Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings | Zendy

Jarohs Sven | Zendy; Kulczycki Tadeusz | Zendy; Salani Paolo | Zendy

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Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

Author(s) -

Jarohs Sven,

Kulczycki Tadeusz,

Salani Paolo

Publication year - 2019

Publication title -

mathematische nachrichten

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.913

H-Index - 50

eISSN - 1522-2616

pISSN - 0025-584X

DOI - 10.1002/mana.201700226

Subject(s) - mathematics , point (geometry) , nonlinear system , set (abstract data type) , open set , fractional laplacian , laplace operator , pure mathematics , discrete mathematics , mathematical analysis , geometry , computer science , physics , quantum mechanics , programming language

We study solutions of the problem 0.1− ( − Δ ) α / 2 u=f ( x , u )inD 0 ∖ D ¯ 1 ,u = 0 inR N ∖ D 0 ,u = 1 inD ¯ 1 ,whereD 1 , D 0 ⊂ R Nare open sets such thatD ¯ 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 pointx ¯ ∈ D ¯ 1 , then the same occurs for every superlevel set of u .

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