Rohit Kumar Mishra, Manmohan Vashisth
We study the inverse problem for determining the time-dependent matrix potential appearing in the wave equation. We prove the unique determination of potential from the knowledge of solution measured on a part of the boundary.
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We consider the inverse problem of determining, the possibly anisotropic, conductivity of a body by means of the so called local Neumann to Dirichlet map on a curved portion $Σ$ of the boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities proved in \cite{Al-dH-G}, we provide a Hölder stability estimate on $Σ$ when the conductivity is a priori known to be a constant matrix near $Σ$.
In [2] we introduced a method combining together an observability inequality and a spectral decomposition to get a logarithmic stability estimate for the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. The present work deals with an adaptation of that method to obtain a logarithmic stability estimate for the inverse problem of determining a boundary damping coefficient from boundary measurements. As in our preceding work, the different boundary measurements are generated by varying one of the initial condi…
We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove H{ö}lder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.
We prove uniqueness and stability for an inverse boundary problem associated to an anisotropic elliptic equation arising in the modeling of prestressed elastic membranes.
Stability inequality for the problem of determining an unbounded potential from boundary measurements
本文在3维空间中，针对势属于L^s（s∈(2,3)）的情况，建立了由边界测量确定薛定谔方程中势的稳定性不等式。
Stability of determining the potential from partial boundary data in a Schrödinger equation in the high frequency limit
本文建立了高频极限下由部分边界数据确定薛定谔方程势的稳定性不等式，该结果在势于边界附近已知时成立。
In this paper we show, in dimension n >=3, that knowledge of the Cauchy data for the Schroedinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential.
We extend results of Dos Santos Ferreira-Kenig-Sjoestrand-Uhlmann (math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schroedinger operator determine uniquely the magnetic field related to a Hoelder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.
We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that the uniqueness holds at the origin provided that the initial condition is appropriately chosen. We show that the choice of the initial condition leading to uniqueness is related to a fine version of unique continuation property for elliptic operators. We also establish a Lipschitz directional stability estimate at the origin, which is obtained by a linearization process.
In this article, for the radiative transport equation, we study inverse problems of determining a time independent scattering coefficient or total attenuation by boundary data on the complementary sub-boundary after making one time input of a pair of a positive initial value and boundary data on a suitable sub-boundary. The main results are Lipschitz stability estimates. We can also prove the reverse inequality, which means that our estimates for the inverse problems are the best possible. The proof is based on a Carleman estimate with a linear weight function.
We consider the following perturbed polyharmonic operator $\Lc(x,D)$ of order $2m$ defined in a bounded domain $Ω\subset \mathbb{R}^n, n\geq 3$ with smooth boundary, as \begin{equation*} \Lc(x,D) \equiv (-Δ)^m + \sum_{j,k=1}^{n}A_{jk} D_{j}D_{k} + \sum_{j=1}^{n}B_{j} D_{j} + q(x), \end{equation*} where $A$ is a symmetric $2$-tensor field, $B$ and $q$ are vector field and scalar potential respectively. We show that the coefficients $A=[A_{jk}]$, $B=(B_j)$ and $q$ can be recovered from the associated Dirichlet-to-Neumann data on the boundary. Note that, this result shows an example of determinin…
We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding "linearized" Dirichlet-to-Neumann map. The key ingredient in our analysis is a parabolic version of an elliptic Carlema…
We study the inverse problem of determining both the source of a wave and its speed inside a medium from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and thermoacoustic tomography, and has important applications in medical imaging. We prove that if the solutions of the wave equation with the source and sound speed $(f_1,c_1)$ and $(f_2,c_2)$ agree on the boundary of a bounded region $Ω$, then \[ \int_Ω(c_2^{-2}-c_1^{-2})φdy=0,\] for every harmonic function $φ\in C(\barΩ)$, which holds without any knowledge of the source. We also show t…
In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $α\in(0,1)$. Our survey covers the following types of inverse problems: 1. determination of time-dependent functions in interior source terms 2. determination of space-dependent functions in interior source terms 3. determination of time-dependent functions appearing in boundary conditions
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model, for example, the orders of the fractional derivative or the source term, are often unknown, which requires one to discuss inverse problems to identify these physical quantities from some additional information that can be observed or measured practically. This chapter investigates several kinds of inverse coefficient problems for the fractional diffusion equation.
