arXiv subjects

Mohammed Barkatou

Publications and source records attributed to Mohammed Barkatou.

14 recordsLinked to original sources

arXiv2026

Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem in Riemannian Geometry

We generalize to the setting of compact Riemannian manifolds a recent result of Barkatou on quadrature surfaces. Following the geometric and variational framework recently developed by Djité and Seck, we formulate the quadrature surface free boundary problem as a shape optimization problem on a compact Riemannian manifold. Using the Riemannian $RC$-GNP condition introduced in \cite{DjiteSeck2026} and the stability results established therein, we prove that the quadrature surface problem $QS(f,k)$ admits a solution strictly containing the totally convex hull $C$ of the support of $f$ if and only if the following integral condition holds: \[ \int_C f(x)\,dv(g) > k |\partial C|_g, \] where $dv(g)$ is the Riemannian volume element and $|\partial C|_g$ is the perimeter of $C$ with respect to the metric $g$. This work extends the results of Barkatou et al. (2005) by replacing the Euclidean space with a Riemannian manifold. We also provide explicit examples on the round sphere where the condition can be verified explicitly. This paper complements the recent works \cite{DjiteSeck2026} and \cite{DjiteSeck2026b} by establishing a sharp necessary and sufficient condition in the spirit of the Euclidean result of Barkatou.

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arXiv2026

Necessary and Sufficient Condition of Existence for the $p$-Laplacian Quadrature Surfaces Free Boundary Problem in Riemannian Geometry

We generalize the necessary and sufficient condition of existence for the quadrature surfaces free boundary problem to the case of the $p$-Laplacian operator in Riemannian geometry. Following the shape optimization approach of Barkatou for the Euclidean case and the Riemannian framework recently developed by Djité and Seck, we formulate the $p$-Laplacian quadrature surface problem as a shape optimization problem on a compact Riemannian manifold. Using the Riemannian $RC$-GNP condition, the stability results, and the $p$-Laplacian shape derivative formula from \cite{DjiteSeck2026b}, we prove that the $p$-Laplacian quadrature surface problem $QS_p(f,k)$ admits a solution strictly containing the totally convex hull $C$ of the support of $f$ if and only if \[ \int_C f(x)\,dv(g) > k^{p-1} |\partial C|_g, \] for $1 < p < \infty$, under suitable regularity and stability assumptions. This extends the Euclidean result of Barkatou for $p=2$ and the Riemannian result of Djité and Seck to the nonlinear $p$-Laplacian setting. Explicit examples on the round sphere are provided, including the computation of the radial $p$-harmonic solution and the boundary flux.

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arXiv2026

Geometric Gradient Flows from Elliptic Level Sets: Normal Decomposition and Reflection Dynamics

We investigate the asymptotic geometry of shifting superlevel sets $Ω_t = \{x \in Ω: u(x) > t\}$ generated by solutions to the elliptic Dirichlet problem $-Δu = f$ in $Ω$, where the non-negative source $f \not\equiv 0$ is compactly supported within a strictly convex inner core $C \subset Ω$. Under a quantitative radial monotonicity condition, each boundary $\partialΩ_t$ is a smooth normal graph over $\partial C$ characterized by a thickness function $d_t \in C^{1,α}(\partial C)$ tracking $d_0$ as $t \to 0$.A central contribution is a rigorous decomposition of the inward unit normal field along the level surfaces: $\mathbf{n}_{Ω_t} = ν- \nabla_{\partial C} d_t + \mathcal{G} + \mathcal{P}$, where $ν$ is the static radial normal, $-\nabla_{\partial C} d_t$ is the kinematic driving vector, and $\mathcal{G}, \mathcal{P}$ are curvature and PDE Hessian remainder operators.In a thin-shell configuration $(\Vert{}d_0\Vert{}_{C^1} \ll 1)$, we formalize a discrete specular point-reflection mapping $F_n$ on $\partial C$. We prove the tangential displacement satisfies $F_n(p) - p = -2d_n(p)\nabla_{\partial C}d_n(p) + R_n(p)$, with quadratic control $\Vert{}R_n\Vert{}_{L^\infty} \le C\Vert{}d_n\Vert{}_{C^1}^2$. Using the energy $\mathcal{E}(t) = \int_{\partial C} d_t^2 \, d\mathcal{H}^{N-1}$, we show these orbits approximate a continuous gradient flow driven by $+\nabla_{\partial C} d_{\tilde{t}}(p)$ to first order. Finite element computations (FEniCS) validate these convergence rates.

