E. Francini, S. Vessella, J. -N. Wang
In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result in [BL] and the arguments in [DcFLVW], we present an elementary method to derive the Carleman estimate under the optimal regularity assumption on the coefficients.
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We consider a fractional radiative transport equation, where the time derivative is of half order in the Caputo sense. By establishing Carleman estimates, we prove the global Lipschitz stability in determining the coefficients of the one-dimensional time-fractional radiative transport equation of half-order.
In this paper we prove a local Carleman estimate for second order elliptic equations with a general anisotropic Lipschitz coefficients having a jump at an interface. Our approach does not rely on the techniques of microlocal analysis. We make use of the elementary method so that we are able to impose almost optimal assumptions on the coefficients and, consequently, the interface.
We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and L…
In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x)$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.
We consider the Kelvin-Voigt model for the viscoelasticity, and prove a Carleman estimate for functions without compact supports. Then we apply the Carleman estimate to prove the Lipschitz stability in determining a spatial varying function in an external source term of Kelvin-Voigt model by a single measurement. Finally as a related system, we consider an isothermal compressible fluid system and apply the Carleman estimate to establish the Lipschitz stability for an inverse source problem for the compressible fluid system.
By using some deep tools from microlocal analysis, J. Le Rousseau and L. Robbiano (Invent. Math., 183 (2011), 245--336) established several Carleman estimates for parabolic operators with isotropic diffusion coefficients which have jumps at interfaces. In this paper, we revisit the same problem but for the general case of anisotropic diffusion coefficients. Our main tools are a pointwise estimate for parabolic operators and a suitable chosen weight function.
In this paper we provide a simple proof of a Carleman estimate for a second order elliptic operator $P$ with Lipschitz leading coefficients. We apply such a Carleman estimate to derive a three sphere inequality for solutions to equation $Pu=0$
An overdetermination is introduced in an initial condition for the second order mean field games system (MFGS). This makes the resulting problem close to the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem and overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. The second estimate the author calls "quasi-Carleman estimate", since it contains two test functions rather than a single one in conventional Carleman estimates. These two estimates play the…
Carleman estimate on a tree for Schrodinger equation and application to an inverse problem
该研究构造权函数建立了树上薛定谔方程的全局Carleman估计，并将其应用于推导系数反问题的Lipschitz稳定性。
We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schrödinger equation can be Lipschitz stably retrieved through finitely many local boundary measurements of the solution. The proof is by means of a specific global Carleman estimate for the Schrödinger equation, established in the first part of the paper.
We consider inverse problems of determining coefficients or time independent factors of source terms in radiative transport equations by means of Carleman estimate. We establish global Lipschitz stability results with an additional condition which requires some strict positivity for initial value or given factor of source, but we need not any extra conditions on domains of velocities, which is the main achievement of this article compared with the existing work by Machida and Yamamoto ({\it Inverse Problems} {\bf 30} 035010, 2014). The proof relies on a Carleman estimate with a piecewise linea…
Carleman estimate for telegrapher's equations on a network and application to an inverse problem
该研究针对树形网络上的电报方程反系数问题，通过建立Carleman估计，在部分边界额外数据的条件下推导出Lipschitz稳定性，为相关反问题的稳定性分析提供了方法支撑。
We consider a half-order time-fractional diffusion equation in an arbitrary dimension and investigate inverse problems of determining the source term or the diffusion coefficient from spatial data at an arbitrarily fixed time under some additional assumptions. We establish the stability estimate of Lipschitz type in the inverse problems and the proofs are based on the Bukhgeim-Klibanov method by using Carleman estimates.
In this paper we prove a Hölder propagation of smallness for solutions to second order parabolic equations whose general anisotropic leading coefficient has a jump at an interface. We assume that the leading coefficient is Lipschitz continuous with respect to the parabolic distance on both sides of the interface. The main effort consists in proving a local Carleman estimate for this parabolic operator.
We consider inverse problems for the first and half order time fractional equation. We establish the stability estimates of Lipschitz type in inverse source and inverse coefficient problems by means of the Carleman estimates.
Lipschitz Stability Estimates for Master Fields with Measure-Variable Regularity in Mean Field Games
本文针对平均场博弈主方程，通过推导非局部抛物型方程并建立Carleman估计，获得了主场及其测度导数沿特征测度流的终端到内部Lipschitz稳定性估计。
We consider a transmission wave equation in two embedded domains in $R^2$, where the speed is $a1 > 0$ in the inner domain and $a2 > 0$ in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and $a1 > a2$ . As a consequence of this inequality, uniqueness and Lip- schitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement.
We consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain, from one Neumann boundary observation of the solution. Assuming that this potential is known outside some fixed compact subset of the waveguide, we prove that it may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schrödinger operator acting in an unbounded cylindrical domain, the Neumann data is measure…
Lipschitz stability and reconstruction in inverse problems for semi-discrete parabolic operators
针对半离散抛物方程的反问题，本文利用受网格尺寸约束的新颖半离散Carleman估计，建立了恢复势函数的稳定性估计并提出了收敛的重构算法。
In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work. We then apply the three-region inequality to study the size estimate problem with one boundary measurement.
