arXiv subjects

Cătălin I. Cârstea

Publications and source records attributed to Cătălin I. Cârstea.

At least 19 recordsLinked to original sources

arXiv2026

The conformal Poisson embedding method for the anisotropic Calderón problem with a potential

We prove that the local Dirichlet-to-Neumann map of $Δ_g+q$ determines a compact connected real-analytic Riemannian manifold of dimension at least three and a real-analytic real-valued potential, up to a diffeomorphism fixing the observed boundary and the conformal gauge of the equation. The result is new even when $q$ is constant, i.e. for the fixed-frequency (Helmholtz) equation $Δ_g u+λu=0$. The measurements are restricted to an arbitrary nonempty open boundary set $Γ$. The proof develops conformal Poisson embeddings method and uses coordinates given by quotients of solutions.

math.AP↗

arXiv2026

Uniqueness for the inverse Schrödinger problem in the plane with $L^p$ potentials, $p>1$

Let $Ω\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-Δ+V$ uniquely determines every complex-valued potential $V\in L^p(Ω)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. The proof uses Bukhgeim's quadratic-phase solutions and an average of Alessandrini's identity over the phase center. After separating the Neumann-series tails, we show that each remaining Born term tends to zero. The fixed-order estimates combine bounds for the absolute kernels with oscillatory cancellation and a duality argument for the product of the two Cauchy transforms.

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arXiv2026

Relative formalization in Isabelle/HOL of a result in inverse problems

This reports on an experiment in autoformalization of arxiv:2606.15977, an inverse problems result for piecewise polynomial anisotropic conductivities, using Isabelle/HOL. The proof of the result is relative to a number of external results which were deemed to be well known and generally accepted to be true. The formalization files are made available at a GitHub repository. Translation issues are discussed.

math.AP↗

arXiv2026

The Calderón problem for piecewise polynomial anisotropic conductivities on partitions with many flat faces

We prove uniqueness for a finite-dimensional anisotropic Calderón problem in dimensions $n\geq 3$. The unknown conductivity is a symmetric uniformly elliptic matrix field whose restriction to each cell of a known finite partition is polynomial; jumps across cell interfaces are allowed. We assume that the cells can be ordered so that each successive cell is accessible through sufficiently many flat faces. Under this assumption, the local Dirichlet-to-Neumann map on an exterior boundary patch determines every polynomial piece. The proof first recovers one polynomial matrix from its boundary invariants on several flat faces and then propagates the required local data through the recovered cells by unique continuation and Runge approximation. A general finite-dimensional analytic stability theorem also gives Hölder stability on compact admissible parameter sets.

math.AP↗

arXiv2026

An inverse free boundary problem

We study inverse problems for the elliptic and parabolic obstacle problems from boundary measurements. For the classical elliptic obstacle problem with strictly superharmonic obstacle function, we show that the Dirichlet-to-Neumann map admits a one-sided linearization at every boundary datum lying strictly above the obstacle. The linearized map is the Dirichlet-to-Neumann map for a rough Dirichlet problem on the a priori unknown non-contact set. We show that these linearized Cauchy data uniquely determine the non-contact set up to set of Sobolev $2$-capacity zero and consequently determine the obstacle. Our result applies to the inverse problem for a parabolic obstacle problem where both the coefficient and the obstacle function are time-independent by reducing to the elliptic inverse problem.

math.AP↗

arXiv2026

An inverse source problem for the Monge--Ampere equation from large boundary data

We study an inverse source problem for the Monge--Ampere equation \[ \det D^2u=f(x) \] on a bounded smooth uniformly convex domain. In the smooth classical regime, we prove that the Dirichlet-to-Neumann map associated with convex solutions determines the positive source uniquely. The proof uses a family of large boundary values and reduces the inverse source problem to the injectivity of the Euclidean X-ray transform.

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arXiv2026

A note on several inverse problems with generally random coefficients

We consider several inverse problems for elliptic equations whose coefficients are random, without imposing a special probabilistic structure on the randomness. The main body treats the Schrödinger equation. We compare what can be recovered from the full law of the Dirichlet-to-Neumann map, from its expectation, from finitely many joint moments of its boundary bilinear form, and from the averaged interior Green's operator. We obtain both positive and negative results. That the full law of the Dirichlet-to-Neumann map determines the law of the random potential is almost trivial. However, the expected Dirichlet-to-Neumann map and, more generally, any fixed finite hierarchy of its boundary moments need not determine even the mean potential. In contrast, the averaged Schrödinger Green's operator determines the pointwise mean and variance of the potential. In a two-atom model it determines all pointwise moments of the two-point law. The appendices contain the corresponding results for the conductivity equation.