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, which requires one to discuss inverse problems of identifying these physical quantities from some indirectly observed information of solutions. Inverse problems in determining these unknown parameters of the model are not only theoretically interesting, but also necessary for finding solutions to initial-boundary value problems and studying pro…
In this paper, we consider an inverse conductivity problem on a bounded domain $Ω\subset\mathbb{R}^n$, $n\geq2$, also known as Electrical Impedance Tomography (EIT), for the case where unknown impenetrable obstacles are embedded into $Ω$. We show that a piecewise-constant conductivity function and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary of the domain $Ω$. The method depends on the well-posedness of a coupled PDE-system constructed for the conductivity equations in the $H^1$-space…
Stability of determining a Dirichlet-Laplace-Beltrami operator from its boundary spectral data
本文针对由边界谱数据确定Dirichlet-Laplace-Beltrami算子的反问题，在完整与部分谱数据两种情形下建立了稳定性不等式。
Inverse boundary value problems of determining nonlinear coefficients for the JMGT equation
本文研究了JMGT方程在非线性声学中的反边界值问题，证明了在特定条件下，西斯特尔型模型中的非线性系数β和库兹内茨型模型中的(β,κ)可被唯一确定。
We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $\partial_t^α u(x,t) = -Au(x,t)$, where $-A = \sum}{i,j=1}^d \partial_i(a_{ij}(x)\partial_j) + \sum{j=1}^d b_j(x)\partial_j + c(x)$. We establish the uniqueness for an inverse problem of determining an order $α$ of fractional derivatives by data $u(x_0,t)$ for $0<t<T$ at one point $x_0$ in a spatial domain $\OOO$. The uniqueness holds even under assumption that $\OOO$ and $A$ are unknown, provided that the initial value does not change signs and is not identically…
Determining time-dependent convection and density terms in convection-diffusion equation using partial data
该研究针对时变对流扩散方程的逆边值问题，提出利用非线性Carleman权函数的方法，仅需在边界极小开子集上测量Dirichlet-to-Neumann映射，即可在自然规范下唯一恢复时变对流项与密度系数，证明了该逆问题的唯一性。
We are concerned with the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. We establish stability estimates of logarithmic type when the measurements are given by the operator who maps the initial condition to Neumann boundary trace of the solution of the corresponding initial-boundary value problem. We build a method combining an observability inequality together with a spectral decomposition. We also apply this method to a clamped Euler-Bernoulli beam equation. Finally, we indicate how the present approac…
For an initial-boundary value problem for a parabolic equation in the spatial variable $x=(x_1,.., x_n)$ and time $t$, we consider an inverse problem of determining a coefficient which is independent of one spatial component $x_n$ by extra lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also we prove similar results for the corresponding inverse source problem.
Uniqueness and stability in determining the wave equation from a single passive boundary measurement
针对多维波动方程，提出利用单次被动边界测量同时恢复波速与未知初始源，并证明唯一性和 Hölder 稳定性。
Determining a parabolic system by boundary observation of its non-negative solutions with biological applications
本文通过边界观测非负解，利用高阶变分方案创新逐次线性化方法，确定耦合非线性抛物型系统系数，并应用于生态种群模型反问题。
Conditional stability in determining source terms of semilinear parabolic partial differential equations
本文提出了一种非线性反源问题的条件Hölder稳定性方法，结合广义微分算子线性化和Carleman估计，用于确定半线性抛物型方程的源项。
We consider a second-order hyperbolic equation on an open bounded domain $Ω$ in $\mathbb{R}^n$ for $n\geq2$, with $C^2$-boundary $Γ=\paΩ=\bar{Γ_0\cupΓ_1}$, $Γ_0\capΓ_1=\emptyset$, subject to non-homogeneous Neumann boundary conditions on the entire boundary $Γ$. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit sub-portion $Γ_1$ of the boundary $Γ$, and over a computable time interval $T>0$. Under sharp conditions on the complementary pa…
We prove a logarithmic stability estimate for the inverse problem of determining the potential in a wave equation from boundary measurements obtained by varying the first component of the initial condition. The novelty of the present work is that no geometric condition is imposed to the sub-boundary where the measurements are made. Our results improve those obtained by the first and second authors in [2]. We also show how the analysis for the wave equation can be adapted to an inverse coefficient problem for the heat equation
We are concerned with the Calderón problem of determining an unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities are piecewise constants supported in a nested polygonal geometry. Our methods in establishing the stability and uniqueness results have a significant technical initiative and a strong potent…
