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Register a new userOn a Curvature Flow in a Band Domain with Unbounded Boundary Slopes
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We consider an anisotropic curvature flow $V= A(mathbf{n})H + B(mathbf{n})$ in a band domain $Omega :=[-1,1]times R$, where $mathbf{n}$, $V$ and $H$ denote the unit normal vector, normal velocity and curvature, respectively, of a graphic curve $Gamma_t$. We consider the case when $A>0>B$ and the curve $Gamma_t$ contacts $partial_pm Omega$ with slopes equaling to $pm 1$ times of its height (which are unbounded when the solution moves to infinity). First, we present the global well-posedness and then, under some symmetric assumptions on $A$ and $B$, we show the uniform interior gradient estimates for the solution. Based on these estimates, we prove that $Gamma_t$ converges as $tto infty$ in $C^{2,1}_{text{loc}} ((-1,1)times R)$ topology to a cup-like traveling wave with {it infinite} derivatives on the boundaries.
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We consider a curvature flow $V=H$ in the band domain $Omega :=[-1,1]times R$, where, for a graphic curve $Gamma_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $Gamma_t$ contacts the two boundaries $partial_pm Omega$ of $Omega$ with constant slopes, in 1993, Altschular and Wu cite{AW1} proved that $Gamma_t$ converges to a {it grim reaper} contacting $partial_pm Omega$ with the same prescribed slopes. In this paper we consider the case where $Gamma_t$ contacts $partial_pm Omega$ with slopes equaling to $pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C^{2,1}_{loc} ((-1,1)times R)$ topology to the {it grim reaper} with span $(-1,1)$.
We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $mgeq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.
In this article we prove a reducibility result for the linear Schrodinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than $1/2$. As far as we know, this is the first reducibility results for an unbounded perturbation of a linear system which is not integrable.
We consider a class of non-trivial perturbations ${mathscr A}$ of the degenerate Ornstein-Uhlenbeck operator in ${mathbb R}^N$. In fact we perturb both the diffusion and the drift part of the operator (say $Q$ and $B$) allowing the diffusion part to be unbounded in ${mathbb R}^N$. Assuming that the kernel of the matrix $Q(x)$ is invariant with respect to $xin {mathbb R}^N$ and the Kalman rank condition is satisfied at any $xin{mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernsteins method we prove that we can associate a semigroup ${T(t)}$ of bounded operators (in the space of bounded and continuous functions) with the operator ${mathscr A}$. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup ${T(t)}$ both in isotropic and anisotropic spaces of (Holder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${mathscr A}$.
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition.
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