Mathematicians find cracks in decades-old theory of smooth functions
Researchers have discovered that a foundational assumption in mathematical analysis—that certain types of smooth functions must be continuous—fails in unusual geometric spaces. The finding challenges 40+ years of accepted theory and could reshape how engineers model complex systems in materials science, signal processing, and artificial intelligence.
Originaltitel: Non-quasicontinuous Newtonian functions and outer capacities based on Banach function spaces
<p>We construct various examples of Sobolev-type functions, defined via upper gradients in metric spaces, that fail to be quasicontinuous or weakly quasicontinuous. This is done with quasi-Banach function lattices X as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case X = Lp with 1 ≤ p < ∞, where all Sobolev-type functions in N1,p are known to be quasicontinuous, provided that the underlying metric space 𝒫 is locally complete. In most of our examples, 𝒫 is a compact subset of R2 and X = L∞. Four particular examples are the damped topologist’s sine curve, the von Koch snowflake curve, the Cantor ternary set and the Sierpiński carpet. We also discuss several related properties, such as whether the Sobolev capacity is an outer capacity, and how these properties are related. A fundamental role in these considerations is played by the lack of the Vitali–Carathéodory property.</p>