Backward Behavior and Determining Functionals for Chevron Pattern Equations
该研究针对齐次Dirichlet边界条件下的人字形图案方程初边值问题，证明了解的渐近行为由有限个泛函的动力学决定，并给出了一维情形下后向问题解有限时间爆破的数值证据。
In this paper we study inverse boundary value problems with partial data for the bi-harmonic operator with first order perturbation. We consider two types of subsets of $\mathbb{R}^{n}(n\geq 3)$, one is an infinite slab, the other is a bounded domain. In the case of a slab, we show that, from Dirichlet and Neumann data given either on the different boundary hyperplanes of the slab or on the same boundary hyperplane, one can uniquely determine the magnetic potential and the electric potential. In the case of a bounded domain, we show the unique determination of the magnetic potential and the el…
We investigate inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. We establish in several generic scenarios that one can uniquely determine the nonlinearity or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and a Runge approximation with higher order linearization for the semilinear hy…
We consider an operator $Δ^2 + A(x)\cdot D+q(x)$ with the Navier boundary conditions on a bounded domain in $R^n$, $n\ge 3$. We show that a first order perturbation $A(x)\cdot D+q$ can be determined uniquely by measuring the Dirichlet--to--Neumann map on possibly very small subsets of the boundary of the domain. Notice that the corresponding result does not hold in general for a first order perturbation of the Laplacian.
Boundary partial regularity for a class of Dirac-harmonic maps
本文研究一类满足特定能量单调不等式的耦合 Dirac-harmonic 映射，证明其在边界附近具有部分正则性。
In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.
Determining evolutionary equations from a single passive boundary observation
针对演化型偏微分方程，提出基于积分恒等式、调和/微局部分析及低高频渐近的统一框架，利用单个被动边界观测同时恢复未知源和系数，克服信息稀缺与未知耦合难题。
We deal with the problem of determining an inclusion within an electrical conductor from electrical boundary measurements. Under mild a priori assumptions we establish an optimal stability estimate.
Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary
本文证明了在域度量沿自由边界以$d^α$速率退化或奇异时，具有自由边界数据的弱平稳调和映照的部分正则性。
We show that the knowledge of the set of the Cauchy data on the boundary of a $C^1$ bounded open set in $\R^n$, $n\ge 3$, for the Schrödinger operator with continuous magnetic and bounded electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives.
We consider two inverse boundary value problems for the time-harmonic Maxwell equations in an infinite slab. Assuming that tangential boundary data for the electric and magnetic fields at a fixed frequency is available either on subsets of one boundary hyperplane, or on subsets of different boundary hyperplanes, we show that the electromagnetic material parameters, the conductivity, electric permittivity, and magnetic permeability, are uniquely determined by these partial measurements.
We improve the preceding results obtained by the first and the second authors in [3]. They concern the stability issue of the inverse problem that consists in determining the potential and the damping coefficient in a wave equation from an initial-to-boundary operator. We partially modify the arguments in [3] to show that actually we have H{ö}lder stability instead of logarithmic stability.
The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, $n\in\bbN$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$, where $$ Lu = - \sum_{j,k=1}^n\partial_j a_{j,k}\partial_k u. $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ ν\cdot A\nabla u + Θ\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partialΩ, $$ where $Θ$ represents an appropriate operator actin…
We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-Δ_x u+q(t,x)u=0$ in $Q=(0,T)\timesΩ$ with $T>0$ and $Ω$ a $ \mathcal C^2$ bounded domain of $\mathbb R^n$, $n\geq2$. We start by considering the unique determination of some singular time-dependent coefficients from observations on $\partial Q$. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first cl…
Restoring the Navier--Stokes dynamics by determining functionals depending on pressure
本文针对有界光滑区域中的二维Navier--Stokes方程，构造了一个由N个仅依赖压力的线性连续泛函和一个固定边界点处的涡度值泛函组成的确定泛函系统，以恢复其动力学。
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining nonclassical symmetry of a PDE and make it easier to solve the system of nonlinear determining equations. As example, new nonclassical symmetries are exhibited for a class of generalized Burgers equation and KdV-type equations are given.