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arXiv2026

Hölder Stability from Exact Uniqueness for Finite-Dimensional Analytic Inverse Problems

We prove a stability theorem for finite-dimensional analytic inverse problems. Let $U\subset\R^m$ be an open parameter set, let $F(p)$ be a boundary measurement operator, and let $R(p)$ be the finite-dimensional quantity to be recovered. If $F$ is real analytic and \[ F(p)=F(q)\quad\Longrightarrow\quad R(p)=R(q), \] then $R$ satisfies a Hölder stability estimate on every compact subset of $U$. The proof uses a Hilbert--Schmidt scalarization of the operator equation $F(p)=F(q)$ and the Łojasiewicz distance inequality. We also prove that, after fixing countable dense families of boundary inputs and tests, finitely many scalar matrix elements of the data give the same Hölder recovery on compact parameter sets. This finite-measurement conclusion is qualitative: the proof does not give an effective measurement list, exponent, or constant. The finite-measurement statement follows from finite determinacy of real analytic zero sets. We apply the result to local Neumann-to-Dirichlet data for piecewise constant anisotropic conductivities and to localized Dirichlet-to-Neumann data for piecewise homogeneous anisotropic elasticity.

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arXiv2026

Inverse boundary value problems for certain doubly nonlinear parabolic and elliptic equations

We consider an inverse boundary value problem for the doubly nonlinear parabolic equation \[ ε(x)\partial_t u^m-\nabla\cdot\bigl(γ(x)|\nabla u|^{p-2}\nabla u\bigr)=0 \quad\text{in }(0,T)\timesΩ, \] where $p\in(1,\infty)\setminus\{2\}$, $m>0$, and the coefficients $ε$ and $γ$ are positive. Our first main result shows that when $m>p-1$, the lateral Cauchy data determine both coefficients. The proof proceeds by reducing the parabolic inverse problem to an inverse problem for the nonlinear elliptic equation \[ -\nabla\cdot\bigl(γ|\nabla w|^{p-2}\nabla w\bigr)+Vw^m=0 \quad\text{in }Ω. \] Our second main result establishes uniqueness for the pair $(γ,V)$ from the nonlinear Dirichlet-to-Neumann map of this elliptic equation. The argument has two steps. First, asymptotic expansions of the elliptic Dirichlet-to-Neumann map recover the weighted $p$-Laplacian Dirichlet-to-Neumann map, and and from it the coefficient $γ$. Second, once $γ$ is known, linearization at a noncritical background solution yields recovery of $V$. In dimension two we work under a simply connectedness assumption on the domain, while in dimensions $n\ge 3$ we assume that the conductivity is invariant in one known direction.

math.AP↗

arXiv2025

Reconstruction of coefficients in the double phase problem

The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $Ω$, $u=f$ on $\partialΩ$, from the Dirichlet to Neumann (DN) map $Λ_a$. We show that this can be achieved, when the coefficient $a$ has Hölder continuous first order derivatives and the exponents satisfy $1<p\neq q<\infty$. Our reconstruction method relies on a careful analysis of the asymptotic behavior of the solution $u$ to the double phase problem with small or large Dirichlet datum $f$ (depending on the ordering of $p$ and $q$) as well as the related DN map $Λ_a$. As is common for inverse boundary value problems, we need a sufficiently rich family of special solutions to a related partial differential equation, which is independent of the coefficient one aims to reconstruct (in our case to the $p$-Laplace equation). We construct such families of solutions by a suitable linearization technique.

