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We study the well-posedness of a semilinear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map.…

Analysis of PDEs · Mathematics 2022-03-30 Li Li

This work deals with the problem of determining a non-homogeneous heat conductivity profile in a steady-state heat conduction boundary-value problem with mixed Dirichlet-Neumann boundary conditions over a bounded domain in $\mathbb{R}^n$,…

Numerical Analysis · Mathematics 2022-08-25 Angel A. Ciarbonetti , Sergio Idelsohn , Ruben D. Spies

This paper concerns the reconstruction of an anisotropic conductivity tensor in an elliptic second-order equation from knowledge of the so-called power density functionals. This problem finds applications in several coupled-physics medical…

Analysis of PDEs · Mathematics 2013-02-15 Guillaume Bal , Chenxi Guo , Francois Monard

We study the inverse problem of determining the magnetic field and the electric potential entering the Schr\"odinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed…

Analysis of PDEs · Mathematics 2016-05-24 Mourad Bellassoued , Yavar Kian , Eric Soccorsi

In this paper we study the inverse conductivity problem with partial data. Moreover, we show that, in dimension $n\geq 3$ the uniqueness of the Calder\'{o}n problem holds for the $C^{1}\bigcap H^{3/2, 2}$ conductivities.

Analysis of PDEs · Mathematics 2015-06-05 Guo Zhang

We investigate global uniqueness for an inverse problem for a nonlocal diffusion equation on domains that are bounded in one direction. The coefficients are assumed to be unknown and isotropic on the entire space. We first show that the…

Analysis of PDEs · Mathematics 2022-11-16 Yi-Hsuan Lin , Jesse Railo , Philipp Zimmermann

We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Amp\`ere equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the…

Analysis of PDEs · Mathematics 2025-10-14 Tony Liimatainen , Yi-Hsuan Lin

We address the stability issue in Calder\'on's problem for a special class of anisotropic conductivities of the form $\sigma=\gamma A$ in a Lipschitz domain $\Omega\subset\mathbb{R}^n$, $n\geq 3$, where $A$ is a known Lipschitz continuous…

Analysis of PDEs · Mathematics 2024-08-08 Sonia Foschiatti , Romina Gaburro , Eva Sincich

In this paper, we study the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data. Our contributions are twofold: first we introduce a Dirichlet-to-Neumann map for this operator and analyze an associated inverse problem;…

Analysis of PDEs · Mathematics 2026-04-09 Ravi Shankar Jaiswal , Pu-Zhao Kow , Suman Kumar Sahoo

We study the inverse problem for a semilinear wave equation on metric tree graphs. From the Dirichlet-to-Neumann map defined at all but one of the boundary vertices, we recover unknown connectivity of the graph, lengths of the edges, the…

Analysis of PDEs · Mathematics 2026-03-30 Sergei Avdonin , Matti Lassas , Jinpeng Lu , Medet Nursultanov , Lauri Oksanen

In this paper, we give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in…

Mathematical Physics · Physics 2015-10-23 Thierry Daudé , Niky Kamran , Francois Nicoleau

We prove that an $L^\infty$ potential in the Schr\"odinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace…

Analysis of PDEs · Mathematics 2019-10-10 Giovanni S. Alberti , Matteo Santacesaria

In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent,…

Analysis of PDEs · Mathematics 2007-08-27 Horst Heck , Jenn-Nan Wang

In this paper, we consider an inverse problem for a nonlinear wave equation with a damping term and a general nonlinear term. This problem arises in nonlinear acoustic imaging and has applications in medical imaging and other fields. The…

Analysis of PDEs · Mathematics 2023-03-01 Yang Zhang

In this paper I consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(\gamma\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $\gamma$ is a smooth, matrix valued, function with a…

Analysis of PDEs · Mathematics 2024-06-24 Cătălin I. Cârstea

We revisit the stability issue of determining the conductivity at the boundary from the corresponding Dirichlet-to-Neumann map. We discuss both the method based on singular solutions and the one built on the localized oscillating solutions.…

Analysis of PDEs · Mathematics 2021-12-30 Mourad Choulli

We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.

Analysis of PDEs · Mathematics 2019-05-07 Katya Krupchyk , Gunther Uhlmann

We study inverse boundary problems for third-order nonlinear tensorial perturbations of biharmonic operators on a bounded domain in $\mathbb{R}^n$, where $n\geq 3$. By imposing appropriate assumptions on the nonlinearity, we demonstrate…

Analysis of PDEs · Mathematics 2023-12-14 Sombuddha Bhattacharyya , Katya Krupchyk , Suman Kumar Sahoo , Gunther Uhlmann

We consider an inverse boundary value problem for the heat equation with a nonsmooth coefficient of conductivity which models the displacement of a moving body inside a nonhomogeneous background. We prove the uniqueness of the moving…

Analysis of PDEs · Mathematics 2022-01-24 Olivier Poisson

This paper is devoted to the investigation of inverse problems related to stationary drift-diffusion equations modeling semiconductor devices. In this context we analyze several identification problems corresponding to different types of…

Analysis of PDEs · Mathematics 2020-11-24 M. Burger , H. W. Engl , A. Leitão , P. A. Markowich