In this paper we present new results related to the ones obtained in our previous papers on the singular semilinear elliptic problem u ≥ 0 in Ω, −div A(x)Du = F(x, u) in Ω, u = 0 on ∂Ω, where F(x, s) is a Carathéodory function which can take the value +∞ when s = 0. Three new topics are investigated. First, we present definitions which we prove to be equivalent to the definition given in our paper Giachetti, Martínez-Aparicio, Murat (2018). Second, we study the set {x ∈ Ω : u(x) = 0}, which is the set where the right-hand side of the equation could be singular in Ω, and we prove that actually, at almost every point x of this set, the right-hand side is non singular since one has F(x, 0) = 0. Third, we consider the case where a zeroth order term μu, with μ a nonnegative bounded Radon measure which also belongs to H−1(Ω), is added to the left-hand side of the singular problem considered above. We explain how the definition of solution given in Giachetti, Martínez-Aparicio, Murat (2018) has to be modified in such a case, and we explicitly give the a priori estimates that every such solution satisfies (these estimates are at the basis of our existence, stability and uniqueness results). Finally we give two counterexamples which prove that when a zeroth order term μu of the above type is added to the left-hand side of the problem, the strong maximum principle in general does not hold anymore.

On the definition of the solution to a semilinear elliptic problem with a strong singularity at u=0 / Giachetti, Daniela; Martínez-Aparicio, Pedro J.; Murat, François. - In: NONLINEAR ANALYSIS: SPECIAL ISSUE. - STAMPA. - (2018).

On the definition of the solution to a semilinear elliptic problem with a strong singularity at u=0

Daniela Giachetti;
2018

Abstract

In this paper we present new results related to the ones obtained in our previous papers on the singular semilinear elliptic problem u ≥ 0 in Ω, −div A(x)Du = F(x, u) in Ω, u = 0 on ∂Ω, where F(x, s) is a Carathéodory function which can take the value +∞ when s = 0. Three new topics are investigated. First, we present definitions which we prove to be equivalent to the definition given in our paper Giachetti, Martínez-Aparicio, Murat (2018). Second, we study the set {x ∈ Ω : u(x) = 0}, which is the set where the right-hand side of the equation could be singular in Ω, and we prove that actually, at almost every point x of this set, the right-hand side is non singular since one has F(x, 0) = 0. Third, we consider the case where a zeroth order term μu, with μ a nonnegative bounded Radon measure which also belongs to H−1(Ω), is added to the left-hand side of the singular problem considered above. We explain how the definition of solution given in Giachetti, Martínez-Aparicio, Murat (2018) has to be modified in such a case, and we explicitly give the a priori estimates that every such solution satisfies (these estimates are at the basis of our existence, stability and uniqueness results). Finally we give two counterexamples which prove that when a zeroth order term μu of the above type is added to the left-hand side of the problem, the strong maximum principle in general does not hold anymore.
2018
Semilinear elliptic problems. Singularity at u = 0. Zeroth order term with coefficient a measure. Strong maximum principle
01 Pubblicazione su rivista::01a Articolo in rivista
On the definition of the solution to a semilinear elliptic problem with a strong singularity at u=0 / Giachetti, Daniela; Martínez-Aparicio, Pedro J.; Murat, François. - In: NONLINEAR ANALYSIS: SPECIAL ISSUE. - STAMPA. - (2018).
File allegati a questo prodotto
File Dimensione Formato  
Giachetti_definition_2018.pdf

accesso aperto

Tipologia: Documento in Post-print (versione successiva alla peer review e accettata per la pubblicazione)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 1.05 MB
Formato Adobe PDF
1.05 MB Adobe PDF
Giachetti_preprint_definition_2018.pdf

solo gestori archivio

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 827.53 kB
Formato Adobe PDF
827.53 kB Adobe PDF   Contatta l'autore

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1147485
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 4
social impact