Linear Stability and Jacobi Kernels of Three-Dimensional Sessile Drops

We consider the linear stability of three-dimensional sessile drops with a free contact line.

The equilibrium surface is an axisymmetric solution of the Young--Laplace equation with gravity, fixed volume, and prescribed contact angle.

We derive the constrained second variation of the gravity--capillary energy and formulate the associated Jacobi problem.

Although variational stability gives nonnegativity, the second variation is necessarily degenerate because horizontal translations preserve the energy.

Our main result identifies this degeneracy completely.

Under the pressure--volume nondegeneracy condition $(dV/d\lambda\neq 0)$, we prove that the kernel of the constrained Jacobi operator is exactly the two-dimensional space generated by horizontal translations.

The proof combines the geometric structure of the Jacobi operator with a Fourier-mode analysis: the axisymmetric mode is ruled out by the pressure--volume condition, the first mode gives translations, and all higher modes are excluded by comparison.

This provides the precise linear nondegeneracy underlying stability of droplet dynamics modulo translations.