The interior Backus problem: local resolution in H\"older spaces

TL;DR

This paper proves an existence result for the nonlinear Backus interior problem in the Euclidean ball, focusing on solutions with symmetry properties to overcome derivative loss issues, using fixed point methods in H"older spaces.

Contribution

It introduces a novel approach to solve the nonlinear Backus problem by linearizing around symmetric solutions and applying fixed point techniques in H"older spaces.

Findings

Existence of solutions for symmetric cases of the Backus problem.

Overcoming derivative loss through symmetry assumptions.

Application of fixed point methods in H"older spaces for nonlinear PDEs.

Abstract

We prove an existence result for the Backus interior problem in the Euclidean ball. The problem consists in determining a harmonic function in the ball from the knowledge of the modulus of its gradient on the boundary. The problem is severely nonlinear. From a physical point of view, the problem can be interpreted as the determination of the velocity potential of an incompressible and irrotational fluid inside the ball from measurements of the velocity field's modulus on the boundary. The linearized problem is an irregular oblique derivative problem, for which a phenomenon of loss of derivatives occurs. As a consequence, a solution by linearization of the Backus problem becomes problematic. Here, we linearize the problem around the vertical height solution and show that the loss of derivatives does not occur for solutions which are either (vertically) axially symmetric or oddly symmetric in the vertical direction. A standard fixed point argument is then feasible, based on ad hoc weighted estimates in H\"older spaces.