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Geophysical intensity problems: the axisymmetric case

Ralf Kaiser

TL;DR

This work analyzes the axisymmetric exterior-intensity problem for harmonic vector fields inside the Earth-scale context, formulating a nonlinear boundary-value problem with prescribed surface intensity. By reducing the 3D axisymmetric field to a 2D complex-analytic framework and introducing a nonlinear elliptic equation with a nonlinear, natural boundary condition, the authors construct a fixed-point scheme around a linearized problem to prove the existence of infinitely many axisymmetric solutions with prescribed decay and prescribed vanishing at selected interior points. They establish uniqueness for the linearized problem, develop Hardy-type weighted estimates to control axis singularities, and show how the data (zeroes and boundary intensity) determine a data function Ω that governs the nonlinear problem. The results yield a rigorous, constructive approach to the geophysical intensity problem in the axisymmetric setting and elucidate how nonuniqueness can arise when nontrivial zero-sets are present.

Abstract

Considering the earth or any other celestial body the main sources of the gravitational as well as of the magnetic field lie inside the body. Above the surface both fields are in good approximation harmonic vector fields determined by their values at the body's surface or any other surface enclosing the body. The intensity problem seeks to determine harmonic vector fields vanishing at infinity and with prescribed intensity of the field at the surface. This problem constitutes a nonlinear boundary value problem, whose general solvability is not yet established. In this paper {\em axisymmetric} harmonic fields ${\bf H}$ outside the unit sphere $S^2$ are studied and, given an axisymmetric Hölder continuous intensity function $I\neq 0$ on $S^2$, the existence of infinitely many solutions of the intensity problem is proved. These solutions can more precisely be characterized as follows: fix a number $\de \in \nat\setminus \{1 \}$ and a meridional plane $M$ through the symmetry axis $S\!A$, and in $M$ a unit circle $S^1$ (symmetric with respect to $S\!A$) and, furthermore, $2\, N$, $N \in \nat_0$, points $z_n \in M$ (symmetric with respect to $S\!A$, avoiding $S\!A$, and outside $S^1$), then the existence of an (up to a sign) unique harmonic field ${\bf H}$ is established that vanishes at (the axisymmetric circles piercing $M$ at) $z_n$ and nowhere else, that has intensity $I$ at $S^2$ and (exact) decay order $\de$ at infinity. The proof is based on the solution of a nonlinear elliptic equation with discontinuous coefficients, which are, moreover, singular at the symmetry axis. Its combination with fixed boundary conditions was the basis of a recent treatment of the ``geomagnetic direction problem'' \cite{KR22}. Here we have instead natural boundary conditions, which provide less information, and which require, therefore, in part new solution techniques and sharper estimates.

Geophysical intensity problems: the axisymmetric case

TL;DR

This work analyzes the axisymmetric exterior-intensity problem for harmonic vector fields inside the Earth-scale context, formulating a nonlinear boundary-value problem with prescribed surface intensity. By reducing the 3D axisymmetric field to a 2D complex-analytic framework and introducing a nonlinear elliptic equation with a nonlinear, natural boundary condition, the authors construct a fixed-point scheme around a linearized problem to prove the existence of infinitely many axisymmetric solutions with prescribed decay and prescribed vanishing at selected interior points. They establish uniqueness for the linearized problem, develop Hardy-type weighted estimates to control axis singularities, and show how the data (zeroes and boundary intensity) determine a data function Ω that governs the nonlinear problem. The results yield a rigorous, constructive approach to the geophysical intensity problem in the axisymmetric setting and elucidate how nonuniqueness can arise when nontrivial zero-sets are present.

Abstract

Considering the earth or any other celestial body the main sources of the gravitational as well as of the magnetic field lie inside the body. Above the surface both fields are in good approximation harmonic vector fields determined by their values at the body's surface or any other surface enclosing the body. The intensity problem seeks to determine harmonic vector fields vanishing at infinity and with prescribed intensity of the field at the surface. This problem constitutes a nonlinear boundary value problem, whose general solvability is not yet established. In this paper {\em axisymmetric} harmonic fields outside the unit sphere are studied and, given an axisymmetric Hölder continuous intensity function on , the existence of infinitely many solutions of the intensity problem is proved. These solutions can more precisely be characterized as follows: fix a number and a meridional plane through the symmetry axis , and in a unit circle (symmetric with respect to ) and, furthermore, , , points (symmetric with respect to , avoiding , and outside ), then the existence of an (up to a sign) unique harmonic field is established that vanishes at (the axisymmetric circles piercing at) and nowhere else, that has intensity at and (exact) decay order at infinity. The proof is based on the solution of a nonlinear elliptic equation with discontinuous coefficients, which are, moreover, singular at the symmetry axis. Its combination with fixed boundary conditions was the basis of a recent treatment of the ``geomagnetic direction problem'' \cite{KR22}. Here we have instead natural boundary conditions, which provide less information, and which require, therefore, in part new solution techniques and sharper estimates.

Paper Structure

This paper contains 10 sections, 13 theorems, 304 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Mathematical setting and results
  3. Uniqueness in the problems $P_\Omega (Q_R)$ and $P_\Omega (Q_\infty)$
  4. The linearized problem
  5. Proofs of theorems 2.1 and 2.2
  6. Proofs of theorems 2.3 and 2.4
  7. A related intensity problem
  8. Hardy-type inequalities
  9. Estimates of the data function $\Omega$
  10. Estimate of the Fredholm constant $\widetilde{C}_F$

Key Result

Theorem 2.1.

Let $R > 1$ and ${\widetilde{\Omega}} \in L^\infty (Q_R)$ with bound for some constant $K > 0$, then problem $P_{\widetilde{\Omega}} (Q_R)$ has a unique weak (in the sense of eq. (2.28)) solution $u \in {\cal H}_0 (Q_R)$ satisfying the bound with $\beta = 1/5$, $\eta = 10^{-3}$, and $C$ denoting some constant that depends on $K$, but does not depend on $R$.

Figures (6)

  • Figure 1: Illustration of an axisymmetric intensity function on a sphere with intensity coded by color.
  • Figure 2: Various coordinates in the meridional cross section $A_R$.
  • Figure 3: Graphs of the lower bounds $\gamma \mapsto L\!B (0.39, \gamma, 10^{-3}, 0.1, 1.2)$ (solid thin line), $\gamma \mapsto L\!B (0.39, \gamma,0 , 0.1, 1.2)$ (dashed line), and the comparison function $C\!F(0.39, \gamma)$ (solid thick line).
  • Figure 4: $A_{R, \varepsilon}^+$ and the curved boundary components $S_{1, \varepsilon}^+$ and $S_{R, \varepsilon}^+$.
  • Figure 5: Graph of the tent function ${\widehat{\varphi}} : [0\, , \pi] \rightarrow [0\, , \pi/2]$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 2.1.
  • Theorem 2.2.
  • Theorem 2.3.
  • Theorem 2.4.
  • Proposition 3.1.
  • Proposition 3.2.
  • Proposition 3.3.
  • Proposition 6.1.
  • Lemma A.1.
  • Lemma B.1.
  • ...and 3 more