Subellipticity on pseudo-convex domains with isolated degeneracies

Area of Science:

• Complex Analysis
• Partial Differential Equations
• Geometric Analysis

Background:

• The [unk]-Neumann problem is crucial in complex analysis and partial differential equations.
• Subellipticity conditions are essential for understanding the regularity of solutions.
• Pseudoconvex domains present unique challenges in analysis.

Purpose of the Study:

• To establish necessary and sufficient conditions for the subellipticity of the [unk]-Neumann problem.
• To introduce a general method for analyzing subellipticity.
• To prove the sufficiency of these conditions for a specific class of domains.

Main Methods:

• Reduction of subellipticity to inequalities between germs of C(infinity) functions.
• Application of the general method to domains with specific boundary properties.
• Analysis of Levi form degeneration at isolated boundary points.

Main Results:

• A general method for subellipticity analysis is outlined.
• Sufficiency of conjectured conditions is proven for domains with real analytic boundaries and isolated Levi form degeneracies.
• The reduction to function inequalities provides a new analytical tool.

Conclusions:

• The study provides a significant advancement in understanding the [unk]-Neumann problem.
• The developed method offers a pathway to analyze subellipticity in more complex settings.
• The findings are particularly relevant for domains with analytic boundaries and specific degeneracy patterns.