Abstract
In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B u =gon Ω∂Г where ω is a domain in ℝn,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary.
The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue, the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet boundary value problem, where Γ=∂Ω, were examined intensively by many authors.
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Pinchover, Y., Saadon (Suez), T. On positivity of solutions of degenerate boundary value problems for second-order elliptic equations. Isr. J. Math. 132, 125–168 (2002). https://doi.org/10.1007/BF02784508
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DOI: https://doi.org/10.1007/BF02784508