The stability for the Cauchy problem for elliptic equations

Abstract

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.

57 pages, review article

References in corpus (2)

• Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates
• Quantitative uniqueness for second order elliptic operators with strongly singular coefficients

Cited by in corpus (73)

• The fractional Calderón problem: low regularity and stability
• Strong unique continuation for general elliptic equations in 2D
• Exponential instability in the fractional Calderón problem
• The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing
• Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities
• Single-logarithmic stability for the Calderón problem with local data
• Stable determination of an immersed body in a stationary Stokes fluid
• Stable determination of an inclusion in an elastic body by boundary measurements (unabridged)
• A modified integral equation method of the nonlinear elliptic equation with globally and locally Lipschitz source
• Uniqueness results for inverse Robin problems with bounded coefficient
• Quantitative uniqueness for elliptic equations with singular lower order terms
• Unique continuation for the Helmholtz equation using stabilized finite element methods
• Stability estimates for Navier-Stokes equations and application to inverse problems
• A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime
• Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: the case of non flat interfaces
• Size estimates for fat inclusions in an isotropic Reissner-Mindlin plate
• Global stability for a coupled physics inverse problem
• Optimal three spheres inequality at the boundary for the Kirchhoff-Love plate's equation with Dirichlet conditions
• Analysis of a model of elastic dislocations in geophysics
• Recovering point sources for the inhomogeneous Helmholtz equation
• Stability Estimates for an Inverse Hyperbolic Initial Boundary Value Problem with Unknown Boundaries
• A note on unique continuation for discrete harmonic functions
• Global logarithmic stability results on the Cauchy problem for anisotropic wave equations
• Smoothness dependent stability in corrosion detection
• Dislocations in a layered elastic medium with applications to fault detection
• Stable determination of surface impedance on a rough obstacle by far field data
• Carleman estimates for the parabolic transmission problem and Hölder propagation of smallness across an interface
• Nodal sets of Robin and Neumann eigenfunctions
• Doubling inequality and nodal sets for solutions of bi-Laplace equations
• A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems
• Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Ill-Posed Elliptic Cauchy Problems
• Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant
• Primal-Dual Weak Galerkin Finite Element Methods for Elliptic Cauchy Problems
• Absence of Critical Points of Solutions to the Helmholtz Equation in 3D
• Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators
• On a nonlinear model in domains with cavities arising from cardiac electrophysiology
• Unique continuation at the boundary for harmonic functions in $C^1$ domains and Lipschitz domains with small constant
• Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data
• Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging
• Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems
• Size estimates for nanoplates
• Stable determination of an anisotropic inclusion in the Schrödinger equation from local Cauchy data
• Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves
• Quantitative estimates of strong unique continuation for anisotropic wave equations
• Hölder stability of quantitative photoacoustic tomography based on partial data
• Approximate Two-Sphere One-Cylinder Inequality in Parabolic Periodic Homogenization
• Doubling inequality at the boundary for the Kirchhoff-Love plate's equation with supported conditions
• Strong unique continuation for second order hyperbolic equations with time independent coefficients
• Lipschitz stability of an inverse boundary value problem for a Schrödinger type equation
• Optimal unique continuation for periodic elliptic equations on large scales
• Propagation of smallness in elliptic periodic homogenization
• Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement
• Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object
• Cloaking using complementary media for electromagnetic waves
• Stabilized nonconforming finite element methods for data assimilation in incompressible flows
• Is the cosmological constant an eigenvalue?
• Stable determination of a rigid inclusion in an anisotropic elastic plate
• Reconstruction of a conformally Euclidean metric from local boundary diffraction travel times
• Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients
• Non-zero constraints in elliptic PDE with random boundary values and applications to hybrid inverse problems
• Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations
• Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients
• Three Balls Theorem for Eigenfunctions of Dirac Operator in Clifford Analysis
• Error estimates for stabilized finite element methods applied to ill-posed problems
• Hausdorff measure bounds for nodal sets of Steklov eigenfunctions
• Doubling inequalities and critical sets of Dirichlet eigenfunctions
• Unique continuation for a non bi-Laplacian fourth order elliptic operator
• Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime
• Acoustic waveguide with a dissipative inclusion
• High frequency stability estimates for a partial data inverse problem
• Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations
• Stable determination of a body immersed in a fluid: the nonlinear stationary case
• Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization