Well-Posedness and Boundary Value Problems for a Class of Quasilinear Divergence-Form Equations Arising in Density Field Dynamics

Abstract (Mathematical Physics; Preprint)

We study the quasilinear elliptic partial differential equation −∇ · [μ(|∇ψ|)∇ψ] = f in Ω ⊆ ℝ³, where μ is a nonlinear constitutive function. Motivated by density-field models of gravitational optics, we develop a rigorous framework for existence, uniqueness, and regularity of weak solutions, extend the analysis to exterior domains with asymptotically flat boundary conditions, and incorporate monotone nonlinear Robin–Neumann conditions modeling photon-spheres and horizons. We further establish stability estimates, continuous dependence on data, and parabolic well-posedness using nonlinear semigroup theory. A variational formulation, catalog of admissible μ-families, and finite element method (FEM) implementation outline are provided. Open problems relevant to global existence and singularity formation are discussed.