A new approach to weighted Sobolev spaces

Area of Science:

• Mathematical Analysis
• Partial Differential Equations
• Functional Analysis

Background:

• Traditional weighted Sobolev spaces require modifications to the domain when weight functions are small.
• Existing methods exclude non-locally integrable functions from analysis.

Purpose of the Study:

• To introduce a new definition for weighted Sobolev spaces accommodating arbitrarily small weight functions.
• To extend the analysis to include non-locally integrable functions.
• To establish conditions for unique solutions to degenerate elliptic partial differential equations.

Main Methods:

• Replacing distributional derivatives with a new weak derivative concept.
• Developing tools such as Poincaré inequality and trace operators.
• Establishing density results for smooth functions within the new spaces.

Main Results:

• A new framework for weighted Sobolev spaces is presented.
• Non-locally integrable functions can now be analyzed within these spaces.
• Conditions for unique solutions to degenerate elliptic partial differential equations are provided.

Conclusions:

• The new definition offers a more flexible and inclusive approach to weighted Sobolev spaces.
• This work advances the study of degenerate elliptic partial differential equations by including non-locally integrable solutions.