The Dirichlet problem for a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow

Abstract

In this lecture we consider the Dirichlet problem associated with a nonlinear singular elliptic equation arising in the level set formulation of the inverse mean curvature flow; namely,We introduce a suitable concept of weak solution, for which we prove existence and uniqueness of the homogeneous Dirichlet problem in a bounded open set of ℝ , in the case 0 ≤ ∈ (Ω), > . Moreover, examples of explicit solutions are shown.

Existence and comparison results for an elliptic equation involving the 1-Laplacian and $$L^1$$ L 1 -data

“…Although the homogeneous problem is not interesting in bounded domains because it leads to the trivial solution, this does not occur in the non-homogeneous case since the source can override the fattening phenomenon (at least when f is not very small). Problem (1) in bounded domains has been considered in [17] for data f ∈ L p (Ω), with p > N , seeking bounded solutions, and in [16] when data belong to the Marcinkiewicz space L N,∞ (Ω), looking for unbounded variational solutions. Existence and uniqueness results have been obtained in both papers for any given non-negative datum.…”

Bounded solutions to the 1-Laplacian equation with a critical gradient term

“…Now apply [27], Proposition 2.2, to get that the Radon-Nikodým derivative of (z, Du) with respect to |Du| is equal to the Radon-Nikodým derivative of (z, De u ) with respect to |De u |. Thus, (9) implies (z, De u ) = |De u |.…”

“…Related developments can be found in [31]; the framework of these works, however, is different since Ω is unbounded and the datum vanishes. It is shown in [27] that the solution behaves very differently from (2). Indeed,…”

“…We will also need the following technical results. The proof of the first one can be found in [27], Proposition 2.3. Proof.…”

Anzellotti's pairing theory and the Gauss–Green theorem

“…Hence, after some simplifications (35) gives Summing together (36) and (37) we get (32). The relations (33) and (34) More precisely, the measure (A, D(uv)) admits the following decomposition: (i) absolutely continuous part: (A, D(uv)) a = A · ∇(uv) L N , with ∇(uv) = u∇v + v∇u; (ii) jump part:…”