The planar Mumford-Shah conjecture (interior regularity of the discontinuity set)

Theorem 1.1: for a reduced absolute minimizer $(u,K)$ with $g\in L^\infty$, every $x\in K$ has a neighborhood where $K$ is one $C^{1,\alpha}$ arc through $x$, one arc ending at $x$, or three arcs meeting at $x$ at $120^\circ$; only finitely many connected components of $K$ meet any $U\Subset\Omega$. Corollary: $\nabla u\in L^{4,\infty}_{\mathrm{loc}}$, sharp because of crack tips. The key step excludes bounded components in blow-up limits. It does not treat behaviour at $\partial\Omega$, higher dimensions, or minimizers in weaker senses than absolute ones.

Theorem 1.1: for a reduced absolute minimizer $(u,K)$ with $g\in L^\infty$, every $x\in K$ has a neighborhood where $K$ is one $C^{1,\alpha}$ arc through $x$, one arc ending at $x$, or three arcs meeting at $x$ at $120^\circ$; only finitely many connected components of $K$ meet any $U\Subset\Omega$. Corollary: $\nabla u\in L^{4,\infty}_{\mathrm{loc}}$, sharp because of crack tips. The key step excludes bounded components in blow-up limits. It does not treat behaviour at $\partial\Omega$, higher dimensions, or minimizers in weaker senses than absolute ones.