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Register a new userThe CMO-Dirichlet problem for the Schrodinger equation in the upper half-space and characterizations of CMO
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Let $mathcal{L}$ be a Schrodinger operator of the form $mathcal{L}=-Delta+V$ acting on $L^2(mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse Holder class ${RH}_q$ for some $qgeq (n+1)/2$. Let ${CMO}_{mathcal{L}}(mathbb{R}^n)$ denote the function space of vanishing mean oscillation associated to $mathcal{L}$. In this article we will show that a function $f$ of ${ CMO}_{mathcal{L}}(mathbb{R}^n) $ is the trace of the solution to $mathbb{L}u=-u_{tt}+mathcal{L} u=0$, $u(x,0)=f(x)$, if and only if, $u$ satisfies a Carleson condition $$ sup_{B: { balls}}mathcal{C}_{u,B} :=sup_{B(x_B,r_B): { balls}} r_B^{-n}int_0^{r_B}int_{B(x_B, r_B)} big|t abla u(x,t)big|^2, frac{ dx, dt } {t} <infty, $$ and $$ lim _{a rightarrow 0}sup _{B: r_{B} leq a} ,mathcal{C}_{u,B} = lim _{a rightarrow infty}sup _{B: r_{B} geq a} ,mathcal{C}_{u,B} = lim _{a rightarrow infty}sup _{B: B subseteq left(B(0, a)right)^c} ,mathcal{C}_{u,B}=0. $$ This continues the lines of the previous characterizations by Duong, Yan and Zhang cite{DYZ} and Jiang and Li cite{JL} for the ${ BMO}_{mathcal{L}}$ spaces, which were founded by Fabes, Johnson and Neri cite{FJN} for the classical BMO space. For this purpose, we will prove two new characterizations of the ${ CMO}_{mathcal{L}}(mathbb{R}^n)$ space, in terms of mean oscillation and the theory of tent spaces, respectively.
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Let $(X,d,mu)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincar{e} inequality. Let $mathscr{L}=mathcal{L}+V$ be a Schrodinger operator on $X$, where $mathcal{L}$ is a non-negative operator generalized by a Dirichlet form, and $V$ is a non-negative Muckenhoupt weight that satisfies a reverse Holder condition $RH_q$ for some $qge (Q+1)/2$. We show that a solution to $(mathscr{L}-partial_t^2)u=0$ on $Xtimes mathbb{R}_+$ satisfies the Carleson condition, $$sup_{B(x_B,r_B)}frac{1}{mu(B(x_B,r_B))} int_{0}^{r_B} int_{B(x_B,r_B)} |t abla u(x,t)|^2 frac{mathrm{d}mumathrm{d} t}{t}<infty,$$ if and only if, $u$ can be represented as the Poisson integral of the Schrodinger operator $mathscr{L}$ with trace in the BMO space associated with $mathscr{L}$.
We derive asymptotic formulas for the solution of the derivative nonlinear Schrodinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.
The most challenging problem in the implementation of the so-called textit{unified transform} to the analysis of the nonlinear Schrodinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called textit{linearizable} boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large $t$, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large $t$ behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given $tau$-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular $tau$-periodic functions, which includes ${a exp(i omega t) , | , a>0, , omega geq a^2}$, then the large $t$ behavior of the Neumann value is given by a $tau$-periodic function which can be computed explicitly.
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $alpha$-stable processes, which help verify the solvability of the original HJB equation.
The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^alpha(mathbb{S}^{d-1}),$ with $alphain mathbb{R}$. We present two characterizations. The first one is written in terms of certain $L^2$-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For $alpha>0$ this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for $alpha<0$ it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.
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