We characterize the behavior of the solutions of linear evolution partial differential equations on the half line in the presence of discontinuous initial conditions or discontinuous boundary conditions, as well as the behavior of the solutions in the presence of corner singularities. The characterization focuses on an expansion in terms of computable special functions.
In this note we consider boundary point principles for partial differential inequalities of elliptic type. Firstly, we highlight the difference between conditions required to establish classical strong maximum principles and classical boundary point lemmas for second order linear elliptic partial differential inequalities. We highlight this difference by introducing a singular set in the domain where the coefficients of the partial differential inequality need not be defined, and in a neighborhood of which, can blow-up. Secondly, as a consequence, we establish a comparison-type boundary point …
We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map in the fractional linear magnetic Calderón problem by using an integral identity. We also determine both the magnetic potential and the nonlinearity in the fractional semilinear magnetic Calderón problem by using a first order linearization.
We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator $\mathcal L_{g,X,q}$, with $X$ being a $W^{1,\infty}$ vector field and $q$ being an $L^\infty$ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann determines $X$ and $q$ uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace-Beltrami operator due to Dos Santos Ferr…
In dimension $n\geq 3$, we prove a local uniqueness result for the potentials $q$ of the Schrödinger equation $-Δu+qu=0$ from partial boundary data. More precisely, we show that potentials $q_1,q_2\in L^\infty$ with positive essential infima can be distinguished by local boundary data if there is a neighborhood of a boundary part where $q_1\geq q_2$ and $q_1\not\equiv q_2$.
We are concerned with the solvability of linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance, in which the nonlinear boundary conditions perturbation is not necessarily required to satisfy Landesman-Lazer conditions or the monotonicity assumption. The nonlinearity may be unbounded. The nonlinearity interact, in some sense with the Steklov spectrum on boundary nonlinearity. The proofs are based on a priori estimates for possible solutions to a homotopy on suitable trace and topological degree arguments.
This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation $c(x)\partial_{t}^{2}u - Δu = 0$ in a bounded smooth domain in $\R^{d}$ from partial (on part of the boundary) dynamic boundary measurements. In this paper we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset $Γ$ of the boundary determines explicitly the coefficient $c$ provided that $c$ is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula…
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 a Plateau problem in codimension $1$ in the non-parametric setting. A Dirichlet boundary datum is given only on part of the boundary $\partial Ω$ of a bounded convex domain $Ω\subset\mathbb{R}^2$. Where the Dirichlet datum is not prescribed, we allow a free contact with the horizontal plane. We show existence of a solution, and prove regularity for the corresponding minimal surface. Finally we compare these solutions with the classical minimal surfaces of Meeks and Yau, and show that they are equivalent when the Dirichlet boundary datum is assigned in at most $2$ disjoint arcs of $…
We show that the elliptic equation with a non-Lipschitz right-hand side, $-Δu = λ|u|^{β-1}u - |u|^{α-1}u$ with $λ>0$ and $0<α<β<1$, considered on a smooth star-shaped domain $Ω$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset $Γ$ of the boundary $\partialΩ$ such that $Γ\neq \partialΩ$.
Determining nonlinear balance laws in product-type domains by a single local passive boundary observation
本文提出一种算子理论范式，通过单一局部被动边界观测，在乘积型区域中唯一确定非线性平衡律相关通量与源算子的实现，揭示了宏观边界数据编码微观动力学信息的全息型原理。
We consider the problem of determining an unaccessible part of the boundary of a conductor by mean of thermal measurements. We study a problem of corrosion where a Robin type condition is prescribed on the damaged part and we prove logarithmic stability estimate.
We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schrödinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.
An inverse boundary value problem for the Maxwell's equations with partial data
本文研究动态麦克斯韦方程组的逆边界问题，在存在严格凸叶状结构的条件下，证明了介电常数、电导率和磁导率的局部唯一确定性。
We describe a simple way of constructing exponentially growing solutions of the second order systems with the Laplacian as the principal term.
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