Quantitative Unique Continuation Research Articles

Quantitative Unique Continuation for Second Order Elliptic Operators with Singular Coefficients

We establish some quantitative strong unique continuation properties for solution of |Pu|≤ C1|x|− 1|∇u| + C0|x|− 2|u| where P is a second order elliptic operator. As in (Rev. Mat. Iberoam. 27: 475–491, 9), our result is of quantitative nature but requires weaker conditions on the coefficients of P.

On Landis’ conjecture in the plane for some equations with sign-changing potentials

In this article, we investigate the quantitative unique continuation properties of real-valued solutions to elliptic equations in the plane. Under a general set of assumptions on the operator, we establish quantitative forms of Landis’ conjecture. Of note, we prove a version of Landis’ conjecture for solutions to −\Delta u + Vu = 0 , where V is a bounded function whose negative part exhibits polynomial decay at infinity. The main mechanism behind the proofs is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing result, we present a new idea for constructing positive multipliers and use it reduce the equation to a Beltrami system. The resulting first-order equation is analyzed using the similarity principle and the Hadamard three-quasi-circle theorem.

Carleman estimates for Baouendi–Grushin operators with applications to quantitative uniqueness and strong unique continuation

ABSTRACT In this paper, we establish some new Carleman estimates for the Baouendi–Grushin operators , in Equation (1). We apply such estimates to obtain: (i) an extension of the Bourgain–Kenig quantitative unique continuation and (ii) the strong unique continuation property for some degenerate sublinear equations.

Quantitative uniqueness of solutions to second-order elliptic equations with singular lower order terms

In this article, we study some quantitative unique continuation properties of solutions to second-order elliptic equations with singular lower order terms. First, we quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when u is a nontrivial solution to in some open, connected subset of where we characterize the vanishing order of solutions in terms of the norms of V and W in their respective Lebesgue spaces. Then, using these maximal order of vanishing estimates, we establish quantitative unique continuation at infinity results for solutions to in The main tools in our work are new versions of Carleman estimates for a range of p and q values.

The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian

In this paper, we establish the bang-bang property of time and norm optimal control problems for parabolic equations governed by time-varying fractional Laplacian, evolved in a bounded domain of ℝd. We firstly get a quantitative unique continuation at one point in time for parabolic equations governed by time-varying fractional Laplacian. Then, we establish an observability inequality from measurable sets in time for solutions of the above-mentioned equations. Finally, with the aid of the observability inequality, the bang-bang property of time and norm optimal control problems can be obtained.

Global Uniqueness and Lipschitz-Stability for the Inverse Robin Transmission Problem

In this paper, we consider the inverse problem of detecting a corrosion coefficient between two layers of a conducting medium from the Neumann-to-Dirichlet map. This inverse problem is motivated by the description of the index of corrosion in non-destructive testing. We show a monotonicity estimates between the Robin coefficient and the Neumann-to-Dirichlet operator. We prove a global uniqueness result and Lipschitz stability estimate, and show how to quantify the Lipschitz stability constant for a given setting. Our quantification of the Lipschitz constant does not rely on quantitative unique continuation or analytic estimates of special functions. Instead of deriving an analytic estimate, we show that the Lipschitz constant for a given setting can be explicitly calculated from the a priori data by solving finitely many well-posed PDEs. Our arguments rely on standard (non-quantitative) unique continuation, a Runge approximation property, the monotonicity result and the method of localized potentials. To solve the problem numerically, we reformulate the inverse problem into a minimization problem using a least square functional. The reformulation of the minimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup formulation. The reconstruction is then performed by means of the BFGS algorithm. Finally, numerical results are presented to illustrate the efficiency of the proposed alogorithm.

Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves

In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru [Tat95, Tat99b], Robbiano–Zuily [RZ98] and Hörmander [Hör97]. We provide local stability estimates that can be propagated, leading to global ones. Then, we specify the previous results to the wave operator on a Riemannian manifold \mathcal M with boundary. For this operator, we also prove Carleman estimates and local quantitative unique continuation from and up to the boundary \partial \mathcal M . This allows us to obtain a global stability estimate from any open set \Gamma of \mathcal M or \partial \mathcal M , with the optimal time and dependence on the observation. As a first application, we compute a sharp lower estimate of the intensity of waves in the shadow of an obstacle. We also provide the cost of approximate controllability on the compact manifold \mathcal M : for any T > 2\: \mathrm {sup}_{x \in \mathcal M} \mathrm {dist}(x,\Gamma) , we can drive any data of H^1_0 \times L^2 in time T to an \varepsilon -neighborhood of zero in L^2 \times H^{-1} , with a control located in \Gamma , at cost e^{C/\varepsilon} . We finally obtain related results for the Schrödinger equation.

Quantitative unique continuation for the heat equations with inverse square potential

In this paper, we investigate the unique continuation properties for multi-dimensional heat equations with inverse square potential in a bounded convex domain Ω of mathbb{R}^{d}. We establish observation estimates for solutions of equations. Our result shows that the value of the solutions can be determined uniquely by their value on an open subset ω of Ω at any given positive time L.

Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.

Quantitative Runge Approximation and Inverse Problems

AbstractIn this short note, we provide a quantitative version of the classical Runge approximation property for second-order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application, we provide a new proof of the result from [8], [2] on stability for the Calderón problem with local data.

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector $\chi_{(-\infty,E]}(H_L)$ of a Schr\"odinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schr\"odinger operators with non-linear parameter-dependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-scale domain.

Quantitative unique continuation for a parabolic equation

We address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - \Delta u = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of vanishing is bounded by $C(\Vert v\Vert_{L^\infty}^{2/3}+\Vert w\Vert_{L^\infty}^2)$ matching the upper bound previously established in the elliptic case. in the elliptic case.

Quantitative unique continuation for the heat equation with Coulomb potentials

In this paper, we establish a Holder-type quantitative estimate of unique continuation for solutions to the heat equation with Coulomb potentials in either a bounded convex domain or a \begin{document}$C^2$\end{document} -smooth bounded domain. The approach is based on the frequency function method, as well as some parabolic-type Hardy inequalities.

Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential

We consider non-ergodic magnetic random Sch\"odinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of the arguments from [Kle13], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [BTV15]. This generalizes Klein's result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold $E_0(\infty) \in (0, \infty]$, it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian with vanishing magnetic field.

Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations

We consider elliptic differential operators on either the entire Euclidean space Rd or subsets consisting of a cube ΛL of integer length L. For eigenfunctions of the operator, and more general solutions of elliptic differential equations, we derive several quantitative unique continuation results. The first result is of local nature and estimates the vanishing order of a solution. The second is a sampling result and compares the L2-norm of a solution over a union of equidistributed δ-balls in space with the L2-norm on the entire space. In the case where the space Rd is replaced by a finite cube ΛL, we derive similar estimates. Particular features of our bound are that they are uniform as long as the coefficients of the operator are chosen from an appropriate ensemble, they are quantitative and explicit with respect to the radius δ, and they are L-independent and stable under small shifts of the δ-balls. Our proof applies to second order terms which have slowly varying coefficients on the relevant length scale. The results can also be interpreted as special cases of uncertainty relations, observability estimates, or spectral inequalities.

Quantitative unique continuation for the linear coupled heat equations

In this paper, we established a quantitative unique continuation results for a coupled heat equations, with the homogeneous Dirichlet boundary condition, on a bounded convex domain Ω of mathbb{R}^{d} with smooth boundary ∂Ω. Our result shows that the value of the solutions can be determined uniquely by its value on an arbitrary open subset ω of Ω at any given positive time T.

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let $Λ_L = (-L/2, L/2)^d$ and $H_L = -Δ_L + V_L$ be a Schrödinger operator on $L^2 (Λ_L)$ with a bounded potential $V_L : Λ_L \to \mathbb{R}^d$ and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type $ \int_{Λ_L} \lvert φ \rvert^2 \leq C_{\rm {sfuc}} \int_{W_δ (L)} \lvert φ \rvert^2,$ where $φ$ is an infinite complex linear combination of eigenfunctions of $H_L$ with exponentially decaying coefficients, $W_δ (L)$ is some union of equidistributed $δ$-balls in $Λ_L$ and $C_{{\rm {sfuc}}} > 0$ an $L$-independent constant. The exponential decay condition on $φ$ can alternatively be formulated as an exponential decay condition of the map $λ \mapsto \lVert χ_{[λ, ∞)} (H_L) φ \rVert^2$. The novelty is that at the same time we allow the function $φ$ to be from an infinite dimensional spectral subspace and keep an explicit control over the constant $C_{{\rm {sfuc}}}$ in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.

Quantitative uniqueness of some higher order elliptic equations

We study the quantitative unique continuation property of some higher order elliptic operators. We prove a lower bound for nontrivial solutions of the equation (−Δ)mu+V(x)u=0, m∈N, V(x) is bounded. The bound shows that nontrivial solutions can not decay faster than e−|x|4/3ln⁡|x| at infinity. Moreover, we obtain an improved lower bound of nontrivial solutions for a special forth order elliptic operators in dimension 2, the bound is shown to be essentially sharp by constructing a Meshkov-type example.

On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates

In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schr\"odinger equations) on a compact, smooth Riemannian manifold, $(M,g)$, without boundary. Moreover, with only slight modifications these results generalize to equations with $C^1$ potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary doubling property, we prove upper bounds on the $\mathcal{H}^{n-1}$-measure of nodal domains of eigenfunctions of the generalized Dirichlet-to-Neumann map on analytic manifolds.

Local behavior of solutions of the stationary Schrödinger equation with singular potentials and bounds on the density of states of Schrödinger operators

ABSTRACTWe study the local behavior of solutions of the stationary Schrödinger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term. Combining a corollary to this result with a quantitative unique continuation principle for singular potentials, we obtain log-Hölder continuity for the density of states outer-measure in one, two, and three dimensions for Schrödinger operators with singular potentials, results that hold for the density of states measure when it exists.