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Inverse problems for elliptic equations on the plane: I
Differential Equations, 2006The author deals with the problem of finding a function \(u(x,y)\) which is a solution of the following boundary value problem \(\Omega\subset\mathbb R^2\) satisfying certain conditions: \[ \begin{gathered} (Lu)(x,y)= F(x,y),\quad (x,y)\in\Omega,\\ u(x,y)= f(x,y),\quad (x,y)\in\partial\Omega,\end{gathered} \] where \[ \begin{multlined} (Lu)(x,y):= a_ ...
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Inverse coefficient problems for elliptic equations in a cylinder: II
Differential Equations, 2013The author studies sufficient conditions for the unique solvability of the inverse coefficient problem. There are obtained various global sufficient conditions in the form of constraints on the signs of the given functions and their derivatives. As a corollary, there are proved statements of inverse coefficient problems with overdetermination on the ...
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The Factorization Method for a Class of Inverse Elliptic Problems
Mathematische Nachrichten, 2005Abstract In this paper the factorization method from inverse scattering theory and impedance tomography is extended to a class of general elliptic differential equations in divergence form. The inverse problem is to determine the interface ∂Ω of an interior change of the material parameters from the Neumann‐Dirichlet map.
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On Elliptic Inverse Heat Conduction Problems
Journal of Heat Transfer, 2017This note is concerned with a new method for the solution of an elliptic inverse heat conduction problem (IHCP). It considers an elliptic system where no information is given at part of the boundary. The method is iterative in nature. Starting with an initial guess for the missing boundary condition, the algorithm obtains corrections to the assumed ...
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On an inverse problem for an elliptic equation
UZBEK MATHEMATICAL JOURNALThe purpose of the paper is to study the well-posedness of an inverse problem for an elliptic equation with unknown coefficients on the right-hand side. The inverse problem is considered in a bounded domain in the case of the Dirichlet boundary condition and with integral additional information.
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Nonlinear inverse problems for elliptic equations
jiip, 2001Abstract - In this paper we prove the existence of regular solutions for some nonlinear coefficient inverse problems for elliptic equations.
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A class of inverse problems for elliptic equations
Siberian Mathematical Journal, 1990The author first considers the problem of seeking a pair of function (u.q) satisfying the following conditions: \[ (1)\quad Au+pg+f,\quad q_{x_ n}=0\quad on\quad \Omega,\quad u=g\quad on\quad \partial \Omega,\quad u_{x_ n}=h\quad on\quad D_ 0 \] where A is the following linear elliptic operator \[ Au=-\sum^{n- 1}_{j,k=1}a^{jk}u_{x_ jx_ k}-u_{x_ nx_ n}+\
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Weak approximation of an elliptic inverse problem
There is a significant body of research work concerning the quantification of uncertainty in the solution of elliptic PDEs with random coefficients. Typically this work makes the assumption that the diffusion coefficient is a random field with a simple representation via a Karhunen-Loeve or polynomial chaos representation. However, in many applicationsMasoumeh Dashti +2 more
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An inverse problem for an elliptic equation with an affine term
Mathematische Annalen, 2000The author proves that for a large class of planar domains \(\Omega\) (including all true ellipses) and any real-valued \(\psi\in C^\infty (\partial \Omega)\) which is not identically constant there exist at most \(k_0\) different pairs of coefficients \((\lambda_k,\mu_k) \in\mathbb{R}^2\) such that the Cauchy problem \[ \Delta u=-\lambda_k u-\mu_k ...
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