The Double Equivalence

The fractional p-Laplacian heat equation combines two generalizations of classical diffusion: nonlocal (fractional) diffusion, where influence extends beyond nearest neighbors, and nonlinear (p-Laplacian) diffusion, where the diffusion rate depends on the gradient magnitude. Each generalization alone is well studied. Together, they resist the standard techniques of both theories.

For the parabolic fractional p-Laplace equation in the degenerate range (p between 2 and 2/(1-s)), three results are established simultaneously. First, weak solutions are Lipschitz continuous in space — and when p exceeds 1/(1-s), also Lipschitz in time. Second, a comparison principle holds: if one solution starts below another, it stays below. Third, weak solutions and viscosity solutions — two fundamentally different ways of defining what “solves” the equation means — are equivalent.

The third result is the structural one. Weak solutions are defined through integration — they satisfy the equation in an averaged sense. Viscosity solutions are defined through comparison — they’re bounded by smooth test functions at every point. These notions arise from different mathematical traditions (variational methods vs. maximum principles) and agree for simple equations but can diverge for nonlinear or nonlocal ones. Proving their equivalence for the fractional p-Laplacian means that any result proved using one framework automatically transfers to the other.

The through-claim: the Lipschitz regularity is not just a regularity result — it’s the bridge that makes the equivalence work. Without knowing that solutions are sufficiently smooth, you can’t translate between the two frameworks. The regularity creates the common ground where two different definitions of “solution” turn out to describe the same objects.