For every f ∈ LN(Ω) defined in an open bounded subset Ω of ℝN, we prove that a solution u ∈ W01,1(Ω) of the 1-Laplacian equation -div (∇u/|Δu|)= f in Ω satisfies ∇u = 0 on a set of positive Lebesgue measure. The same property holds if f ∉ LN(Ω) has small norm in the Marcinkiewicz space of weak-LNfunctions or if u is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.
Flat solutions of the 1-Laplacian equation / Orsina, Luigi; Ponce, Augusto C.. - In: RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE. - ISSN 0049-4704. - STAMPA. - 49:(2017), pp. 41-51. [10.13137/2464-8728/16204]
Flat solutions of the 1-Laplacian equation
Orsina, Luigi;
2017
Abstract
For every f ∈ LN(Ω) defined in an open bounded subset Ω of ℝN, we prove that a solution u ∈ W01,1(Ω) of the 1-Laplacian equation -div (∇u/|Δu|)= f in Ω satisfies ∇u = 0 on a set of positive Lebesgue measure. The same property holds if f ∉ LN(Ω) has small norm in the Marcinkiewicz space of weak-LNfunctions or if u is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.| File | Dimensione | Formato | |
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