Document Zbl 1417.58010 - zbMATH Open

Let \(E\) be a Dirac bundle over a Riemannian manifold \(M^m\) (\(m \ge 2\)) with boundary \(\partial M\), and let \(\nabla_0\) be a fixed smooth Dirac connection on \(E\). Recall that a Dirac bundle is a Hermitian vector bundle of left Clifford modules over the Clifford bundle \(\text{Cl}(M)\) such that multiplication by unit vectors in \(TM\) is orthogonal and the covariant derivative is a module derivation. We define the Dirac connection space \(\mathfrak{D}^p(E)\) by the completetion of subspace of \(\Omega^1(\mathrm{Ad}(E))\) commuting with Clifford multiplication with respect to the norm \(\Vert\Gamma\Vert_p := \Vert\Gamma\Vert_{L^{2p}(M)} + \Vert d\Gamma\Vert_{L^p(M)}\) so that \(\nabla= \nabla_0 + \Gamma\) becomes another Dirac connection on \(E\) for \(\Gamma \in \mathfrak{D}^p(E)\). The Dirac operator associated with the Dirac connection \(\nabla\) is defined by \({D\!\!\!\!/} := e_i \cdot \nabla_{e_i}\), where \(e_i\cdot\) denotes Clifford multiplication, and \(\{e_i\}\) is a local orthonormal frame on \(M\). In this paper, the authors first consider the existence and uniqueness for Dirac operators under a class of local elliptic boundary value conditions \(\mathcal{B}\) including chiral conditions, MIT bag boundary conditions and J-boundary conditions: \[ \begin{cases} {D\!\!\!\!/} \psi = \varphi &\text{in }M\\ \mathcal{B}\psi = \mathcal{B}\psi_0 &\text{on }\partial M, \end{cases} \] where \(\varphi \in L^p(M)\) and \(\mathcal{B}\psi_0\in W^{1-1/p, p}(E|_{\partial M})\). The Sobolev spaces of sections of \(E\) are associated with the fixed smooth Dirac connection \(\nabla_0,\) and assume \(p^*>1\) if \(m=2\) and \(p^*\ge (3m-2)/4\) if \(m >2\). The authors prove that if \(\Gamma \in \mathfrak{D}^{p^*}(E)\), then for any \(1 <p<p^*\) the above Dirac equation with boundary condition admits a unique solution \(\psi \in W^{1,p}(E)\), and \(\psi\) satisfies the following estimate \[ \Vert \psi\Vert_{W^{1,p}(E)} \le c (\Vert \varphi\Vert_{L^p(E)} + \Vert\mathcal{B}\psi_0\Vert_{W^{1-1/p,p}(E|_{\partial M})}), \] where \(c = c(p, \Vert \Gamma\Vert_{p^*})>0\). This estimate is optimal in dimension 2 in the sense that the exponents cannot be improved. It also improves the known estimates in higher dimensions in [R. A. Bartnik and P. T. Chruściel, J. Reine Angew. Math. 579, 13–73 (2005; Zbl 1174.58305)] and [D. Jerison, Adv. Math. 62, 118–134 (1986; Zbl 0627.35008)].
Applying this result, the authors also derive the existence and uniqueness for boundary value problems for Dirac operators along a map. Let \(M\) be a compact Riemannian spin manifold with boundary \(\partial M\), \(N\) be a compact Riemannian manifold and \(\Phi: M\to N\) be a smooth map. Given a fixed spin structure on \(M\), let \(\Sigma M\) be the spin bundle of \(M\). On the twisted bundle \(\Sigma M \otimes \Phi^{-1}TN\), the Dirac operator \({D\!\!\!\!/}\) along a map \(\Phi\) is defined by \({D\!\!\!\!/}\Psi := {\partial\!\!\!/}\psi^\alpha \otimes \theta_\alpha + e_i\cdot \psi^\alpha \otimes \nabla_{\Phi_*(e_i)}^{TN} \theta_\alpha\), where \(\Psi= \psi^\alpha\otimes \theta_\alpha\), \(\{\theta_\alpha\}\) are local cross-section of \(\Phi^{-1}(TN)\), \({\partial\!\!\!/} = e_i\cdot \nabla_{e_i}\) is the usual Dirac operator on the spin bundle over \(M\). We say that \(\Psi\) is a harmonic spinor along the map \(\Phi\) if \({D\!\!\!\!/}\Psi = 0\).
