Related papers: Analysis of degenerate elliptic operators of Gru\v…

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider matrix elliptic second order differential operators $\mathcal{A}_{D,\varepsilon}$ and…

In this paper we study the decay estimates of the fourth order Schr\"{o}dinger operator $H=\Delta^{2}+V(x)$ on $\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the…

In this paper we prove strong unique continuation for the following degenerate elliptic equation \begin{equation}\label{e0} \Delta_zu +|z|^2\partial_t^2u = Vu,\quad (z,t) \in \mathbb{R}^N \times \mathbb{R} \end{equation} where the potential…

Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…

In this paper, the authors investigate non-homogeneous Hamiltonian operators composed of a first-order Dubrovin-Novikov operator and an ultralocal one. The study of such operators turns out to be fundamental for the inverted system of…

Let ${X_j},{Y_j}(j = 1, \cdot \cdot \cdot,n)$ be vector fields satisfying H\"{o}rmander's condition and ${\Delta_L} = \sum\limits_{j = 1}^n {(X_j^2 + Y_j^2)}$. In this paper, we establish some inequalities of Dirichlet eigenvalues for…

We put together a general framework to deal with elliptic and parabolic equations associated with (nonlinear) nonlocal (fractional order) operators. Many well-known nonlocal operators enter into our framework, and in addition one may…

This paper is about elliptic and parabolic partial differential operators with discontinuities in the gradient which are compatible with a Finsler norm in a sense to be made precise. Examples of this type of problems arise in a number of…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is an inverse square potential. In this paper we obtain sharp decay estimates of the operator norms of $e^{-tH}$ and $\nabla e^{-tH}$ in…

Let $h$ be a quadratic form with domain $W_0^{1,2}(\Ri^d)$ given by \[ h(\varphi)=\sum^d_{i,j=1}(\partial_i\varphi,c_{ij}\,\partial_j\varphi) \] where $c_{ij}=c_{ji}$ are real-valued, locally bounded, measurable functions and…

Let $\Delta$ be the Laplace--Beltrami operator acting on a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$ with $m > n \ge 3$. Let $\frak{h}_t(x,y)$ be the kernels of the semigroup $e^{-t\Delta}$ generated by $\Delta$.…

This paper is mainly devoted to study time decay estimates of the higher-order Schr\"{o}dinger type operator $H=(-\Delta)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive…

Let $L = -{\rm div}( A(x) \cdot \nabla ) + V(x)$ be a second-order uniformly elliptic operator on $\mathbb{ R }^{n}$ $(n\geq 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative…

For a sequence of uniformly bounded, degenerate semigroups on a Hilbert space, we compare various types of convergences to a limit semigroup. Among others, we show that convergence of the semigroups, or of the resolvents of the generators,…

In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in H\"{o}lder spaces. Our context is that of open sets $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, satisfying…

We consider the degenerate elliptic operator acting on $C^2$ functions on $[0,\infty)^d$: \[ L f(x)=\sum_{i=1}^d a_i(x) x_i^{\alpha_i} \frac{\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x), \] where…

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order $X^{s-1,q}_D(\Omega)$ for $s > 0$ small, including…

Let $n,m\ge 1$, $\alpha\in(0,1)$, and $\beta\ge 0$. For the Grushin-type operator \[ L=-\nabla_x\!\cdot\!\bigl(|x|^{2\alpha}\nabla_x\bigr)+|x|^{2\beta}\Delta_y \qquad \text{on } \mathbb R^n\times \mathbb R^m, \] we prove the isoperimetric…

We investigate second order elliptic equations \[F(\mathcal{H}u) = 0\] where the function $F\colon S(n)\to\mathbb{R}$ on the space of symmetric $n\times n$ matrices is assumed to be sublinear. There is very little to be found in the…

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/ \varepsilon$, $\varepsilon>0$. We study the…