting a pathway to proving their continuity.