Communications in Partial Differential Equations· 2026Q1
Another regularizing property of the 2D eikonal equation
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- 2026year
Short summary
Solutions to the 2D eikonal equation with borderline Besov regularity (p=6) are shown to be locally Lipschitz, extending a known regularizing effect.
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Key points
- Solutions to the 2D eikonal equation with Besov regularity Bp,∞1/3 for p=6 are proven to be locally Lipschitz.
- This regularizing effect extends a known property that applies to solutions with lower regularity.
- For disk domains with tangent boundary conditions, the result holds for p slightly below 6.
- The findings address a long-standing conjecture by Aviles and Giga regarding solution regularity.
AI-generated from the title and abstract; the full text is not read.
Abstract
A weak solution of the two-dimensional eikonal equation amounts to a vector field m:Ω⊂R2→R2 such that |m|=1 a.e. and divm=0 in D′(Ω). It is known that, if m has some low regularity, e.g., continuous or W1/3,3, then m is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption m∈Bp,∞1/3 for p>3. In this note we establish that regularizing effect in the borderline case p=6, above which the Besov regularity assumption implies continuity. If the domain is a disk and m satisfies tangent boundary conditions, we also prove this for p slightly below 6.
The authors' abstract, as published at the source. Communications in Partial Differential Equations, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics