Weak solutions and simulations to a square phase‐field crystal model with Neumann boundary conditions

2022 ◽  
Author(s):  
Fan Wu ◽  
Zixian Zhu
Keyword(s):  
Boundary Conditions ◽  
Phase Field ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Crystal Model ◽  
Phase Field Crystal

scholarly journals Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Ghasem A. Afrouzi ◽  
Z. Naghizadeh ◽  
Nguyen Thanh Chung
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Weak Solutions ◽  
Multiple Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Nonlocal Problems ◽  
Mapping Theory ◽  
Nonlinear Neumann Boundary Conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

scholarly journals Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

2015 ◽  
Vol 17 (03) ◽  
pp. 1450016
Author(s):  
Futoshi Takahashi
Keyword(s):  
Boundary Conditions ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Extremal Solutions ◽  
Smooth Bounded Domain ◽  
Suitable Notion ◽  
Extremal Parameter ◽  
Nonlinear Neumann Boundary Conditions ◽  
Increasing Function

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions [Formula: see text] where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

Sign in / Sign up

Export Citation Format

Share Document