Unique Continuation Principle Research Articles

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.

Lorentzian Calderón problem near the Minkowski geometry

We study a Lorentzian version of the well-known Calderón problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map on its timelike boundary. In the earlier work of the authors it was shown that zeroth order coefficients can be uniquely determined under a two-sided spacetime curvature bound and the additional assumption that there are no conjugate points along null or spacelike geodesics. In this paper we show that uniqueness for the zeroth order coefficient holds for manifolds satisfying a weaker curvature bound and for spacetime perturbations of such manifolds. This relies on a new enhanced optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calderón problem near the Minkowski geometry.

Low-regularity Seiberg–Witten moduli spaces on manifolds with boundary

For a compact spin^{c} manifold X with boundary b_1(partial X)=0, we consider moduli spaces of solutions to the Seiberg–Witten equations in a generalized double Coulomb slice in W^{1,2} Sobolev regularity. We prove they are Hilbert manifolds, prove denseness and “semi-infinite-dimensionality” properties of the restriction to partial X, and establish a gluing theorem. To achieve these, we prove a general regularity theorem and a strong unique continuation principle for Dirac operators, and smoothness of a restriction map to configurations of higher regularity on the interior, all of which are of independent interest.

Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators

We prove a scale-free quantitative unique continuation estimate for the gradient of eigenfunctions of divergence-type operators, i.e., operators of the form −divA∇, where the matrix function A is uniformly elliptic. The proof uses a unique continuation principle for elliptic second-order operators and a lower bound on the L2-norm of the gradient of eigenfunctions corresponding to strictly positive eigenvalues.As an application, we prove an eigenvalue lifting estimate that allows us to prove a Wegner estimate for random divergence-type operators. Here our approach allows us to get rid of a restrictive covering condition that was essential in previous proofs of Wegner estimates for such models.

Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant

Let Omega subset {mathbb {R}}^d be a C^1 domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d times d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Omega , or with greater generality u solves {text {div}}(A(x)nabla u)=0 in Omega , and u vanishes on Sigma = partial Omega cap B for some ball B. We study the dimension of the singular set of u in Sigma , in particular we show that there is a countable family of open balls (B_i)_i such that u|_{B_i cap Omega } does not change sign and K backslash bigcup _i B_i has Minkowski dimension smaller than d-1-epsilon for any compact K subset Sigma . We also find upper bounds for the (d-1)-dimensional Hausdorff measure of the zero set of u in balls intersecting Sigma in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of Sigma is bounded except for a set of Hausdorff dimension at most d-1-epsilon .

Unique continuation and inverse problem for an anisotropic beam bending equation

This article studies the unique continuation of the principal and inverse problem of a perturbed anisotropic fourth-order elliptic operator arising in elastic beam theory, where the principle part of the operator is modelled by the non-rotational symmetric operator ∑j=1nDxj4. We discuss the three-ball inequality, stability estimate, strong unique continuation principle, and finally the inverse boundary value problem of recovering the coefficients featuring the bending stiffness of the material.

Uniqueness and stability for inverse source problem for fractional diffusion-wave equations

Abstract This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at a subboundary. A uniqueness result is obtained by using the analyticity and the newly established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation.

Conductivity reconstruction from power density data in limited view

In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.

Unique Continuation for the Multi-term Time Fractional Diffusion Equation and its Numerical Simulation

This paper deals with the unique continuation principle of solutions for a one-dimensional anomalous diffusion equation with multi-term time fractional derivatives. The proof is mainly based on the Laplace transform and Theta function method. Numerically, we reformulate the unique continuation as an optimization problem, and propose an iterative thresholding algorithm to simulate it numerically. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Properties of the support of solutions of a class of nonlinear evolution equations

AbstractIn this work we consider equations of the form where P is any polynomial without constant term, and G is any polynomial without constant or linear terms. We prove that if u is a sufficiently smooth solution of the equation, such that for some , then there exists such that for every . Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation, and for the generalized KdV hierarchy

Localisation for Delone operators via Bernoulli randomisation

Delone operators are Schrödinger operators in multi-dimensional Euclidean space with a potential given by the sum of all translates of a given “single-site potential” centred at the points of a Delone set. In this paper, we use randomisation to study dynamical localisation for families of Delone operators. We do this by suitably adding more points to a Delone set and by introducing i.i.d. Bernoulli random variables as coupling constants at the additional points. The resulting non-ergodic continuum Anderson model with Bernoulli disorder is accessible to the latest version of the multiscale analysis. The novel ingredient here is the initial length-scale estimate whose proof is hampered due to the non-periodic background potential. It is obtained by the use of a quantitative unique continuation principle. As applications we obtain both probabilistic and topological statements about dynamical localisation. Among others, we show that Delone sets for which the associated Delone operators exhibit dynamical localisation at the bottom of the spectrum are dense in the space of Delone sets.

