Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings. Issue 5 (17th January 2019)

• Record Type: Journal Article
• Authors: Jarohs, Sven; Kulczycki, Tadeusz; Salani, Paolo
• Is Part Of: Mathematische Nachrichten. Volume 292:Issue 5(2019)
• Page Start: 1008
• Page End: 1021
• Publication Date: 2019-01-17
• Subjects: fractional Laplacian -- starshaped superlevel sets -- 35B06 -- 35B50; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201700226
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: We study solutions of the problem 0.1 − ( − Δ ) α / 2 u = f ( x, u ) in D 0 ∖ D ¯ 1, u = 0 in R N ∖ D 0, u = 1 in D ¯ 1, where D 1, D 0 ⊂ R N are open sets such that D ¯ 1 ⊂ D 0, α ∈ ( 0, 2 ), and f is a nonlinearity. Under different assumptions on f we prove that, if D 0 and D 1 are starshaped with respect to the same point x ¯ ∈ D ¯ 1, then the same occurs for every superlevel set of u .