arXiv subjects

Anna Rozanova-Pierrat

Publications and source records attributed to Anna Rozanova-Pierrat.

At least 19 recordsLinked to original sources

arXiv2026

Tangential and normal traces for extension domains with non-Lipschitz boundaries

We generalize the classical vector-valued tangential and normal trace theory on Lipschitz domains to the setting of non-Lipschitz $H^1$-extension domains, which includes domains with fractal boundaries such as the Koch snowflake. We define and study these operators based on the surjectivity of the trace operator for elements of $H^1(Ω)$ and the Hilbert structure of the associated trace space. The normal trace operator is defined on $\Hdiv$ in $\mathbb{R}^n$. A generalized Stokes formula allows us to introduce the tangential trace operator on $\Hcurl$ and $\Hc^1(Ω)$ in two and three dimensions. Following the approach of Buffa, Costabel and Sheen (2002) for Lipschitz domains, we define two abstract tangential boundary spaces as images of these trace operators, establish their Hilbert structure, and construct an abstract rotation operator linking them, in place of the geometric rotation based on the normal vector. We also extend Costabel's (1991) coercive bilinear form approach for Maxwell's equations to non-Lipschitz domains, yielding a theory that supports the treatment of the Hodge-Dirac operator and Green's formulas and enables the solution of boundary-value problems for the $\rotv \rotv +1$ operator within the $H^1$-extension domains framework.

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arXiv2026

Integral Equation Methods for Scattering by Multifractal Obstacles

Caetano et al. (Proc. R. Soc. A. 481:20230650, 2025) have proposed a formulation for sound-soft acoustic scattering by a compact scatterer O $\subset$ Rn, in which the scattered field is represented as an acoustic Newtonian potential whose density is the solution of an operator equation on a compact set $Γ$ $\subset$ O. In the case that $Γ$ is Ahlfors-David d-regular (a d-set), for some d $\in$ (n--2, n], they show, moreover, that the operator equation can be interpreted as an integral equation, the integration with respect to d-dimensional Hausdorff measure, and present a convergent Galerkin scheme for numerical computation. In this paper we make a substantial extension of these results so that they apply to more realistic fractal scatterers that are multifractal, in the sense that they have spatially varying fractal dimension. Firstly, we provide, inspired by Claret et al. (J. Math. Pures Appl. 212:103888, 2026), an interpretation of this operator equation as an equation between a trace space on $Γ$ and its dual, and, in many cases, relate the density to a notion of the normal derivative of the scattered field on $Γ$. Secondly, we show that the operator equation is equivalent to an integral equation on $Γ$ whenever $Γ$ is the support of a Radon measure $μ$ such that: (i) the trace operator from H1(Rn) to L2($Γ$, $μ$) is continuous and; (ii) certain canonical singular integrals with respect to $μ$ are finite; and we characterise a large class of measures for which (i) and (ii) hold. Finally, we show that Galerkin methods based on finite element subspaces of L2($Γ$, $μ$) are convergent if and only if, additionally, C$\infty$\_0 (Rn\$Γ$) is dense in the kernel of the trace operator. These results apply, in particular, if $Γ$ is a finite union of d-sets with different values of d. In the case that each d-set is the attractor of an iterated function system of contracting similarities, we establish rates of convergence for the Galerkin method.

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arXiv2026

Layer potential operators for transmission problems on extension domains

We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincar{é} operators. These classes of domains contain Lipschitz domains, and also domains with fractal boundaries. Although our variational formulation does not involve any measures on the boundary, we recover the classical results in smooth domains by considering the surface measure on the boundary. We discuss properties of these operators and generalize basic results in imaging beyond the Lipschitz case.

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arXiv2025

Convergence of layer potentials and Riemann-Hilbert problem on extension domains

We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{é} operators, Calder{ó}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets.

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arXiv2025

On analysis of problems of mathematical physics with non-Lipschitz boundaries

We review recent advances in solving problems of mathematical physics on domains with irregular boundaries in Rn. We distinguish two frameworks: a measure-free approach in the image of the trace operator spaces for extension domains and an L2-approach depending on a d-upper regular boundary measure. In both cases, the domains can have boundaries with different Hausdorff dimensions inside the interval (n -- 2, n). The generalization of the Poincar{é}-Steklov/Dirichlet-to-Neumann operator for these two contexts is given. To illustrate the established convergence of spectral problems for elliptic operators with Robin boundary conditions, we give a numerical example of the stability of localized eigenfunctions, using results of M. Graffin.

