Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measurement

In this paper, we consider the inverse Robin transmission problem with one electrostatic measurement. We prove a uniqueness result for the simultaneous determination of the Robin parameter p, the conductivity k, and the subdomain D, when D is a ball. When D and k are fixed, we prove a uniqueness result and a directional Lipschitz stability estimate for the Robin parameter p. When p and k are fixed, we give an upper bound to the subdomain D. For the reconstruction purposes of the Robin parameter p, we set the inverse problem under an optimization form for a Kohn–Vogelius cost functional. We prove the existence and the stability of the optimization problem. Finally, we show some numerical experiments that agree with the theoretical considerations. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiscale model for moisture transport with adsorption phenomenon in concrete materials

Abstract

We consider a new two-scale problem which is given as a mathematical model for moisture transport arising in a concrete carbonation process. In research for moisture transport, it is a crucial step how to describe the relationship between the relative humidity and the degree of saturation, mathematically. Here, we have proposed the two-scale model consisting of the diffusion equation for the relative humidity in a macro domain and the free boundary problems describing the relationship in infinitely micro domains. Accordingly, the structures of the micro domains are unknown in our model. This is a significant feature of our new model to emphasize. The aim of this paper is to establish local existence in time and uniqueness of a solution to the model.     

Lipschitz stability estimate in the inverse Robin problem for the Stokes system

We are interested in the inverse problem of recovering a Robin coefficient defined on some non-accessible part of the boundary from available data on another part of the boundary in the non-stationary Stokes system. We prove a Lipschitz stability estimate under the a priori assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois and which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework.     

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite difference equations with random, possibly non symmetric coe?cients. Under the assumption that the coe?cients are stationary and ergodic in the quantitative form of a logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d≥2. Similar estimates have recently been obtained in the special case of diagonal coe?cients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green’s function in weighted spaces. In the critical case d = 2, our argument for the estimate on the gradient of the elliptic Green’s function uses a Calderón–Zygmund estimate in discrete weighted spaces, which we state and prove. As applications, we provide a quantitative two-scale expansion and a quantitative approximation of the homogenized coe?cients.     

On a reaction–advection–diffusion equation with Robin and free boundary conditions

ABSTRACT

A reaction–advection–diffusion equation with variable intrinsic growth rate, Robin and free boundary conditions is investigated in this paper. Firstly, we present a spreading–vanishing dichotomy for the asymptotic behavior of the solutions of the equation. Then, we obtain criteria for spreading and vanishing, and get an estimate for the asymptotic spreading speed of the spreading front. Moreover, numerical simulation is also given to illustrate the impact of the expansion capacity on the free boundary.     

Solving a nonlinear inverse Robin problem through a linear Cauchy problem

ABSTRACT

Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton's law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton's law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems

We study the approximation of a multiscale reaction–diffusion system posed on both macroscopic and microscopic space scales. The coupling between the scales is done through micro–macro flux conditions. Our target system has a typical structure for reaction–diffusion flow problems in media with distributed microstructures (also called, double porosity materials). Besides ensuring basic estimates for the convergence of two-scale semi-discrete Galerkin approximations, we provide a set of a priori feedback estimates and a local feedback error estimator that help in designing a distributed-high-errors strategy to allow for a computationally e?cient zooming in and out from microscopic structures. The error control on the feedback estimates relies on two-scale-energy, regularity, and interpolation estimates as well as on a fine bookeeping of the sources responsible with the propagation of the (multiscale) approximation errors. The working technique based on a priori feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions.     

Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients.

Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L ² space for solution to the inverse problem under consideration.     

Stability estimate for an inverse boundary coefficient problem in thermal imaging

We establish a stability estimate for an inverse boundary coefficient problem in thermal imaging. The inverse problem under consideration consists in the determination of a boundary coefficient appearing in a boundary value problem for the heat equation with Robin boundary condition (we note here that the initial condition is assumed to be a priori unknown). Our stability estimate is of logarithmic type and it is essentially based on a logarithmic estimate for a Cauchy problem for the Laplace equation.     

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H ^∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ?. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H ^k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.     

Stability analysis in the inverse Robin transmission problem

In this paper, we consider the conductivity problem with piecewise‐constant conductivity and Robin‐type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non‐monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius‐type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first‐order and second‐order shape derivative of the proposed cost functional, which is performed rigorously by means of shape‐Lagrangian formulation without using the shape sensitivity of the states variables. © 2016 The Author. Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.     

Stability Estimates for the Inverse Conductivity Problem for Less Regular Conductivities

A log-type stability estimate for the inverse conductivity problem in space dimension n ≥ 3, if the conductivity has C ^3/2+ε regularity is proven.     

Existence of solutions for semilinear elliptic boundary value problems on arbitrary open sets

We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set Ω. We make no assumptions about the open set Ω and very mild regularity assumptions on the semilinearity f, plus a coerciveness assumption which depends on the optimal Poincaré–Steklov constant λ₁. The proof is based on Schaefer’s fixed point theorem applied to a sequence of truncated problems. We state a simple uniqueness result. We also generalize the results to Robin boundary conditions [17].     

Note on local quadratic growth estimates in bang–bang optimal control problems

Abstract

Strong second-order conditions in mathematical programming play an important role not only as optimality tests but also as an intrinsic feature in stability and convergence theory of related numerical methods. Besides of appropriate firstorder regularity conditions, the crucial point consists in local growth estimation for the objective which yields inverse stability information on the solution. In optimal control, similar results are known in case of continuous control functions, and for bang–bang optimal controls when the state system is linear. The paper provides a generalization of the latter result to bang–bang optimal control problems for systems which are affine-linear w.r.t. the control but depend nonlinearly on the state. Local quadratic growth in terms of L₁ norms of the control variation are obtained under appropriate structural and second-order sufficient optimality conditions.     

Lipschitz behavior of convex semi-infinite optimization problems: a variational approach

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.      

Stability estimate in determination of a coefficient in transmission wave equation by boundary observation

In this paper, we study the global stability in determination of a coefficient in the transmission wave equation from data of the solution in a subboundary over a time interval. Providing regular initial data, we prove a hölder stability estimate in the inverse problem with a single measurement. Moreover, the exponent in the stability estimate depends on the regularity of initial data.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

Uniqueness and stability in multidimensional hyperbolic inverse problems

Under a weak regularity assumption, we prove the uniqueness in multidimensional hyperbolic inverse problems with a single measurement. Moreover we show that our uniqueness results yield the best possible Lipschitz stability in L²-space in the inverse problems by means of the exact observability inequality.     

Quasilinear systems with memory kernels

We consider a quasilinear integrodifferential system in non-normal form. Such a system is a generalization of a phase-field model with memory and includes, as a particular case, the system describing the combustion of a material with memory. In this paper, we study both the direct and the inverse problems. Our fundamental tools are: the theory of analytic semigroups, optimal regularity results and fixed point arguments.     

A Carleman estimate for the linear magnetoelastic waves system and an inverse source problem in a bounded conductive medium

ABSTRACT

In this paper, we consider an inverse problem for the simultaneous diffusion process of elastic and electromagnetic waves in an isotropic heterogeneous elastic body which is identified with an open bounded domain. From the mathematical point of view, the system under consideration can be viewed as the coupling between the hyperbolic system of elastic waves and a parabolic system for the magnetic field. We study an inverse problem of determining the external source terms by observations data in a neighborhood of the boundary and we prove the Hölder stability. For the proof, we show a Carleman estimate for the displacement and the magnetic field of the magnetoelastic system.