[Paper Review] Variational formulation of the earth's elastic-gravitational deformations under low regularity conditions

Katharina Brazda, Maarten V. de Hoop+1 more|arXiv (Cornell University)|Feb 15, 2017

Geophysics and Gravity Measurements41 references3 citations

TL;DR

This paper establishes a rigorous variational formulation for the Earth's elastic-gravitational deformations under minimal regularity conditions using Sobolev spaces and calculus of variations. It proves Fréchet differentiability of the action functional, yielding well-defined Euler-Lagrange equations for a rotating, self-gravitating, composite Earth model with discontinuous material parameters and fluid-solid boundaries.

ABSTRACT

We present a construction of the action, in the framework of the calculus of variations and Sobolev spaces, describing deformations and the oscillations of a uniformly rotating, elastic and self-gravitating earth. We establish the Fréchet differentiability of the action under minimal regularity assumptions, which constrain the possible composition of an earth model. Thus we obtain well-defined Euler-Lagrange equations, weakly and strongly, that is, the system of elastic-gravitational equations.

Motivation & Objective

• To develop a mathematically rigorous variational framework for modeling the Earth’s elastic-gravitational oscillations under minimal regularity assumptions.
• To ensure the action functional is Fréchet differentiable even when material parameters and boundaries (e.g., Moho, CMB) are discontinuous or only Lipschitz regular.
• To derive consistent Euler-Lagrange equations for a composite, rotating, self-gravitating Earth model with fluid and solid subdomains.
• To incorporate physically accurate fluid-solid interface conditions—specifically, frictionless tangential slip—via first-principles derivation from Newton’s third law.
• To lay the foundation for existence and uniqueness proofs of solutions in a subsequent paper, by ensuring the variational formulation is well-posed under realistic physical constraints.

Proposed method

• Formulates the action functional in the framework of Sobolev spaces, using volume and surface Lagrangian densities derived from nonlinear elasticity and gravity physics.
• Imposes minimal regularity: particle motions are positively oriented Lipschitz, material parameters (density, stiffness tensor) are in $L^∞$, and boundaries are Lipschitz-regular.
• Applies linearization to small perturbations around an equilibrium rotating state, leading to quadratic volume and surface Lagrangian densities.
• Derives the first variation of the action and proves its Fréchet differentiability using distribution theory and trace theorems on Lipschitz domains.
• Uses the divergence theorem and the fundamental lemma of calculus of variations to derive weak-form Euler-Lagrange equations and natural boundary conditions.
• Incorporates fluid-solid interface contributions explicitly in the action, ensuring consistency with physical laws such as momentum balance and Newton’s third law.

Experimental results

Research questions

• RQ1 What minimal regularity conditions on material parameters and domain boundaries allow for a well-defined variational formulation of elastic-gravitational deformations in a rotating Earth?
• RQ2 How can the action functional be rigorously constructed and shown to be Fréchet differentiable under low-regularity assumptions, including discontinuities at internal boundaries?
• RQ3 What is the correct mathematical formulation of fluid-solid interface conditions—specifically, frictionless tangential slip—within a variational framework?
• RQ4 How do the resulting Euler-Lagrange equations relate to the classical equations of normal mode seismology, and under what conditions do they reduce to them?
• RQ5 What are the implications of the lack of coercivity in the spatial operator and the presence of an essential spectrum due to a fluid outer core?

Key findings

• The action functional is Fréchet differentiable under minimal regularity: particle motions are Lipschitz continuous, and material parameters (density and stiffness tensor) are in $L^∞$.
• The resulting Euler-Lagrange equations are derived in weak form, ensuring mathematical consistency even with discontinuous interior boundaries such as the core-mantle boundary or Moho.
• The fluid-solid interface contributes explicitly to the action, and the derived boundary conditions enforce frictionless tangential slip via first-principles mechanics.
• The variational formulation is consistent with underlying conservation laws, including momentum and energy conservation, as verified through the structure of the Lagrangian.
• The lack of coercivity in the spatial operator implies that standard energy estimates may not hold directly, necessitating further analysis—addressed in a follow-up paper.
• The formulation allows for straightforward extension to include shear ruptures with nonlinear friction laws, by modifying the surface Lagrangian density at fault interfaces.