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arXiv2024

Prescribed nonlinearity helps in an anisotropic Calderón-type problem

In this paper I consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(γ\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $γ$ is a smooth, matrix valued, function with a uniform lower bound. I show that boundary Dirichlet and Neumann data for this equation, in the form of a Dirichlet-to-Neumann map, determine the coefficient matrix uniquely, in dimension 3 and higher. This stands in contrast to the classical linear anisotropic Calderón problem where there is a known obstruction to uniqueness due to the invariance of the boundary data under transformations of the equation via any boundary fixing diffeomorphism.

math.AP↗

arXiv2024

The Calderón problem on Riemannian surfaces and of minimal surfaces

In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $Δ_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a priori that the surface is a planar domain [36] or that the geometry is a priori known [29]. We will then apply this result to study a geometric inverse problem for determining minimal surfaces embedded in $3$-dimensional Riemannian manifolds. In particular we will show that knowledge of the volumes of embedded minimal surfaces determine not only their topological and differential structure but also their Riemannian structure as an embedded hypersurface. Such geometric inverse problems are partially inspired by the physical models proposed by the AdS/CFT correspondence. The crucial ingredient in removing the planar domain assumption is the determination of the boundary trace of holomorphic functions from knowledge of the Dirichlet-Neumann map of $Δ_g +q$. This requires a new type of argument involving Carleman estimates and construction of CGO whose phase functions are not Morse as in the case of [29]. We anticipate that these techniques could be of use for studying other inverse problems in geometry and PDE.

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arXiv2024

Two uniqueness results in the inverse boundary value problem for the weighted p-Laplace equation

In this paper we prove a general uniqueness result in the inverse boundary value problem for the weighted p-Laplace equation in the plane, with smooth weights. We also prove a uniqueness result in dimension 3 and higher, for real analytic weights that are subject to a smallness condition on one of their directional derivatives. Both results are obtained by linearizing the equation at a solution without critical points. This unknown solution is then recovered, together with the unknown weight.

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arXiv2023

An inverse problem for general minimal surfaces

In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if $Σ$ is a $2$-dimensional embedded minimal surface, then the knowledge of the Dirichlet-to-Neumann map associated to the minimal surface equation determines $Σ$ up to an isometry. Without the topological condition, we show that a conformal factor of a general minimal surface $Σ$ can be recovered. We develop a semiclassical nonlinear calculus for complex geometric optics solutions, which allows an efficient error analysis for multiplication of the correction terms of the solutions. The calculus is independent of the application to the minimal surface equation and we expect it to have applications in various inverse problems for nonlinear equations in dimension $2$, in both $\mathbb{R}^2$ and geometric settings. Other applications of the results include generalized boundary rigidity problem and the AdS/CFT correspondence in physics.

math.AP↗

arXiv2023

An inverse boundary value problem for the inhomogeneous porous medium equation

In this paper we establish uniqueness in the inverse boundary value problem for the two coefficients in the inhomogeneous porous medium equation $ε\partial_tu-\nabla\cdot(γ\nabla u^m)=0$, with $m>1$, in dimension 3 or higher, which is a degenerate parabolic type quasilinear PDE. Our approach relies on using a Laplace transform to turn the original equation into a coupled family of nonlinear elliptic equations, indexed by the frequency parameter ($1/h$ in our definition) of the transform. A careful analysis of the asymptotic expansion in powers of $h$, as $h\to\infty$, of the solutions to the transformed equation, with special boundary data, allows us to obtain sufficient information to deduce the uniqueness result.

math.AP↗

arXiv2022

Remarks on the anisotropic Calderón problem

We show uniqueness results for the anisotropic Calderón problem stated on transversally anisotropic manifolds. Moreover, we give a convexity result for the range of Dirichlet-to-Neumann maps on general Riemannian manifolds near the zero potential. Finally, we present results for Calderón type inverse problems associated to semilinear elliptic equations on general Riemannian manifolds.

math.AP↗

arXiv2022

An inverse problem for the Riemannian minimal surface equation

In this paper we consider determining a minimal surface embedded in a Riemannian manifold $Σ\times \mathbb{R}$. We show that if $Σ$ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine $Σ$ up to an isometry.

math.AP↗

arXiv2021

An inverse problem for the porous medium equation with partial data and a possibly singular absorption term

In this paper we prove uniqueness in the inverse boundary value problem for the three coefficient functions in the porous medium equation with an absorption term $ε\partial_t u-\nabla\cdot(γ\nabla u^m)+λu^q=0$, with $m>1$, $m^{-1}<q<\sqrt{m}$, with the space dimension 2 or higher. This is a degenerate parabolic type quasilinear PDE which has been used as a model for phenomena in fields such as gas flow (through a porous medium), plasma physics, and population dynamics. In the case when $γ=1$ a priori, we prove unique identifiability with data supported in an arbitrarily small part of the boundary. Even for the global problem we improve on previous work by obtaining uniqueness with a finite (rather than infinite) time of observation and also by introducing the additional absorption term $λu^q$.

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