arXiv subjects

Siavash Sadeghi

Publications and source records attributed to Siavash Sadeghi.

6 recordsLinked to original sources

arXiv2026

A counterexample to Kenig's conjecture for the Laplace double-layer operator

Layer potentials provide a classical approach to boundary value problems for Laplace's equation on Lipschitz domains. Kenig's 1994 spectral-radius conjecture for the double-layer operator would ensure operator-norm convergence of the associated Neumann series on mean-zero $L^2$ densities when the boundary is connected. We disprove this conjecture by constructing a bounded simply connected planar Lipschitz domain whose double-layer operator on arclength $L^2$ has essential spectral radius strictly greater than $1/2$. More precisely, for every $t>1/2$ sufficiently close to $1/2$, we obtain such a domain with $\pm i t$ in its Fredholm essential spectrum. The construction starts from smooth graphs whose shapes repeat under translation. In the limit of separated scales, refinement makes solutions of adjoint resolvent equations grow with fixed forcing. The graph slopes remain uniformly bounded. A computer-assisted certificate proves this growth through an inequality for Hermitian $2\times2$ matrices. Its strict margin at $- i/2$ persists at nearby spectral parameters. Normalisation and a Floquet transform then give compactly supported densities with small residuals on the full graphs. We insert rescaled segments of successive graphs into one bounded boundary, where these densities form a weakly null sequence of approximate eigenvectors. The same spectral conclusion holds on a single periodic Lipschitz graph. The certificate combines continuous estimates, exact rational arithmetic and rigorous interval enclosures.

math.AP↗

arXiv2026

A sharp isoperimetric inequality for the Neumann--Poincaré operator in every dimension

Let $Ω\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,α}$, where $0<α<1$. For the adjoint Neumann--Poincaré operator $K^*_{\partialΩ}$, normalised so that its distinguished eigenvalue is $1/2$, let $λ_j^+(Ω)$ denote the upper min--max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}λ_j^+(Ω)\ge \frac{d-2}{2}.$ It follows that $λ_1^+(Ω)\ge \frac{d-2}{2d},$ with equality if and only if $Ω$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin--Shtrikman inequality $|Ω|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partialΩ}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $Ω$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.

math.AP↗

arXiv2026

A computer-assisted counterexample to the planar Berenstein conjecture

Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $Ω$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in(27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^ω(\overlineΩ)$ satisfying $(Δ+k^2)u=0$ in $Ω$, with $u=0, \partial_νu=\text{constant}\ne0$ on $\partialΩ$. Thus the overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{σ_{\partialΩ}}(kω)=0$ for $ω\in\mathbb S^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton--Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu--Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior--boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.

math.AP↗

arXiv2026

The cut-off resolvent can grow arbitrarily fast in obstacle scattering

We consider time-harmonic acoustic scattering by a compact sound-soft obstacle $Γ\subset \mathbb{R}^n$ ($n\geq 2$) that has connected complement $Ω:= \mathbb{R}^n\setminus Γ$. This scattering problem is modelled by the inhomogeneous Helmholtz equation $Δu + k^2 u = -f$ in $Ω$, the boundary condition that $u=0$ on $\partial Ω= \partial Γ$, and the standard Sommerfeld radiation condition. It is well-known that, if the boundary $\partial Ω$ is smooth, then the norm of the cut-off resolvent of the Laplacian, that maps the compactly supported inhomogeneous term $f$ to the solution $u$ restricted to some ball, grows at worst exponentially with $k$. In this paper we show that, if no smoothness of $Γ$ is imposed, then the growth can be arbitrarily fast. Precisely, given some modestly increasing unbounded sequence $0<k_1<k_2<\ldots$ and some arbitrarily rapidly increasing sequence $0<a_1<a_2<\ldots$, we construct a compact $Γ$ such that, for each $j\in \mathbb{N}$, the norm of the cut-off resolvent at $k=k_j$ is greater than $a_j$.

math.AP↗

arXiv2026

Integral equation methods for scattering by general compact obstacles: wavenumber-explicit estimates

There has been significant recent interest in understanding the dependence on the wavenumber, $k$, of boundary integral operators (BIOs), supported on some set $Γ\subset \mathbb{R}^n$, that arise in the solution of the Helmholtz equation, $Δu + k^2 u=0$. Recently, for the Dirichlet boundary value problem with data $g$, Caetano et al (Proc. R. Soc. A, 481:20230650, 2025) have proposed a novel integral equation $A_kϕ=g$ that applies for arbitrary compact $Γ$. In this paper we study the dependence of $A_k$ on $k$, showing that, for $k\geq k_0>0$, $\|A_k\|\leq ck$ while $\|A_k^{-1}\| \leq c'k$ if $Γ$ is star-shaped, where $c, c'>0$ depend only on $k_0$ and $Γ$. Amongst other bounds we show that: (i) on the one hand, given any mildly increasing unbounded positive sequence $(k_m)$ and any unbounded sequence $(a_m)$, there exists $Γ$, with connected complement, such that $\|A_{k_m}^{-1}\|\geq a_m$ for every $m$; (ii) on the other hand, for every $Γ\subset \mathbb{R}^n$ and $k_0,\varepsilon, δ>0$, there exists $c>0$ and $E\subset [k_0,\infty)$, with Lebesgue measure $m(E)\leq \varepsilon$, such that $\|A_{k}^{-1}\|\leq c k^{2n+2+δ}$ on $[k_0,\infty)\setminus E$, i.e., the growth of $\|A_{k}^{-1}\|$ is at worst polynomial in $k$ if one avoids a set $E$ of arbitrarily small measure. As a corollary we obtain the first $k$-explicit bounds on the condition number of $S_k$, where $S_k$ is the standard single-layer BIO on $Γ$ when $Γ$ is the boundary of a Lipschitz domain, and analogous estimates when $Γ$ is a $d$-set (and so of Hausdorff dimension $d$), for non-integer values of $d$.

math.AP↗

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.

math.AP↗