A class of degenerate elliptic equations with nonlinear boundary conditions

2016 ◽  
pp. 1-29
Author(s):  
Zhuoran Du ◽  
Yanqin Fang ◽  
Changfeng Gui
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic

OBSTACLE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS WITH NONHOMOGENEOUS NONLINEAR BOUNDARY CONDITIONS

2008 ◽  
Vol 18 (11) ◽  
pp. 1869-1893 ◽  
Author(s):  
FUENSANTA ANDREU ◽  
NOUREDDINE IGBIDA ◽  
JOSÉ M. MAZÓN ◽  
JULIÁN TOLEDO
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Maximal Monotone ◽  
Nonlinear Boundary Conditions ◽  
Obstacle Problems ◽  
Laplacian Operator ◽  
Usual Sense ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic

In this paper we study the questions of existence and uniqueness of solutions for equations of type - div a(x,Du) + γ(u) ∋ ϕ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x,Du) · η + β(u) ∋ ψ. The nonlinear elliptic operator div a(x,Du) modeled on the p-Laplacian operator Δp(u) = div (|Du|p-2Du), with p > 1, γ and β maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) ∩ β(0), [Formula: see text] and the data ϕ ∈ L1(Ω) and ψ ∈ L1(∂ Ω). Since D(γ) ≠ ℝ, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solution has to be introduced.

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