Uniqueness of weak DLSS solutions in dimensions four and higher

Establish uniqueness for weak solutions of the Derrida–Lebowitz–Speer–Spohn equation in dimensions d≥4, and thereby determine whether the contraction semigroup constructed in L² coincides with weak solutions generated by the approximation schemes of Jüngel–Pinnau, Gualdani–Jüngel–Toscani, and Jüngel–Matthes.

Background

The paper constructs a canonical contraction semigroup for the DLSS equation in the square-root variable on a bounded convex domain with Neumann boundary conditions, for arbitrary nonnegative L² initial data and in every space dimension. It identifies this semigroup with the established weak-solution theory only in dimensions d≤3, using Fischer’s uniqueness theorem in the relevant regularity class and, for the Neumann setting, an additional conditional boundary-adaptation issue.

For dimensions d≥4, no uniqueness theory for the relevant weak solutions is available. Consequently, it is unresolved whether the canonical semigroup solution agrees with weak solutions obtained through the earlier approximation schemes; resolving this would connect the paper’s operator-theoretic construction to those approximation-based weak formulations in higher dimensions.