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arXiv2026

Global Convergence of the Return Dynamics in the Class $\mathcal{O}_C$

This research investigates a geometric dynamical mechanism within a specific class of domains that contain a fixed convex core. By using a radial structure that links the boundaries of the core and the outer domain via a thickness function, the authors introduce a "return map." This map is constructed by projecting a point from the core to the outer boundary and then returning to the core by following the inward normals. The main results demonstrate that this motion behaves, to a first-order approximation, like an adaptive gradient descent for the domain's thickness. In other words, the system naturally evolves toward areas where the thickness is minimized. The study establishes that the fixed points of this dynamics coincide with the critical points of the thickness function. Additionally, the authors quantify the convergence rate, prove the regularity of the thickness function in relation to the boundary geometry, and establish a structural equivalence between the two surfaces under specific curvature conditions. Ultimately, this work links the dynamical properties of the system to the geometric smoothness of the studied shapes.

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arXiv2026

Thickness functions for elliptic level sets: gradient formulas, normal expansions, and geometric remarks

We study the geometry of superlevel sets $Ω_t = \{u > t\}$ of solutions to $-Δu = μ$ with $μ\geq 0$ compactly supported in a convex core $C \subset Ω$. Under the radial monotonicity lemma (due to Shahgholian), each level set is a normal graph over $\partial C$ with thickness function $d_t$. We derive an exact formula for the tangential gradient of $d_t$ and, under a quantitative small-thickness hypothesis $\|d_t\|_{C^1(\partial C)} \ll 1$, an asymptotic expansion of the unit normal to $Γ^t = \partial Ω_t$. We discuss the relation with Shahgholian's theorem and give examples showing that the geometric normal property (GNP) does not imply that $d_t$ is constant, even in the small-thickness regime. This work provides a geometric language for studying elliptic level sets, without claiming a new proof of Shahgholian's theorem.

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arXiv2026

Sharp existence conditions and geometric inheritance for overdetermined free boundary problems of Laplacian and bi-Laplacian type

This paper provides necessary and sufficient conditions for the existence of free boundaries in overdetermined problems for the Laplacian, and sufficient conditions for the bi-Laplacian, when the overdetermined boundary condition is non-constant. Using classical integral inequalities (Cauchy-Schwarz, Hölder, Hardy, eigenvalue bounds, Pohozaev and Reilly identities), we derive existence results for a broad class of free boundary problems arising in potential theory, plate theory, electromagnetism, and shape optimization. A regularity result for minimizers in the $C$-GNP class is established using the thickness function and the Wiener criterion, based on the geometric description of cusp points given in \cite{Barkatou2002}. We provide a new, self-contained geometric result: for almost every $t$, the level sets of the solution inherit the $C$-GNP property. This inheritance theorem justifies the variational framework and guarantees that the entire foliation generated by the state function remains within the admissible class. New results include refined estimates via interpolation inequalities, stability under perturbations, and connections with isoperimetric inequalities. The physical interpretation of the bi-Laplacian problem $\mathcal{B}(f,g)$ in the Kirchhoff-Love theory of thin plates is emphasized.

