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Discovery Projects - Grant ID: DP140100649

Title
Discovery Projects - Grant ID: DP140100649
Funding
ARC | Discovery Projects
Contract (GA) number
DP140100649
Start Date
2014/01/01
End Date
2016/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP140100649

 

  • Weighted estimates for powers and Smoothing estimates of Schr\"odinger operators with inverse-square potentials

    Bui, The Anh; D'Ancona, Piero; Duong, Xuan Thinh; Li, Ji; Ly, Fu Ken (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP140100649 (DP140100649)
    Let $\mathcal{L}_a$ be a Schr\"odinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.

    Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data

    Bui, The Anh; Duong, Xuan Thinh (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP140100649 (DP140100649)
    Consider the following nonlinear elliptic equation of $p(x)$-Laplacian type with nonstandard growth \begin{equation*} \left\{ \begin{aligned} &{\rm div} a(Du, x)=\mu \quad &\text{in}& \quad \Omega, &u=0 \quad &\text{on}& \quad \partial\Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is a Reifenberg domain in $\mathbb{R}^n$, $\mu$ is a Radon measure defined on $\Omega$ with finite total mass and the nonlinearity $a: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n$ is modeled upon t...

    Global Marcinkiewicz estimates for nonlinear parabolic equations with nonsmooth coefficients

    Bui, The Anh; Duong, Xuan Thinh (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP140100649 (DP140100649)
    Consider the parabolic equation with measure data \begin{equation*} \left\{ \begin{aligned} &u_t-{\rm div} \mathbf{a}(D u,x,t)=\mu&\text{in}& \quad \Omega_T, &u=0 \quad &\text{on}& \quad \partial_p\Omega_T, \end{aligned}\right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $\Omega_T=\Omega\times (0,T)$, $\partial_p\Omega_T=(\partial\Omega\times (0,T))\cup (\Omega\times\{0\})$, and $\mu$ is a signed Borel measure with finite total mass. Assume that the nonlinearity ${\b...

    Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

    Bui, The Anh; Duong, Xuan Thinh (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP140100649 (DP140100649)
    Consider the nonlinear parabolic equation in the form $$ u_t-{\rm div} \mathbf{a}(D u,x,t)={\rm div}\,(|F|^{p-2}F) \quad \text{in} \quad \Omega\times(0,T), $$ where $T>0$ and $\Omega$ is a Reifenberg domain. We suppose that the nonlinearity $\mathbf{a}(\xi,x,t)$ has a small BMO norm with respect to $x$ and is merely measurable and bounded with respect to the time variable $t$. In this paper, we prove the global Calder\'on-Zygmund estimates for the weak solution to this parabolic problem in th...
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