scholarly journals Asymptotic behavior of radial minimizers of a p-energy functional with nonvanishing Dirichlet boundary condition

2009 ◽  
pp. 237-252
Author(s):  
Yaqin Zheng ◽  
Yutian Lei
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Energy Functional

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.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

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