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arXiv2026

The Return Map in the Class $\mathcal{O}_C$: Geometry, Dynamics, and Thickness Regularity

We investigate a geometric dynamical mechanism arising in the class $\mathcal{O}_C$ of domains containing a fixed convex set $C$ and satisfying two geometric normals properties introduced by Barkatou \cite{Barkatou2002}. The first property induces a radial structure linking the boundaries $\partial C$ and $\partial Ω$ through a thickness function $d:\partial C\to \R_{+}$. Using this structure, we introduce a natural return map obtained by composing the radial projection from $\partial C$ to $\partial Ω$ with the map that follows inward normals from $\partial Ω$ back to $C$. This construction generates a discrete dynamical system on $\partial C$. We prove that the return map admits the first-order expansion \[ F(c) = c - 2d(c)\nablaTCd(c) + \text{higher order terms}, \] with explicit remainder estimates. This reveals that the induced dynamics behaves, to leading order, like an adaptive gradient descent for the thickness function. The expansion incorporates curvature corrections arising from the convex core $\partial C$ \cite{Schneider2014}. Consequently, the fixed points of the dynamics coincide with the critical points of $d$, and the iteration admits a natural Lyapunov structure \cite{Smale1961}. We further quantify the convergence rate, provide a rigorous error bound between the discrete and continuous gradient flows, and show that the product condition $dκ_i < 1$ can be relaxed. We then analyze the regularity of the thickness function and its relationship to the regularity of the outer boundary $\partial Ω$. We show that the thickness function inherits the regularity of $\partial Ω$ and vice versa, and we establish a bilipschitz equivalence between the two boundaries under a quantitative curvature condition. These results link the dynamical properties of the return map to the geometric smoothness of the admissible domains.

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arXiv2026

Inverse Problems for the Return Map in the Class ( $\mathcal{O}_C$ ): Reconstruction and Identifiability

We analyze the inverse problem of recovering geometric information from the return map induced by a round-trip between a convex core C and an admissible domain. This process defines a discrete dynamical system on the boundary of C governed by a thickness function d. We prove that the return map determines the gradient structure of d, including its critical points, Morse indices, and basin decomposition. At second order, the geometry is encoded indirectly through a curvature-dependent operator acting on the Hessian of d, revealing a coupling between thickness and curvature. This leads to intrinsic non-uniqueness in the inverse problem, due to scaling and dynamical equivalences. However, uniqueness (up to these ambiguities) can be recovered under additional geometric constraints such as symmetry or isotropy.

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arXiv2026

Symmetry and Qualitative \& Quantitative Stability for a Class of Overdetermined Problems in C-GNP Domains with Source Supported in the Core

We introduce a unified geometric framework for domains satisfying a geometric normal property (C-GNP) relative to a strictly convex set $C$. Under the fundamental assumption that the source $f$ is supported within the core $C$, we establish the stability of superlevel sets for elliptic equations and prove a rigid symmetry result for a classical Serrin-type problem via a method that avoids moving planes, relying instead on geometric monotonicity and the Hopf boundary lemma. We then extend this analysis to a coupled biharmonic overdetermined problem $\mathrm{P}(κ)$ with source supported in the core. Using the compactness properties of the C-GNP class and the stability of thickness functions under Hausdorff convergence, we prove a qualitative stability theorem: if the overdetermined condition is approximately satisfied in $L^2$ norm, the domain converges in the Hausdorff sense to the unique ball solution. Furthermore, we establish a quantitative stability estimate: there exists a constant $C$ such that \[ ρ_e - ρ_i \le C \big\| |\nabla u| |\nabla v| - κ\big\|_{L^2(\partial Ω)}^{τ_N}, \] with $τ_2 = 1$, $τ_3$ arbitrarily close to 1, and $τ_N = 2/(N-1)$ for $N \ge 4$ in the general case. For convex domains, we improve the exponent to $τ_N = 4/(N+1)$ via a weighted Reilly identity. The proof relies on Reilly-type integral identities adapted to the coupled system and Hardy-Poincaré inequalities tailored to the geometry.