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arXiv2025

Fractal curvatures and short-time asymptotics of heat content

The aim of our paper is twofold. First, we present new mathematical developments on the analysis of de Gennes' hypothesis on the short-time asymptotics of the heat content for bounded domains with smooth boundary and with fractal boundary. Second, we discuss new findings and concepts related to fractal curvatures for domains with fractal boundary. We conjecture that fractal curvatures and their scaling exponents will emerge in the short-time heat content asymptotics of domains with fractal boundary and the results discussed here are small initial contributions towards a resolution.

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arXiv2025

Poincar{é}-Steklov operator and Calder{ó}n's problem on extension domains

We consider Calder{ó}n's problem on a class of Sobolev extension domains containing non-Lipschitz and fractal shapes. We generalize the notion of Poincar{é}-Steklov (Dirichlet-to-Neumann) operator for the conductivity problem on such domains. From there, we prove the stability of the direct problem for bounded conductivities continuous near the boundary. Then, we turn to the inverse problem and prove its stability at the boundary for Lipschitz conductivities, which we use to identify such conductivities on the domain from the knowledge of the Poincar{é}-Steklov operator. Finally, we prove the stability of the inverse problem on the domain for W^{2,$\infty$} conductivities constant near the boundary. The last two results are valid in dimension n $\ge$ 3.

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arXiv2025

Parametric shape optimization for the convected Helmholtz equation with a generalized Myers boundary condition

We consider the convected Helmholtz equation with a generalized Myers boundary condition (a boundary condition of the second-order) and characterize the set of physical parameters for which the problem is weakly well-posed. The model comes from industrial applications to absorb acoustic noise in jet engines filled with absorbing liners (porous material). The problem is set on a 3D cylinder filled with a d-upper regular boundary measure, with a real 1 < d $\le$ 2. This setup leads to a parametric shape optimization problem, for which we prove the existence of at least one optimal distribution for any fixed volume fraction of the absorbing liner on the boundary that minimizes the total acoustic energy on any bounded wavenumber range.

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arXiv2024

Frequency range non-Lipschitz parametric optimization of a noise absorption

In the framework of the optimal wave energy absorption, we solve theoretically and numerically a parametric shape optimization problem to find the optimal distribution of absorbing material in the reflexive one defined by a characteristic function in the Robin-type boundary condition associated with the Helmholtz equation. Robin boundary condition can be given on a part or the all boundary of a bounded ($ε$, $\infty$)-domain of R n . The geometry of the partially absorbing boundary is fixed, but allowed to be non-Lipschitz, for example, fractal. It is defined as the support of a d-upper regular measure with d $\in$]n -2, n[. Using the well-posedness properties of the model, for any fixed volume fraction of the absorbing material, we establish the existence of at least one optimal distribution minimizing the acoustical energy on a fixed frequency range of the relaxation problem. Thanks to the shape derivative of the energy functional, also existing for non-Lipschitz boundaries, we implement (in the two-dimensional case) the gradient descent method and find the optimal distribution with 50% of the absorbent material on a frequency range with better performances than the 100% absorbent boundary. The same type of performance is also obtained by the genetic method.

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arXiv2023

Existence of optimal shapes for heat diffusions across irregular interfaces

We consider a heat transmission problem across an irregular interface -- that is, non-Lipschitz or fractal -- between two media (a hot one and a cold one). The interface is modelled as the support of a d-upper regular measure. We introduce the proprieties of the interior and exterior trace operators for two-sided extension domains, which allow to prove the well-posedness (in the sense of Hadamard) of the problem on a large class of domains, which contains regular domains, but also domains with variable boundary dimension. Then, we prove the convergence in the sense of Mosco of the energy form connected to the heat content of one of the domains and the heat transfer for ($ε$, $\infty$)-domains. Finally, we prove the existence of an optimal shape maximizing the heat energy transfer in a class of ($ε$, $\infty$)-domains, allowing fractal boundaries, while that optimum can generally not be reached in the class of Lipschitz domains.

