Interior Backus Problem with Expanded Data

Abstract

We consider a special formulation of the Backus geomagnetic problem in two and higher dimension. Given a domain $\Omega \subset \mathbb{R}^n$, we seek to determine whether there exists a unique harmonic function $u$ satisfying $|\nabla u|^2 = P$ and $\frac{\partial}{\partial \nu}(|\nabla u|^2)= q$ on the boundary $\partial \Omega$ for given functions $P$ and $q$ which we refer to as the Backus problem with expanded data. In two dimensional case, for a function $u$ satisfying the Laplace equation and the above boundary conditions, we derive a system of ordinary differential equations for the tangential and normal components of $\nabla u$ with the coefficients in terms of $P$ and $q$ on the boundary. We study the explicit solutions of the ODE system and establish conditions for existence and uniqueness of solutions for the problem involving the PDE on bounded domains. To achieve this goal we introduce the notion of generalized Hilbert transform and use representation formulas for solutions of the boundary value problems. In addition, we perform numerical experiments to corroborate our well-posedness results. For the higher dimensional problem, our approach is markedly different. For harmonic functions $u$ in $\mathbb{R}^{n+1}$, we derive a quasilinear elliptic equation with coefficients involving the expanded data for the Backus problem satisfied by the restriction of $u$ on $n$-dimensional hyperplanes. The Leray-Schauder fixed point theory relates the solvability of the Dirichlet boundary value problem to apriori estimates for solutions of a related family of problems. This theory is not applicable to the derived equation directly due to the restriction on the gradient of admissible function. To work around this restriction, we introduce a regularization of the operator. The Apriori Estimates Program is fulfilled by establishing the comparison and maximum principles, which allows the estimation of $\displaystyle \sup_\Omega |u|$ in terms of $\displaystyle \sup_{\partial \Omega} |u|$ and additive constants; boundary gradient estimates, that is an estimation of $\displaystyle \sup_{\partial \Omega} |Du|$ in terms of $\displaystyle \sup_\Omega |u|$; interior gradient bounds, by which we estimate $\displaystyle \sup_{\Omega} |Du|$ in terms of $\displaystyle \sup_{\partial \Omega} |Du|$ and $\displaystyle \sup_\Omega |u|$; and H\"older estimates for the gradient, that is an estimation of $[Du]_{\alpha ; \Omega}$ in terms of $\displaystyle \sup_\Omega |Du|$ and $\displaystyle \sup_\Omega |u|$. We eventually obtain the existence of solutions of the regularized equation with Dirichlet boundary conditions.