Asymptotic Expansion of Three-Dimensional Laplace-Type Integrals with a Nonstationary Boundary Minimum Point via Directional Geometric Reduction

We study the asymptotic expansion of three-dimensional Laplace-type integrals with a non-stationary boundary minimum point. The presence of two tangential directions introduces additional geometric complexity in the asymptotic analysis. We develop a geometric approach based on the successive combination of a tubular transformation, an angular reduction, and a polar parametrization of the tangential variables, reducing the original three-dimensional problem to a family of direction-dependent two-dimensional Laplace-type integrals. The main contribution is the introduction of a directional order of contact describing the local interaction between the boundary and the level surface of the phase function in each tangential direction. Under suitable regularity assumptions and a constant directional contact order, a complete asymptotic expansion is obtained, together with a remainder estimate uniform with respect to the tangential direction. The leading term has order λ−(1+2/p) and an explicit leading coefficient. The framework covers both the classical non-degenerate case p=2 and degenerate cases with higher even contact order. An application to the Gibbs equilibrium of a confined particle is presented, for which the leading coefficient is computed explicitly and the resulting asymptotic formula is numerically verified.