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arXiv2022

Boundary value problems on non-Lipschitz uniform domains: Stability, compactness and the existence of optimal shapes

We study boundary value problems for bounded uniform domains in $\mathbb{R}^n$, $n\geq 2$, with non-Lipschitz (and possibly fractal) boundaries. We prove Poincaré inequalities with trace terms and uniform constants for uniform $(\varepsilon,\infty)$-domains within bounded common confinements. We then introduce generalized Dirichlet, Robin and Neumann problems for Poisson type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, and this also implies the norm convergence of the associated resolvents and the convergence of the corresponding eigenvalues and eigenfunctions. Based on our earlier work, we prove compactness results for parametrized classes of admissible domains, energy functionals and weak solutions. Using these results, we can verify the existence of optimal shapes in these classes.

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arXiv2021

Mixed boundary valued problem for linear and nonlinear wave equations in domains with fractal boundaries

The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in the largest natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in R^2 or R^3 we also validate the approximation of the solution of the Wester-velt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.

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arXiv2021

Non-Lipschitz uniform domain shape optimization in linear acoustics

We introduce new parametrized classes of shape admissible domains in R^n , n $\ge$ 2, and prove that they are compact with respect to the convergence in the sense of characteristic functions, the Hausdorff sense, the sense of compacts and the weak convergence of their boundary volumes. The domains in these classes are bounded ($ε$, $\infty$)-domains with possibly fractal boundaries that can have parts of any non-uniform Hausdorff dimension greater or equal to n -- 1 and less than n. We prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acous-tics. A by-product of our proof is the result that the class of bounded ($ε$, $\infty$)-domains with fixed $ε$ is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains.

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arXiv2020

Optimal absorption of acoustical waves by a boundary

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with d-set boundaries (N -- 1 $\le$ d < N). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume).

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arXiv2020

Dirichlet boundary valued problems for linear and nonlinear wave equations on arbitrary and fractal domains

The weak well-posedness results of the strongly damped linear wave equation and of the non linear Westervelt equation with homogeneous Dirichlet boundary conditions are proved on arbitrary three dimensional domains or any two dimensional domains which can be obtained by a limit of NTA domains caractarized by the same geometrical constants. The two dimensional result is obtained thanks to the Mosco convergence of the functionals corresponding to the weak formulations for the Westervelt equation with the homogeneous Dirichlet boundary condition. The non homogeneous Dirichlet condition is also treated in the class of admissible domains composed on Sobolev extension domains of $\mathbb{R}^n$ with a $d$-set boundary $n-1\leq d<n$ preserving Markov's local inequality.The obtained Mosco convergence also alows to approximate the solution of the Westervelt equation on an arbitrary domain by solutions on a converging sequence of domains without additional conditions on their boundary regularity in $\mathbb{R}^3$, or on a converging sequence of NTA domains in $\mathbb{R}^2$.

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arXiv2020

Models of Nonlinear Acoustics Viewed as Approximations of the Kuznetsov Equation

We relate together different models of non linear acoustic in thermo-ellastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the L 2 norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.

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arXiv2020

Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

We present a survey of recent results of the functional analysis allowing to solve PDEs in a large class of domains with irregular boundaries. We extend the previously introduced concept of admissible domains with a d-set boundary on the domains with the boundaries on which the measure is not necessarily Ahlfors regular d-measure. This gives a generalization of Rellich-Kondrachov theorem and the compactness of the trace operator, allowing to obtain, as for a regular classical case the unicity/existence of weak solutions of Poisson boundary valued problem with the Robin boundary condition and to obtain the usual properties of the associated spectral problem.

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arXiv2018

Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropicsystems

The derivation of different models of non linear acoustic in thermo-ellastic media as the Kuznetsov equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) from an isentropic Navier-Stokes/Euler system is systematized using the Hilbert type expansion in the corresponding perturbative and (for the KZK and NPE equations) paraxial ansatz . The use of small, to compare to the constant state perturbations, correctors allows to obtain the approximation results for the solutions of these models and to estimate the time during which they keep closed in the L2 norm. The KZK and NPE equations are also considered as paraxial approximations of the Kuznetsov equation, which is a model obtained only by perturbations from the Navier-Stokes/Euler system. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. In the aim to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Navier-Stokes system and the Kuznetsov equation in a half-space with periodic in time initial and boundary data) were obtained.

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