The second main result in this paper is the following. Under these situations, if \(\Phi \in W^{1, 2p^*}(M;N)\), then for any \(1<p<p^*, \eta \in L^p(M; \Sigma M\otimes \Phi^{-1}TN)\) and \(\mathcal{B}\psi \in W^{1-1/p, p}(\partial M; \Sigma M \otimes \phi^{-1}TN)\), the boundary value problem for the Dirac equation \[ \begin{cases} {D\!\!\!\!/} \Psi = \eta &\text{in }M\\ \mathcal{B}\Psi = \mathcal{B}\psi &\text{on }\partial M, \end{cases} \] admits a unique solution \(\Psi\in W^{1,p}(M; \Sigma M \otimes \Phi^{-1}TN)\), and \[ \Vert \Psi\Vert_{W^{1,p}(M)} \le c (\Vert \eta\Vert_{L^p(M)} + \Vert\mathcal{B}\psi\Vert_{W^{1-1/p,p}(\partial M)}). \] Finally, the authors introduce the notion of Dirac-harmonic map heat flow and obtian the local existence and uniqueness of this heat flow for Dirac-harmonic maps by applying the elliptic estimates for Dirac equations with boundary conditions mentioned above. More precisely, consider the functional \[ L(\Phi, \Psi) = \frac{1}{2}\int_M \left(\Vert d\Phi\Vert^2 + (\Psi, {D\!\!\!\!/}\Psi)\right), \] where \((\,\,,\,\,) = \mathrm{Re} \langle \,\,,\,\,\rangle\) is the real part of the Hermitian inner product on \(\Sigma M \otimes \Phi^{-1}TN\). A Dirac-harmonic map is defined to be a critial point \((\Phi, \Psi)\) of \(L\). The Euler-Lagrange equations for this functional are \[ \tau(\Phi) = \frac{1}{2}(\psi^\alpha, e_i \cdot \psi^\beta) R^N(\theta_\alpha, \theta_\beta)\Phi_*(e_i) =:\mathcal{R}(\Phi, \Psi) \quad \text{and}\quad, {D\!\!\!\!/}\Psi = 0. \] Consider the following flow for Dirac-harmonic maps: for \(\Phi \in C^{2, 1.\alpha} (M \times (0, T]; N)\) and \(\Psi \in C^{1, 0.\alpha}(M\times [0, T]; \Sigma M \times \Phi^{-1}TN)\), \[ \begin{cases} \partial_t \Phi = \tau(\Phi) - \mathcal{R}(\Phi, \Psi) &\text{in }M\times (0, T]\\ {D\!\!\!\!/} \Psi =0 &\text{in } M\times [0, T] \end{cases} \] with the boundary-initial data \[ \begin{cases} \Phi = \phi &\text{in }M \times \{0\}\cup \partial M \times [0, T]\\ \mathcal{B}\Psi = \mathcal{B}\psi &\text{on }\partial M\times [0, T], \end{cases} \] where \(\phi \in C^{2, 1.\alpha}(M \times \{0\}\cup \partial M\times [0, T])\) and \(\psi \in C^{1, 0.\alpha}(\partial M\times [0, T]; \Sigma M \times \phi^{-1}TN)\). Here \(f \in C^{2, l.\alpha}\) means that \(f(x, \cdot) \in C^{l+\alpha/2}\) and \(f(\cdot, t) \in C^{k+\alpha}\). The authors show that this heat flow for Dirac-harmonic maps admit a unique solution provided that \[ \phi \in \bigcap_{T>0} C^{2, 1.\alpha}(\bar M \times [0, T]; N) \] and \[ \mathcal{B}\psi \in \bigcap_{T>0} C^{1, 0.\alpha}(\partial M \times [0, T]; \Sigma M \otimes \phi^{-1}TN) \] for some \(0< \alpha <1\).

MSC:

58E20	Harmonic maps, etc.
35J56	Boundary value problems for first-order elliptic systems
35J57	Boundary value problems for second-order elliptic systems
53C27	Spin and Spin\({}^c\) geometry

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