scholarly journals THE GINZBURG–LANDAU FUNCTIONAL WITH A DISCONTINUOUS AND RAPIDLY OSCILLATING PINNING TERM. PART I: THE ZERO DEGREE CASE

2011 ◽  
Vol 13 (05) ◽  
pp. 885-914 ◽  
Author(s):  
MICKAËL DOS SANTOS ◽  
PETRU MIRONESCU ◽  
OLEKSANDR MISIATS
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Ginzburg Landau ◽  
Zero Degree

We consider minimizers of the Ginzburg–Landau energy with pinning term and zero degree Dirichlet boundary condition. Without any assumptions on the pinning term, we prove that these minimizers do not develop vortices in the limit ε → 0. We next consider the specific case of a periodic discontinuous pinning term taking two values. In this setting, we determine the asymptotic behavior of the minimizers as ε → 0.

scholarly journals Parabolic equations with natural growth approximated by nonlocal equations

2020 ◽  
Vol 23 (01) ◽  
pp. 1950088
Author(s):  
Tommaso Leonori ◽  
Alexis Molino ◽  
Sergio Segura de León
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Comparison Principle ◽  
Dirichlet Boundary Condition ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Natural Growth ◽  
Nonlocal Problems ◽  
Nonlocal Equations ◽  
Bounded Smooth

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is [Formula: see text] where we take, as the most important instance, [Formula: see text] with [Formula: see text] as well as [Formula: see text], [Formula: see text] is a smooth symmetric function with compact support and [Formula: see text] is either a bounded smooth subset of [Formula: see text], with nonlocal Dirichlet boundary condition, or [Formula: see text] itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation.

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