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arXiv2026

From Tangency to Fractals: Quadratic Dynamics in Nested Convex Geometry

We study the dynamics generated by return maps associated with nested convex bodies and growing domains satisfying the geometric normal property in the plane. These maps are defined by transporting boundary points along normal directions to the surrounding domain and projecting them back onto the boundary of a subsequent convex set. We introduce a tangency condition between consecutive convex sets and show that it cancels the linear term in the local expansion of the transition operators. As a result, the dynamics near tangency points is governed by a quadratic normal form with an explicit coefficient depending on curvature and second order geometric data. This quadratic tangency law constitutes the central mechanism of the system. We prove that this nonlinear contraction leads to super exponential convergence toward the tangency set. In logarithmic coordinates, the dynamics becomes approximately affine, which allows for an interpretation in terms of iterated function systems (IFS) and explains the emergence of fractal limit sets. The theory is illustrated by several geometric configurations. Ford circles reveal a connection with continued fractions, nested ellipses yield Cantor-type limit sets, and configurations such as stadia and rounded triangles demonstrate the coexistence of linear and quadratic regimes. In purely quadratic settings with m independent branches, the limit set has similarity dimension in logarithmic coordinates, and an estimation of its Hausdorff dimension. From a broader perspective, the combination of symbolic branching and nonlinear contraction suggests potential connections with geometry, in particular in hybrid classical quantum information processing frameworks.

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arXiv2026

Geometry, Dynamics and Topology of Thickness Landscape: A Morse-Theoretic Analysis of the Return-Map in the Class $\mathcal{O}_{C}$

We study the geometric and dynamical structure induced by the return map associated with domains in the class $\mathcal{O}_{C}$. This map, defined through a geometric round-trip between the convex core and the outer boundary, generates a discrete dynamical system on the boundary $\partial C$. Building on previous results establishing global convergence of the return dynamics, we show that equilibria of the return map coincide with the critical points of the thickness function. This identification allows us to apply Morse-theoretic tools to derive global constraints on the dynamics. In particular, we obtain lower bounds on the number of equilibria in terms of the Betti numbers of $\partial C$, as well as a global balance relation governed by the Euler characteristic. We further analyze the local behavior of the return map near equilibria. Using the differentiability of the return map inherited from the radial and reciprocal constructions, we derive a first-order expansion in which the linearization is governed by the Hessian of the thickness function and an operator arising from the geometry of the return map. This leads to an operator-valued generalization of the previously observed scalar structure, revealing that the dynamics behaves as an anisotropic gradient-like iteration rather than a purely isotropic descent. Near nondegenerate minima, we prove a quantitative descent estimate and local linear convergence under a spectral condition. Under aligned nonlocal geometry, the sign of the curvature gap between the convex core and the outer boundary determines whether the induced dynamics is contracting, neutral, or expanding in each principal direction. Finally, we discuss extensions beyond the Morse setting, including the Morse-Bott case, and highlight connections between the geometry of the domain, the topology of $\partial C$, and the structure of the induced dynamics.

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arXiv2023

Existence of free boundaries for some overdetermined-value problems

The aim of this paper is first to give necessary and sufficient condition of existence (of free boundaries) for both Laplacian and bi-Laplacian operators in the case where the overdetermined condition is not constant. second, by using some classical ineqhalities, we get existence for several (other) overdetermined free boundary problems.

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arXiv2020

Existence and Symmetry results for some overdetermined free boundary problems

In this paper, we prove that a domain which verifies some integral inequality is either (strictly) contained in the solution of some free boundary problem, or it coincides with an $N$-ball. We also present new overdetermined value problems which have an $N$-ball as a solution. To reach our results, we use an integral identity which involves the domain derivative of the solution of Dirichlet problem.

math.AP↗

arXiv2020

Quelques propriétés qualitatives pour le problème de surfaces de quadrature dans le plan

In this paper, we begin by giving a necessary and sufficient condition of existence for the quadrature surfaces problem in the case where the term source is a uniform density supported by a segment. Then, we use the Steiner continuous symmetrization to prove that the obtained solution is symmetric with respect the x-coordinate axis and that its boundary is analytic. ----------- Dans ce papier, nous commençons par donner une condition nécessaire et suffisante d'existence de solutions pour le problème de surfaces de quadrature dans le cas où le terme source est une densité supportée par un segment. Ensuite, en utilisant la symétrisation de Steiner continue, nous montrons que la solution obtenue est symétrique par rapport à l'axe des abscisses et que son bord est analytique.

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