Isomorphism theorems and their applications for elliptic boundary-value problems with discontinuous coefficients, and boundary conditions and conjugacy conditions which are not normal

1977 ◽  
Vol 28 (4) ◽  
pp. 412-416
Author(s):  
Yu. V. Kostarchuk
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Discontinuous Coefficients ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Isomorphism Theorems

scholarly journals Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions

2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Ciprian G. Gal ◽  
Mahamadi Warma
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Fredholm Alternative ◽  
Elliptic Boundary Value ◽  
Nonlinear Elliptic ◽  
Nonlinear Elliptic Boundary

Given a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω and p,q∈(1,+∞), we consider the quasilinear elliptic equation -Δpu+α1u=f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form bx∇up-2∂nu-ρbxΔq,Γu+α2u=g on ∂Ω. In the first part of the article, we give necessary and sufficient conditions in terms of the given functions f, g and the nonlinearities α1, α2, for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of “nonlinear Fredholm alternative” for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an L∞-setting.

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