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.

scholarly journals Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Jin-soo Hwang
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Continuous Dependence ◽  
Dirichlet Boundary ◽  
Strong Solutions ◽  
Function Spaces ◽  
Well Posedness ◽  
Continuous Dependence On Data ◽  
Strongly Damped ◽  
Weak And Strong Solutions

We consider a strongly damped quasilinear membrane equation with Dirichlet boundary condition. The goal is to prove the well-posedness of the equation in weak and strong senses. By setting suitable function spaces and making use of the properties of the quasilinear term in the equation, we have proved the fundamental results on existence, uniqueness, and continuous dependence on data including bilinear term of weak and strong solutions.

Existence of solution for elliptic equations with supercritical Trudinger–Moser growth

2021 ◽  
pp. 1-20
Author(s):  
Luiz F. O. Faria ◽  
Marcelo Montenegro
Keyword(s):  
Boundary Condition ◽  
Unit Ball ◽  
Positive Solution ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Existence Of Solution

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.

Symmetry breaking in the minimization of the second eigenvalue for composite membranes

2015 ◽  
Vol 145 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Claudia Anedda ◽  
Fabrizio Cuccu
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Symmetry Breaking ◽  
Arbitrary Function ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Composite Membranes ◽  
Radial Symmetry ◽  
Connected Set ◽  
Second Eigenvalue

Let Ω ⊂ ℝN be an open bounded connected set. We consider the eigenvalue problem –Δu = λρu in Ω with Dirichlet boundary condition, where ρ is an arbitrary function that assumes only two given values 0 < α < β and is subject to the constraint ∫Ωρ dx = αγ + β(|Ω| – γ) for a fixed 0 < γ < |Ω|. Cox and McLaughlin studied the optimization of the map ρ ⟼ λk(ρ), where λk is the kth eigenvalue. In this paper we focus our attention on the case when N ≥ 2, k = 2 and Ω is a ball. We show that, under suitable conditions on α, β and γ, the minimizers do not inherit radial symmetry.

scholarly journals Existence of radial solutions for a $p(x)$-Laplacian Dirichlet problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Maria Alessandra Ragusa ◽  
Abdolrahman Razani ◽  
Farzaneh Safari
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Variational Methods ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Radial Solution ◽  
Radial Solutions ◽  
Positive Radial Solution ◽  
Radial Functions

AbstractIn this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized $p(x)$ p ( x ) -Laplacian problem $$ -\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \vert u \vert ^{q(x)-2} u- b(x) \vert u \vert ^{r(x)-2} u $$ − Δ p ( x ) u + R ( x ) u p ( x ) − 2 u = a ( x ) | u | q ( x ) − 2 u − b ( x ) | u | r ( x ) − 2 u with Dirichlet boundary condition in the unit ball in $\mathbb{R}^{N}$ R N (for $N \geq 3$ N ≥ 3 ), where a, b, R are radial functions.

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.

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