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Trace theory for parabolic boundary value problems with rough boundary conditions
Robert Denk1, Floris B Roodenburg2
1Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany.
Journal of Evolution Equations
|June 8, 2026
Summary
This study introduces trace spaces for weighted Sobolev functions, essential for boundary value problems with boundary singularities. We demonstrate their application by proving the heat equation
Area of Science:
- Mathematical analysis
- Partial differential equations
- Function spaces
Background:
- Weighted Sobolev spaces are crucial for analyzing differential equations with solutions exhibiting boundary singularities.
- Trace theory is fundamental for understanding the behavior of functions and their derivatives at the boundaries of domains.
- Boundary value problems often require specialized function spaces to handle complex boundary conditions and solution behaviors.
Purpose of the Study:
- To characterize trace spaces derived from intersections of weighted, vector-valued Sobolev spaces.
- To apply the developed trace theory to establish well-posedness for the heat equation with challenging boundary data.
- To extend the applicability of Sobolev space methods to domains with specific regularity properties.
Main Methods:
- Characterization of trace spaces using intersections of weighted Sobolev spaces with power-type weights.
- Development of trace theory tailored for functions with boundary singularities.
- Application of trace theory to analyze the heat equation using functional analysis techniques.
Main Results:
- The study successfully characterizes trace spaces arising from weighted Sobolev spaces, considering powers of the distance to the boundary.
- A novel trace theory is established, suitable for boundary value problems where solution derivatives may diverge at the boundary.
- Well-posedness for the heat equation with rough, inhomogeneous boundary data in higher regularity Sobolev spaces is proven for C^(1,κ) domains.
Conclusions:
- The characterized trace spaces provide a robust framework for addressing boundary value problems with singular behavior.
- The established trace theory enhances the understanding of function behavior at domain boundaries.
- The results offer a significant advancement in the analysis of the heat equation with complex boundary conditions and rough data.
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