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 study a partial differential inclusion, driven by the p-Laplacian operator, involving a p-superlinear nonsmooth potential, and subject to Neumann boundary conditions. By means of nonsmooth critical point theory, we prove the existence of at least two constant sign solutions (one positive, the other negative). Then, by applying the nonsmooth Morse relation, we find a third non-zero solution.
We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-Δu+u^q=0$ ($q>1$) in $Ω\times (0,\infty)$, $u=\infty$ on $\partialΩ\times (0,\infty)$ and $u(.,0)\in L^1(Ω)$, when $Ω$ is a bounded domain in $\mathbb R^N$. We construct a maximal solution, prove that this maximal solution is a large solution whenever $q<N/(N-2)$ and it is unique if $\partialΩ=\partial\barΩ^c$. If $\partialΩ$ has the local graph property, we prove that there exists at most one solution to problem (P)
Consider the parabolic equation with measure data \begin{equation*} \left\{ \begin{aligned} &u_t-{\rm div} \mathbf{a}(D u,x,t)=μ&\text{in}& \quad Ω_T, &u=0 \quad &\text{on}& \quad \partial_pΩ_T, \end{aligned}\right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^n$, $Ω_T=Ω\times (0,T)$, $\partial_pΩ_T=(\partialΩ\times (0,T))\cup (Ω\times\{0\})$, and $μ$ is a signed Borel measure with finite total mass. Assume that the nonlinearity ${\bf a}$ satisfies a small BMO-seminorm condition, and $Ω$ is a Reifenberg flat domain. This paper proves a global Marcinkiewicz estimate for the SOLA…
We discuss stability of square root domains for uniformly elliptic partial differential operators $L_{a,Ω,Γ} = -\nabla\cdot a \nabla$ in $L^2(Ω)$, with mixed boundary conditions on $\partial Ω$, with respect to additive perturbations. We consider open, bounded, and connected sets $Ω\in \mathbb{R}^n$, $n \in \mathbb{N} \backslash\{1\}$, that satisfy the interior corkscrew condition and prove stability of square root domains of the operator $L_{a,Ω,Γ}$ with respect to additive potential perturbations $V \in L^p(Ω) + L^{\infty}(Ω)$, $p>n/2$. Special emphasis is put on the case of uniformly ellipt…
We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the $W^1_p$ estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdo…
We show existence of solutions to the least gradient problem on the plane for boundary data in $BV(\partialΩ)$. We also provide an example of a function $f \in L^1(\partialΩ) \backslash (C(\partialΩ) \cup BV(\partialΩ))$, for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.
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.
This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (Bouligand (B) stationarity), which is derived from the directional differentiability of the solution operator. By using a local regu…
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 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…
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…
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…
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 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…
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.
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…
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 coefficient problem of simultaneously determining the space dependent electric potential, the zero-th order coupling term and the first order coupling vector of a two-state Schrödinger equation in an infinite cylindrical domain of $\mathbb{R}^n$, $n \ge 2$, from finitely many partial boundary measurements of the solution. We prove that these $n+1$ unknown scalar coefficients can be Hölder stably retrieved by $(n+1)$-times suitably changing the initial condition attached at the system.
We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schrödinger equation in a bounded domain of $\mathbb R^n$, $n\geq 2$, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be Hölder stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.
In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$, $\varepsilon>0$, can be determined from the full and partial Dirichlet-to-Neumann map. This extends the results from [M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam. (2021)] where this is shown for Hölder continuous potentials. Also we show that when the Dirichlet-to-Neumann map is restricted to one point on the boundary…
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.
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.
In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension $n\geq 2$, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.
We study the inverse problem of determining the vector and scalar potentials $\mathcal{A}(t,x)=\left(A_{0},A_{1},\cdots,A_{n}\right)$ and $q(t,x)$, respectively, in the relativistic Schrödinger equation \begin{equation*} \Big{(}\left(\partial_{t}+A_{0}(t,x)\right)^{2}-\sum_{j=1}^{n}\left(\partial_{j}+A_{j}(t,x)\right)^{2}+q(t,x)\Big{)}u(t,x)=0 \end{equation*} in the region $Q=(0,T)\timesΩ$, where $Ω$ is a $C^{2}$ bounded domain in $\mathbb{R}^{n}$ for $n\geq 3$ and $T>\mbox{diam}(Ω)$ from partial data on the boundary $\partial Q$. We prove the unique determination of these potentials modulo a …
We consider the multidimensional Borg-Levinson theorem of determining both the magnetic field $dA$ and the electric potential $V$, appearing in the Dirichlet realization of the magnetic Schrödinger operator $H=(-{\rm i}\nabla+A)^2+V$ on a bounded domain $Ω\subset\mathbb R^n$, $n\geq2$, from partial knowledge of the boundary spectral data of $H$. The full boundary spectral data are given by the set $\{(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ}):\ k\geq1\}$, where $\{ λ_k:\ k\in \mathbb N^* \}$ is the non-decreasing sequence of eigenvalues of $H$, $\{ ϕ_k:\ k\in \mathbb N^* \}$ an associated Hilberti…
Given a compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for full symbol of the thermoelastic Dirichlet-to-Neumann map $Λ_g$ with variable coefficients $λ,μ,α,β\in C^{\infty}(\bar{M})$. We prove that $Λ_g$ uniquely determines partial derivatives of all orders of the coefficients on the boundary. Moreover, for a nonempty open subset $Γ\subset\partial M$, suppose that the manifold and the coefficients are real analytic up to $Γ$, we show that $Λ_g$ uniquely determines the coefficients on the whole manifold $\bar{M}$.
Let $Ω\subset\R^n$, $n\ge 3$, be a smooth bounded domain and consider a coupled system in $Ω$ consisting of a conductivity equation $\nabla \cdot γ(x) \nabla u(t,x)=0$ and an anisotropic heat equation $κ^{-1}(x)\partial_tψ(t,x)=\nabla\cdot (A(x)\nabla ψ(t,x))+(γ\nabla u(t,x))\cdot \nabla u(t,x), \quad t\ge 0$. It is shown that the coefficients $γ$, $κ$ and $A=(a_{jk})$ are uniquely determined from the knowledge of the boundary map $u|_{\partialΩ}\mapsto ν\cdot A\nabla ψ|_{\partialΩ}$, where $ν$ is the unit outer normal to $\partialΩ$. The coupled system models the following physical phenomenon…
We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth Riemannian metric $g$, and mean-convex boundary $\partial M$ then knowledge of least areas circumscribed by simple closed curves $γ\subset \partial M$ uniquely determines the metric $g$, under some additional geometric assumptions. These are that $g$ is either a) $C^3$-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. %sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves $…
Let $Ω\subset \Bbb R^2$ be a bounded domain with $\partialΩ\in C^\infty$ and $L$ be a positive number. For a three dimensional cylindrical domain $Q=Ω\times (0,L)$, we obtain some uniqueness result of determining a complex-valued potential for the Schrödinger equation from partial Cauchy data when Dirichlet data vanish on a subboundary $(\partialΩ\setminus\widetildeΓ) \times [0,L]$ and the corresponding Neumann data are observed on $\widetildeΓ\times [0,L]$, where $\widetildeΓ$ is an arbitrary fixed open set of $\partialΩ.$
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,…
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 s…
