Nonlinear diffusion in transparent media: The resolvent equation

Abstract

We consider the partial differential equationwith f nonnegative and bounded and m ∈ R. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the H N −1 Hausdorff measure. Results and proofs extend to more general nonlinearities.

The Dirichlet problem for singular elliptic equations with general nonlinearities

“…this fact was already noticed in [30,26] under some additional assumptions. Formula (2.2) follows by the Chain rule formula (see [3,Theorem 3.99]) observing that…”

An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

“…An example is u t = (u m u x /|u x |) x for m ≥ 0, which is called the total variation flow for m = 0, or the heat equation in transparent media for m = 1. See [12] and references therein. However, there are two delicate issues to obtain a Lyapunov function.…”

Anzellotti's pairing theory and the Gauss–Green theorem

“…Since J u is a countably H N −1 -rectifiable set, a straightforward consequence of Proposition 3.1 is the following result (see also [27,Lemma 2.5]).…”

“…Hence (20) and (21) we conclude that (22) follows. Finally, the general case u * ∈ L 1 loc (R N , div A) follows using the previous step on the truncated functions u k := T k (u), where, given k > 0, T k is defined by (27) T k (s) := max{min{s, k}, −k} , s ∈ R.…”

“…We extract the following fact from the proof of Theorem 4.12. Let A ∈ DM ∞ loc (Ω), u ∈ BV loc (Ω) ∩ L 1 loc (Ω, div A), and let u k := T k (u) be the truncated functions of u, where T k is defined in (27). If we define We conclude this section with an approximation result in the spirit of [11,Theorem 1.2].…”