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Register a new userOn the Dirichlet problem for the Schrodinger equation with boundary value in BMO space
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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 $mathscr{L}$.
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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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In this paper we consider the initial boundary value problem (IBVP) for the nonlinear biharmonic Schrodinger equation posed on a bounded interval $(0,L)$ with non-homogeneous Navier or Dirichlet boundary conditions, respectively. For Navier boundary IBVP, we set up its local well-posedness if the initial data lies in $H^s(0, L)$ with $sgeq 0$ and $s eq n+1/2, nin mathbb{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H_{loc}^{(s+3-j)/4}(mathbb {R}^+)$, for $j=0,2$. For Dirichlet boundary IBVP the corresponding local well-posedness is obtained when $s>10/7$ and $s eq n+1/2, nin mathbb{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H_{loc}^{(s+3-j)/4}(mathbb {R}^+)$, for $j=0,1$.
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