Unique continuation from a crack’s tip under Neumann boundary conditions

We derive local asymptotics of solutions to second order elliptic equations at the edge of a (N−1)-dimensional crack, with homogeneous Neumann boundary conditions prescribed on both sides of the crack. A combination of blow-up analysis and monotonicity arguments provides a classification of all possible asymptotic homogeneities of solutions at the crack’s tip, together with a strong unique continuation principle.

Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations

We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulæ and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points.

Anderson–Bernoulli localization on the three-dimensional lattice and discrete unique continuation principle

We consider the Anderson model with Bernoulli potential on the three-dimensional (3D) lattice Z3, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework of Bourgain–Kenig and Ding–Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points. Its proof relies on geometric arguments about the 3D lattice.

Identification of time-varying source term in time-fractional diffusion equations

This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in $(0,2)$. We examine two different cases. In the first one, the source is the product of two spatial and temporal terms, and we prove that both of them can be retrieved by knowledge of one arbitrary internal measurement of the solution for all times. In the second case, we assume that the first term of the product varies with one fixed space variable, while the second one is a function of all the remaining space variables and the time variable, and we show that both terms are uniquely determined by two arbitrary lateral measurements of the solution over the entire time span. These two source identification results boil down to a weak unique continuation principle in the first case and a unique continuation principle for Cauchy data in the second one, that are preliminarily established. Finally, numerical reconstruction of spatial term of source terms in the form of the product of two spatial and temporal terms, is carried out through an iterative algorithm based on the Tikhonov regularization method.

Unique continuation properties for polyharmonic maps between Riemannian manifolds

AbstractPolyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher-order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic cases: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on an open subset, then they agree everywhere; and (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping

<p style='text-indent:20px;'>In this paper we prove the exponential decay of the energy for the high-order Kadomtsev-Petviashvili II equation with localized damping. To do that, we use the classical dissipation-observability method and a unique continuation principle introduced by Bourgain in [<xref ref-type="bibr" rid="b3">3</xref>] here extended for the high-order Kadomtsev-Petviashvili. A similar result is also obtained for the two-dimensional Zakharov-Kuznetsov (ZK)equation. The method of proof works better for the ZK equation, so we were led to make some subtle modifications on it to include KP type equations. In fact, to reach a key estimate we use an anisotropic Gagliardo-Nirenberg inequality to drop the <inline-formula><tex-math id="M1">\begin{document}$ y $\end{document}</tex-math></inline-formula>-derivative of the norm.</p>

Carleman estimate for a 1D linear elastic problem involving interfaces: Application to an inverse problem

The aim of this work is to study the stability of the reconstruction of some mechanical parameters that characterize interfaces within linear elastic bodies. The model considered consists of two bonded elastic solids. The adhesive elastic layer between them is a thin interphase that is approximated, by asymptotic analysis, as an interface. The resulting interface model is in turn characterized by a typical spring‐type linear elastic behavior. In this context and on a 1D configuration, we investigate a Carleman‐type estimate that relates to a unique continuation principle. The proof is based on the construction of suitable weight functions: their gradients are non‐zero, the jumps of the derivatives are positive across the interface, and the averages of the derivatives vanish. The design of such weight functions enables the control of the interface terms in the estimates. With this at hand, we establish a Lipschitz stability estimate for the inverse problem of identifying an interface stiffness parameter from measurements that are available on both sides of the external boundary.

Dirac–Coulomb operators with general charge distribution II. The lowest eigenvalue

Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ is minimized, at fixed mass $\mu(\mathbb{R}^3)=\nu$, when $\mu$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-i\alpha\cdot\nabla+\beta-\mu\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $\mu$ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass $\nu_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $\mu\geq0$ with $\mu(\mathbb{R}^3)<\nu_1$. We first show that $\nu_1$ is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all $0\leq\nu<\nu_1$, there exists an optimal measure $\mu\geq0$ giving the lowest possible eigenvalue at fixed mass $\mu(\mathbb{R}^3)=\nu$, which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.

Unique continuation principles for a higher order fractional Laplace equationThe authors are partially supported by the INDAM-GNAMPA 2018 grant ‘Formula di monotonia e applicazioni: problemi frazionari e stabilità spettrale rispetto a perturbazioni del dominio’. V Felli is partially supported by the PRIN 2015 grant ‘Variational methods, with applications to problems in mathematical physics and geometry’.

In this paper we prove the strong unique continuation principle and the unique continuation from sets of positive measure for solutions of a higher order fractional Laplace equation in an open domain. Our proofs are based on the Caffarelli–Silvestre (2007 Commun. PDE 32 1245–60) extension method combined with an Almgren type monotonicity formula. The corresponding extended problem is formulated as a system of two second order equations with singular or degenerate weights in a half-space, for which asymptotic estimates are derived by a blow-up analysis.