arXiv subjects
Publications and source records attributed to Jenn-Nan Wang.
In this work we consider stability of recovery of the conductivity and attenuation coefficients of the stationary Maxwell and Schrödinger equations from a complete set of (Cauchy) boundary data. By using complex geometrical optics solutions we derive some bounds which can be viewed as an evidence of increasing stability in these inverse problems when frequency is growing.
In this paper we derive quantitative uniqueness estimates at infinity for solutions to an elliptic equation with unbounded drift in the plane. More precisely, let $u$ be a real solution to $Δu+W\cdot\nabla u=0$ in ${\mathbf R}^2$, where $W$ is real vector and $\|W\|_{L^p({\mathbf R}^2)}\le K$ for $2\le p<\infty$. Assume that $\|u\|_{L^{\infty}({\mathbf R}^2)}\le C_0$ and satisfies certain a priori assumption at $0$. Then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge \exp(-C_1R^{1-2/p}\log R)\quad\text{if}\quad 2 0$ depends on $p, K, C_0$, while $C_2>0$ depends on $K, C_0$ . Using the scaling argument in [BK05], these quantitative estimates are easy consequences of estimates of the maximal vanishing order for solutions of the local problem. The estimate of the maximal vanishing order is a quantitative form of the strong unique continuation property.
In this paper we prove a quantitative form of Landis' conjecture in the plane. Precisely, let $W(z)$ be a measurable real vector-valued function and $V(z)\ge 0$ be a real measurable scalar function, satisfying $\|W\|_{L^{\infty}({\mathbf R}^2)}\le 1$ and $\|V\|_{L^{\infty}({\mathbf R}^2)}\le 1$. Let $u$ be a real solution of $Δu-\nabla(Wu)-Vu=0$ in ${\mathbf R}^2$. Assume that $u(0)=1$ and $|u(z)|\le\exp(C_0|z|)$. Then $u$ satisfies $\underset{|z_0|=R}{\inf}\,\underset{|z-z_0|<1}{\sup}|u(z)|\ge \exp(-CR\log R)$, where $C$ depends on $C_0$. In addition to the case of the whole plane, we also establish a quantitative form of Landis' conjecture defined in an exterior domain.
In this paper, we prove in two dimensions global identifiability of the viscosity in an incompressible fluid by making boundary measurements. The main contribution of this work is to use more natural boundary measurements, the Cauchy forces, than the Dirichlet-to-Neumann map previously considered in \cite{IY} to prove the uniqueness of the viscosity for the Stokes equations and for the Navier-Stokes equations.
We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovering the potential coefficient in the Schrödinger equation from the Dirichlet-to-Neumann map in the presence of attenuation, when energy level/frequency is growing. These bounds hold under certain a-priori regularity constraints on the unknown coefficient. Proofs use complex and bounded complex geometrical optics solutions.
In this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials. We derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions. The lower bounds depend on asymptotic behaviors of magnetic and electric potentials. The proof is carried out by the Carleman method and the bootstrapping arguments.
In this paper we study the inverse problem of determining the residual stress in Man's model using tomographic data. Theoretically, the tomographic data is obtained at zero approximation of geometrical optics for Man's residual stress model. For compressional waves, the inverse problem is equivalent to the problem of inverting the longitudinal ray transform of a symmetric tensor field. For shear waves, the inverse problem, after the linearization, leads to another integral geometry operator which is called the mixed ray transform. Under some restrictions on coefficients, we are able to prove the uniqueness results in these two cases.
We study the size estimate problem for the two-phase shallow shell equations in this paper. Our aim is to derive bounds on the volume fraction of each phase assuming that the material properties of the two phases are given. The approach in this paper is based on the translation method. One of the key steps is to connect the shallow shell equations to the thin plate equation.
In this paper, we prove that the inverse problems for 2D elasticity and for the thin plate with boundary data (finite or full measurements) are equivalent. Having proved this equivalence, we can solve inverse problems for the plate equation with boundary data by solving the corresponding inverse problems for 2D elasticity, and vice versa. For example, we can derive bounds on the volume fraction of the two-phase thin plate from the knowledge of one pair of boundary measurements using the known result for 2D elasticity \cite{ml}. Similarly, we give another approach to the size estimate problem for the thin plate studied by Morassi, Rosset, and Vessella \cite{mrv07, mrv09}.
In this work we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz estimate for large frequencies.
We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The problem is modeled by the stationary Navier-Stokes equations in an exterior domain in $\R^n$ with $n\ge 2$. We will show that, under some assumptions, any nontrivial velocity field obeys a minimal decaying rate $\exp(-Ct^2\log t)$ at infinity. Our proof is based on appropriate Carleman estimates.
In this paper we study the local behavior of a solution to the Lamé system with \emph{Lipschitz} coefficients in dimension $n\ge 2$. Our main result is the bound on the vanishing order of a nontrivial solution, which immediately implies the strong unique continuation property. This paper solves the open problem of the strong uniqueness continuation property for the Lamé system with Lipschitz coefficients in any dimension.
In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation property. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial \emph{optimal} three-ball inequalities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequalities.
In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.
In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.
In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem in which log-log-type estimates were derived.
We consider the inverse conductivity problem of identifying embedded objects in unbounded domains. The main tool is a set of special solutions to the Schroedinger equation, the complex spherical waves, which are constructed by a Carleman estimate. Using these solutions we can treat the inverse problem in the unbounded domain like in the bounded case, and it is possible to localize the boundary measurements to a finite part of the boundary.