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Genqian Liu

Publications and source records attributed to Genqian Liu.

32 records · Page 2Linked to original sources
arXiv2020

Determining the viscosity from the boundary information for incompressible fluid

For the Stokes equations in a compact connected Riemannian $n$-manifold $(Ω,g)$ with smooth boundary $\partial Ω$, we give an equivalent new system of elliptic equations with $(n+1)$ independent unknown functions on $Ω$. We show that the Dirichlet-to-Neumann map ${\tildeΛ}_{\tildeε, μ,g}$ associated with this new system is also equivalent to the original Dirichlet-to-Neumann map $Λ_{μ,g}$ associated with the Stokes equations. We explicitly give the full symbol expression for the ${\tildeΛ}_{\tildeε,μ,g}$ by a method of factorization, and prove that Dirichlet-to-Neumann map ${\tildeΛ}_{\tildeε,μ,g}$ (or equivalently, $Λ_{μ, g}$) uniquely determines viscosity $μ$ and all tangential and normal derivatives of $μ$ on $\partial Ω$. In particular, combining this result, Lai-Uhlmann-Wang's theorem and Heck-Li-Wang's theorem, we completely solve a long-standing open problem that asks whether one can determine the viscosity for the Stokes equations and for the Navier-Stokes equations by boundary measurements on an arbitrary bounded domain in ${\mathbb{R}}^n$, ($n=2,3$).

math.AP↗
arXiv2020

Determining anisotropic real-analytic metric from boundary electromagnetic information

For a compact, connected, oriented Riemannian $3$-manifold $(M, g)$ with smooth boundary $\partial M$, we explicitly give a local representation and a full symbol expression for the electromagnetic Dirichlet-to-Neumann map by factorizing Maxwell's equations and using an isometric transform. We prove that one can reconstruct a compact, connected, real-analytic Riemannian $3$-manifold $M$ with boundary from the set of tangential electric fields and tangential magnetic fields, given on a non-empty open subset $Γ$ of the boundary, of all electric and magnetic fields with tangential electric data supported in $Γ$. We note that for this result we need no assumption on the topology of the manifold other than compactness and connectedness, nor do we need a priori knowledge of all of $\partial M$. In addition, as a by-product of the explicit symbol expression of $Λ_{g,Γ}$, we show that for a given smooth Riemannian metric $g$, the electromagnetic Dirichlet-to-Neumann map $Λ_{g,Γ}$ uniquely determines all order tangential and normal derivatives of electromagnetic parameters $μ$ and $σ$ on $Γ$. Therefore, $μ$ and $σ$ are completely determined in $M$ by $Λ_{g,Γ}$ if these two parameter functions and metric $g$ are all real analytic in $M$ up to $Γ$.

math.AP↗
arXiv2015

The vibrational frequencies of the elastic body and its geometric quantities

For a bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the trace of the strongly continuous semigroup associated with the Navier-Lamé operator on $Ω$ as $t\to 0^+$. These coefficients (i.e., spectral invariants) provide precise information for the volume of the elastic body $Ω$ and the surface area of the boundary $\partial Ω$ in terms of the spectrum of the Navier-Lamé problem. As an application, we show that an $n$-dimensional ball is uniquely determined by its Navier-Lamé spectrum among all bounded elastic body with smooth boundary.

math.AP↗
arXiv2015

Pólya conjecture for the Neumann eigenvalues

For a given bounded domain $Ω\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} where $μ_k$ is the $k$-th Neumann eigenvalue of the Laplacian for $Ω$.

math.AP↗
arXiv2015

The generalised Pólya conjecture for the Dirichlet eigenvalues

In this paper, we prove the Generalized Pólya conjecture for the Dirichlet eigenvalues. In other words, we show that $λ_k(α) \ge \frac{(2π)^α k^{α/n}}{\big(ω_n \cdot {vol}(Ω)\big)^{α/n}}, \quad\, {for}\;\; k=1,2,3,....$ where $λ_k(α)$ is the $k$-th Dirichlet eigenvalue for the fractional Laplacian $(-Δ)^{α/2}$ with $α\in (0,2]$ in a bounded domain $Ω\subset {\Bbb R}^n$.

