For a class of systems of semi-linear elliptic equations, including (Formula presented.), for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L<sup>∞</sup> boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→+∞, lthat is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one.

Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

SOAVE, NICOLA;
2015

Abstract

For a class of systems of semi-linear elliptic equations, including (Formula presented.), for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→+∞, lthat is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one.
Analysis; Mathematics (miscellaneous); Mechanical Engineering
File in questo prodotto:
File Dimensione Formato  
11311-1010801_Soave.pdf

accesso aperto

: Post-Print (DRAFT o Author’s Accepted Manuscript-AAM)
Dimensione 843.53 kB
Formato Adobe PDF
843.53 kB Adobe PDF Visualizza/Apri

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: http://hdl.handle.net/11311/1010801
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 34
  • ???jsp.display-item.citation.isi??? 34
social impact