In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [FVW].
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois-Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally …
Global Lipschitz stability for an inverse coefficient problem for a mean field game system
本文研究平均场博弈系统Hamiltonian中状态变化因子的逆系数问题，利用空间数据与任意子域内部数据，基于不同范数的Carleman估计证明了全局Lipschitz稳定性。
In this paper we establish a global Carleman estimate for the fourth order Schrödinger equation posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from boundary measurements.
We consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time $θ$ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-deg…
We consider an inverse problem of reconstructing two spatially varying coefficients in an acoustic equation of hyperbolic type using interior data of solutions with suitable choices of initial condition. Using a Carleman estimate, we prove Lipschitz stability estimates which ensures unique reconstruction of both coefficients. Our theoretical results are justified by numerical studies on the reconstruction of two unknown coefficients using noisy backscattered data.
In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.
We consider an anisotropic hyperbolic equation with memory term: $$ \partial_t^2 u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_ju) + \int^t_0 \sum_{| α| \le 2} b_α(x,t,η)\partial_x^αu(x,η) dη+ F(x,t) $$ for $x \in Ω$ and $t\in (0,T)$ or $\in (-T,T)$, which is a model equation for viscoelasticity. First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary $Γ\times (-T,T)$. Second we apply the Carleman estimate to establish a both-sided estimate of $| u(\cdot,0)|_{H^3(Ω)}$ by $\partial_νu|_{Γ\times (0,T)}$ under the assump…
We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator.
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.
This paper concerns the inverse problem of retrieving a stationary potential for the Schrödinger evolution equation in a bounded domain of RN with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the in…
Carleman estimates for backward Cahn-Hilliard-reaction-diffusion problems
针对耦合Cahn-Hilliard-反应-扩散系统的向后逆问题，通过建立四阶/二阶抛物系统的Carleman估计，推导出从单一最终时间观测重构过去状态的Hölder稳定性和对数稳定性，并应用于肿瘤生长模型获得早期状态的向后唯一性和定量稳定性。
Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications
本文为一维随机抛物方程的空间半离散格式建立齐次与非齐次边界下的Carleman估计，并分别用于证明离散反随机源问题的Lipschitz稳定性和离散Cauchy问题的Hölder稳定性。
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch du…
Carleman estimate with piecewise weight and applications to inverse problems for first-order transport equations
本文通过按区域分解构造关于空间变量分段光滑的权函数，在更宽松的主系数条件下证明一阶输运方程的 Carleman 估计，并用于建立初值与源项空间因子反演的 Lipschitz 稳定性。
Inverse problems for first-order hyperbolic equations with time-dependent coefficients
本文针对系数依赖时空的一阶线性双曲方程，利用全局Carleman估计并创新性地通过选择向量场积分曲线长度构造权函数，证明了逆源问题和系数问题的全局Lipschitz稳定性。
On Inverse Problems for Mean Field Games with Common Noise via Carleman Estimate
该文针对带共同噪声的平均场博弈的两类逆问题，借助两个新的Carleman估计，分别建立了耦合系统解的稳定性与逆源问题的唯一性定理。
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates, seems hard to apply to the case of Grushin-type operators studied in this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse coefficient problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a strategy that combines Fourier decomposition …
Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.
Lipschitz Stability of an Inverse Problem of Transmission Waves with Variable Jumps
针对主系数沿严格凸内部界面可变跳跃的透射波方程，本文借助新的单参数全局 Carleman 不等式和 Bukhgeim-Klibanov 方法，证明了零阶系数反演的 Lipschitz 稳定性。
Lipschitz stability for an inverse source problem of the wave equation with kinetic boundary conditions
本文通过改进的Carleman估计，解决了带有动态边界条件的波动方程逆源问题的全局Lipschitz稳定性问题，并改进了早期结果。
We consider a parabolic equation in a bounded domain $\OOO$ over a time interval $(0,T)$ with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary $Γ\subset \ppp\OOO$. Then, we discuss an inverse problem of determining a zeroth-order spatially varying coefficient by extra data of solution $u$: $u\vert_{Γ\times (0,T)}$ and $u(\cdot,t_0)$ in $\OOO$ with $t_0=0$ or $t=T$. First we establish a conditional Lipschitz stability estimate as well as the uniqueness for the case $t_0=T.$ Second, under additional condition for $Γ$, we prove the uniqueness for the case $t_0=0$. T…
An inverse Cauchy problem of a stochastic hyperbolic equation
本文研究随机双曲型方程的反柯西问题，利用 Carleman 恒等式建立观测性估计并构造 Tikhonov 型泛函求正则化近似，结合核学习理论提出数值重构算法，实验验证了其可行性与准确性。
We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity result lies in the fact that it allows discontinuous coefficients in all but one variable.
Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients
在Besov正则性假设下，通过有限差分商技术证明具有线性增长的椭圆方程的解具有几乎Lipschitz连续性，并给出Lebesgue空间尺度下的最优性例子。
Lipschitz stability in the simultaneous recovery of wave speed and initial source
针对双曲方程反问题，提出同时恢复波速系数与初始位移的方法，利用变换与求导得到耦合系统，结合加权能量与Carleman估计，在Sobolev范数下建立Lipschitz稳定性，将系数辨识推广至联合辨识。
Carleman estimates for the Korteweg-de Vries equation with piecewise constant main coefficient
研究具有不连续主系数和界面条件的KdV方程的可观测性，提出新的双参数Carleman估计，应用于局部精确可控性和Lipschitz稳定性反问题。
Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $Ω$. The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz-Morrey spaces as well as Hölder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $W^{1,p}_{0}(Ω)$ for some $p>1$ but not necessarily in $W^{1,2}_{0}(Ω)$) to inhomogeneous equations with coefficients that …
In this article we prove quantitative unique continuation results for wave operators of the form $\partial$ 2 t -- div(c(x)$\nabla$$\bullet$) where the scalar coefficient c is discontinuous across an interface of codimension one in a bounded domain or on a compact Riemannian manifold. We do not make any assumptions on the geometry of the interface or on the sign of the jumps of the coefficient c. The key ingredient is a local Carleman estimate for a wave operator with discontinuous coefficients. We then combine this estimate with the recent techniques of Laurent-L{é}autaud [LL19] to propagate …
Null controllability of n-dimensional parabolic equations degenerated on partial boundary
本文针对部分边界退化且边界非光滑的高维抛物型方程，通过构造特殊权函数并转化退化区域积分项，建立了可用的Carleman估计，从而证明了相应方程的零能控性，推广了已有结果并给出新例子。
An elliptic problem involving critical Choquard and singular discontinuous nonlinearity
本文研究结合临界Hartree型与奇异间断非线性的椭圆问题，应用变分方法和广义梯度证明了弱解的存在性与多重性，并通过$L^\infty$估计确立了其正则性。
We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the initial function has a finite number of jump discontinuities, the corresponding discontinuous viscosity solution of the corresponding Cauchy problem on the real line is unique. Uniqueness follows from a comparison theorem for semicontinuous viscosity sub- and supersolutions, using the barrier effect of spatial discontinuities of a solution. We also prove an exis…
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacob…
Lipschitz regularity for parabolic fractional $p$-Laplace equations
研究抛物分数 $p$-拉普拉斯方程局部弱解的空间局部利普希茨连续性，证明在 $sp > p-1$ 时的均匀时间正则性，适用于对称、平移不变核，包括可能不连续的核。
This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P)} \end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a discontinuous function and asymptotically linear at infinity, $λ=0$ is in a spectral gap of $-Δ+V$, and $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$. Here, by employing Variational Methods for Locally Lipschitz Functionals, we es…
In this paper we study the local behavior of a solution to the Lamé system when the Lamé coefficients $λ$ and $μ$ satisfy that $μ$ is Lipschitz and $λ$ is essentially bounded in dimension $n\ge 2$. One of the main results is the \emph{local} doubling inequality for the solution of the Lamé system. This is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen weights. Furthermore, we also prove the \emph{global} doubling inequality, which is useful in some inverse problems.
We study cavitation type equations, $\text{div}(a_{ij}(X) \nabla u) \sim δ_0(u)$, for bounded, measurable elliptic media $a_{ij}(X)$. De Giorgi-Nash-Moser theory assures that solutions are $α$-Hölder continuous within its set of positivity, $\{u>0\}$, for some exponent $α$ strictly less than one. Notwithstanding, the key, main result proven in this paper provides a sharp Lipschitz regularity estimate for such solutions along their free boundaries, $\partial \{u>0 \}$. Such a sharp estimate implies geometric-measure constrains for the free boundary. In particular, we show that the non-coinciden…
Using a multiple critical points theorem for locally Lipschitz continuous functionals, we establish the existence of at least three distinct solutions for a parametric discrete differential inclusion problem involving a real symmetric and positive definite matrix. Applications to tridiagonal, fourth-order and partial difference inclusions are presented.
A strong unique continuation result for the Baouendi operator
本文研究了Baouendi算子在零阶扰动下的强唯一持续性，证明了在退化流形上无穷阶消失的解必为零，并扩展到变系数算子。
We consider the EBT algorithm (a particle method) for the non-local equation with a discontinuous interaction kernel. The main difficulty lies in the low regularity of the kernel which is not Lipschitz continuous, thus preventing the application of standard arguments. Therefore, we use the radial symmetry of the problem instead and transform it using spherical coordinates. The resulting equation has a Lipschitz kernel with only one singularity at zero. We introduce a new weighted flat norm and prove that the particle method converges in this norm. We also comment on the two-dimensional case wh…
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