math.AP↗
arXiv2014

Heat invariants of the perturbed polyharmonic Steklov problem

For a given bounded domain $Ω$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we establish a procedure to get all the coefficients of the asymptotic expansion of the trace of the heat kernel associated with the perturbed polyharmonic Dirichlet-to-Neumann operator $Λ_m$ ($m\ge 1$) as $t\to 0^+$. We also explicitly calculate the first four coefficients of this asymptotic expansion. These coefficients (i.e., heat invariants) provide precise information for the area and curvatures of the boundary $\partial Ω$ in terms of the spectrum of the perturbed polyharmonic Steklov problem. In particular, when $m=1$ and $q\equiv 0$ our work recovers the previous corresponding results in \cite{PS} and \cite{Liu3}.

math.AP↗
arXiv2013

One can hear the area and curvature of boundary of a domain by hearing the Steklov eigenvalues

For a given bounded domain $Ω$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type upper-estimate of the heat kernel associated to the Dirichlet-to-Neumann operator, the Sobolev trace inequality, the Log-Sobolev trace inequality, the Nash trace inequality, and the Rozenblum-Lieb-Cwikel type inequality are all equivalent. Upon decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferntial operator and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as $t\to 0^+$. In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients give precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.

math.AP↗
arXiv2013

Sharp k-order Sobolev inequalities in the hyperbolic space ${\Bbb H}^n$

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$. In addition, we prove that the associated Sobolev constants are optimal.

math.AP↗
arXiv2012

Sharp Weyl-Type Formulas of the Spectral Functions for Biharmonic Steklov Eigenvalues

In this paper, by explicitly calculating the principal symbols of pseudodifferential operators and by applying Hömander's spectral function theorem, we obtain the Weyl-type asymptotic formulas with sharp remainder estimates for the counting functions of the two classes of biharmonic Steklov eigenvalues $λ_k$ and $μ_k$ in a smooth bounded domain of a Riemannian manifold. This solves a longstanding challenging problem.

math.AP↗
arXiv2012

A sharp asymptotic remainder estimate for biharmonic Steklov eigenvalues on Riemannian manifolds

Let $Ω$ be a bounded domain with $C^\infty$ boundary in an $n$-dimensional $C^\infty$ Riemannian manifold, and let $\varrho$ be a non-negative bounded function defined on $\partial Ω$. It is well-known that for the biharmonic equation $Δ^2 u=0$ in $Ω$ with the 0-Dirichlet boundary condition, there exists an infinite set $\{u_k\}$ of biharmonic functions in $Ω$ with positive eigenvalues $\{λ_k\}$ satisfying $Δu_k+ λ_k \varrho \frac{\partial u_k}{\partial ν}=0$ on the boundary $\partial Ω$. In this paper, we give the Weyl-type asymptotic formula with a sharp remainder estimate for the counting function of the biharmonic Steklov eigenvalues $λ_k$.

math.AP↗
arXiv2011

The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues with Dirichlet boundary condition on Riemannian manifolds

Let $Ω$ be a bounded domain with $C^2$-smooth boundary in an $n$-dimensional oriented Riemannian manifold. It is well-known that for the bi-harmonic equation $Δ^2 u=0$ in $Ω$ with the $0$-Dirichlet boundary condition, there exists an infinite set $\{u_k\}$ of biharmonic functions in $Ω$ with positive eigenvalues $\{λ_k\}$ satisfying $Δu_k+ λ_k \varrho \frac{\partial u_k}{\partial ν}=0$ on the boundary $\partial Ω$. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Stekloff eigenvalues $λ_k$.

math.AP↗
arXiv2009

Some inequalities and asymptotic formulas for eigenvalues on Riemannian manifolds

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case.

math.AP↗
arXiv2008

Matzoh ball soup in spaces of constant curvature

In this paper, we generalize Magnanini-Sakaguchi's result [MS3] from Euclidean space to spaces of constant curvature. More precisely, we show that if a conductor satisfying the exterior geodesic sphere condition in the space of constant curvature has initial temperature 0 and its boundary is kept at temperature 1 (at all times), if the thermal conductivity of the conductor is inverse of its metric, and if the conductor contains a proper sub-domain, satisfying the interior geodesic cone condition and having constant boundary temperature at each given time, then the conductor must be a geodesic ball. Moreover, we show similar result for the wave equations and the Schrödinger equations in spaces of constant curvature.

math.AP↗