Global boundedness of weak solutions to a class of nonuniformly elliptic equations

• Published in: Mathematische annalen Vol. 392; no. 2; pp. 1519 - 1539
• Main Authors: Cupini, Giovanni, Marcellini, Paolo
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.06.2025
• Subjects: Boundary conditions; Differential equations; Dirichlet problem; Divergence; Elliptic functions; Ellipticity; Fields (mathematics); Upper bounds
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-025-03126-5

We consider second order elliptic equations in divergence form ∑i=1n∂∂xiaix,u,Du=bx,u,Du,x∈Ω,where Ω is a bounded open set in Rn and u:Ω→R. Our aim is to give conditions on the vector field ax,u,Du=aix,u,Dui=1,…,n and on the right hand side bx,u,Du in order to obtain the global boundedness in Ω¯ of weak solutions u to the Dirichlet problem associated to the previous differential equation, when a boundary condition u=u0∈L∞Ω has been fixed on ∂Ω. We do not assume structure conditions on the vector field ax,u,Du, nor sign assumptions on bx,u,Du; we only consider ellipticity and growth conditions on a and b. A main novelty with respect to the literature about this subject is that we assume general p,q-growth conditions for the principal part of the differential equation; however we do not need an upper bound for the ratioqp, but nothing more than 1≤p≤q.