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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

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Published on: August 30, 2013

Gradient regularity for widely degenerate elliptic partial differential equations.

Michael Strunk¹

¹Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria.

SN Partial Differential Equations and Applications

|September 30, 2025

Summary

This summary is machine-generated.

This study analyzes weak solutions to elliptic equations with degenerate ellipticity. The key finding is the regularity of K(Du) for continuous functions K vanishing on a specific set E.

Keywords:

Gradient regularity Weak solutions Widely degenerate elliptic PDEs

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Area of Science:

Partial Differential Equations
Mathematical Analysis
Calculus of Variations

Background:

Investigates weak solutions to elliptic equations with degenerate ellipticity.
Focuses on equations where the ellipticity degenerates within a bounded convex set E.
Considers the impact of the function F(x, ξ) properties on solution regularity.

Purpose of the Study:

To establish the regularity of weak solutions to a class of elliptic equations.
To analyze the behavior of solutions where ellipticity degenerates in a specific set.
To determine conditions under which K(Du) is a continuous function.

Main Methods:

Analysis of weak solutions to elliptic partial differential equations.
Utilizing properties of the function F(x, ξ) and the degenerate set E.
Applying techniques from the calculus of variations and regularity theory.

Main Results:

Established the continuity of K(Du) for weak solutions u.
Demonstrated regularity for elliptic equations with ellipticity degenerating in a set E.
Showed that for any continuous function K vanishing on E, K(Du) is continuous.

Conclusions:

The regularity of weak solutions is established under degenerate ellipticity conditions.
The properties of the function F and the set E are crucial for solution regularity.
The findings contribute to the understanding of elliptic equations with complex degeneracy.