= Fixed-edge Euler-Lagrange equation for Mumford–Shah
{title2=$u-\alpha\Delta u=g,\quad\partial_nu=0$}
For a fixed sufficiently regular <image edge> set, the <Mumford–Shah functional> is a <strictly convex> quadratic energy on the <Sobolev space> of its complement. A <first variation> gives $u-\alpha\Delta u=g$ within each region and separate one-sided <Neumann boundary conditions> on free edges. A jump need not vanish across an edge. The weak solution is unique by the <Lax-Milgram theorem>; this does not prove uniqueness when the edge set is also optimized.
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