The defining boundary conditions and the degenerate problem for elliptic boundary-value problems with a small parameter in the highest derivatives

2003 ◽  
Vol 194 (5) ◽  
pp. 641-668 ◽  
Author(s):  
S A Golopuz
Keyword(s):  
Boundary Conditions ◽  
Small Parameter ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Degenerate Problem ◽  
Elliptic Boundary Value

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.

scholarly journals On solvability of elliptic boundary value problems via global invertibility

2020 ◽  
Vol 40 (1) ◽  
pp. 37-47
Author(s):  
Michał Bełdziński ◽  
Marek Galewski
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Dirichlet Boundary Conditions ◽  
Elliptic Boundary Value ◽  
Global Invertibility

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.

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