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Spyros Alexakis

Publications and source records attributed to Spyros Alexakis.

At least 19 recordsLinked to original sources
arXiv2025

Lens rigidity in 2D: The reconstruction of a Riemann surface from its geodesic lengths

We address the question of whether a Riemannian manifold-with-boundary (M,g) in dimension two is uniquely determined from knowledge of the distances between points on its boundary. An affirmative answer is called boundary rigidity for (M,g); it is closely related to lens rigidity. The latter question originates in the problem of reconstructing the speed of sound in an unknown medium from measurements of the travel time of sound waves that are sent in and ultimately return to the boundary. We prove essentially optimal results on these rigidity questions: Our first result answers proves rigidity locally, near a convex portion of the boundary. Our second result proves rigidity globally, for manifolds with convex boundary, in the absence of trapping (closed geodesics), thus confirming a conjecture of Uhlmann. Our final result proves the optimal reconstruction for convex boundaries even in the presence of trapping, showing rigidity up to outermost trapped geodesics. Our results thus extend the classical work of Pestov and Uhlmann on rigidity of simple 2-manifolds, as well as the many prior results on injectivity of the X-ray transform, which address linearized versions of the rigidity problem. \par Our method is to treat the (non-linear) rigidity problem directly, where we simultaneously re-cast the lens data as generalized Riemannian circles, and obtain rigidity for these ``pseudo-circles'', by studying a system of equations that we show these objects must satisfy. The rigidity we obtain ultimately is proven via {novel} estimates that are reminiscent of energy-type estimates for hyperbolic equations.

math.DG↗
arXiv2025

Evaluating singular and near-singular integrals on $C^2$ smooth surfaces with quadratic geometric approximation and closed form expressions

Most Fredholm integral equations involve integrals with weakly singular kernels. Once the domain of integration is discretized into flat triangular elements, these weakly singular kernels become strongly singular or near-singular. Common methods to compute these integrals when the kernel is a Green's function include the Duffy transform, polar coordinates with closed analytic formulas, and singularity extraction. However, these methods do not generalize well to the normal derivatives of Green's functions due to the strongly singular behavior of these functions on triangular elements. We provide methods to integrate both the Green's function and its normal derivative on smooth surfaces discretized by triangular elements in three dimensions for many commonly encountered differential operators. For strongly singular integrals involving normal derivatives of Green's functions, we introduce a more refined approximation of the derivatives of the Green's function on flat triangles. This method uses geometric information of the true surface of integration to approximate the original integral on the true domain using push-forward maps. This is better than simply setting the singular integrals to zero, while being faster than adaptive refinement methods. We provide an algorithm for explicit computations on triangles, and present necessary analytic formulas that the algorithm requires in the appendix.

math.NA↗
arXiv2024

Inverse scattering problems for non-linear wave equations on Lorentzian manifolds

We show that an inverse scattering problem for a semilinear wave equation can be solved on a manifold having an asymptotically Minkowskian infinity, that is, scattering functionals determine the topology, differentiable structure, and the conformal type of the manifold. Moreover, the metric and the coefficient of the non-linearity are determined up to a multiplicative transformation. The manifold on which the inverse problem is considered is allowed to be an open, globally hyperbolic manifold which may have non-trivial topology or several infinities (i.e., ends) of which at least one has to be of the asymptotically Minkowskian type. To formulate the inverse problems we define a new type of data, non-linear scattering functionals, which are defined also in the cases where the classically defined scattering operators are not well-defined. This makes it possible to solve inverse problems also in cases where some of the incoming waves lead to a blow-up of the scattered solution. We use non-linear interaction of waves as a beneficial tool that helps to solve the inverse problem. The corresponding inverse problem for the linear wave equation still remains unsolved.

math.AP↗
arXiv2024

The behaviour of the DC current density at the edge of electrodes

We study the complete electrode model boundary condition for second order elliptic PDE. A specific case of this is the PDE describing the electrostatic potential for a conductive body into which current is injected through electrodes that touch the boundary. We obtain the optimal description of the gradient of the electrostatic potential upon approach to the edge of the electrodes.

math.AP↗
arXiv2023

A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on experimental data

We develop a boundary integral equation-based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, and currents are injected through the electrodes and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single layer potentials; the problem is thus re-cast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography (EIT) which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.

math.NA↗
arXiv2022

Fast Computation of Electrostatic Potentials for Piecewise Constant Conductivities

We present a novel numerical method for solving the elliptic partial differential equation problem for the electrostatic potential with piecewise constant conductivity. We employ an integral equation approach for which we derive a system of well-conditioned integral equations by representing the solution as a sum of single layer potentials. The kernel of the resulting integral operator is smooth provided that the layers are well-separated. The fast multiple method is used to accelerate the generalized minimal residual method solution of the integral equations. For efficiency, we adapt the grid of the Nyström method based on the spectral resolution of the layer charge density. Additionally, we present a method for evaluating the solution that is efficient and accurate throughout the domain, circumventing the close-evaluation problem. To support the design choices of the numerical method, we derive regularity estimates with bounds explicitly in terms of the conductivities and the geometries of the boundaries between their regions. The resulting method is fast and accurate for solving for the electrostatic potential in media with piecewise constant conductivities.

math.NA↗
arXiv2022

Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry

We prove uniform finite-time existence of solutions to the vacuum Einstein equations in polarized U(1) symmetry which have uniformly positive incoming $H^1$ energy supported on an arbitrarily small set in the 2 + 1 spacetime obtained by quotienting by the U(1) symmetry. We also construct a subclass of solutions for which the energy remains concentrated (along a U(1) family of geodesics) throughout its evolution. These results rely on three innovations: a direct treatment of the 2 + 1 Einstein equations in a null geodesic gauge, a novel parabolic scaling of the Einstein equations in this gauge, and a new Klainerman-Sobolev inequality on rectangular strips.

gr-qc↗
arXiv2021

Lorentzian Calder\'{o}n problem near the Minkowski geometry

We study a Lorentzian version of the well-known Calder\'{o}n problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map. In the earlier work of the authors it was shown that zeroth order coefficients can be uniquely determined under a two-sided spacetime curvature bound and the additional assumption that there are no conjugate points along null or spacelike geodesics. In this paper we show that uniqueness for the zeroth order coefficient holds for manifolds satisfying a weaker curvature bound as well as spacetime perturbations of such manifolds. This relies on a new optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calder\'{o}n problem near the Minkowski geometry.

math.AP↗
arXiv2020

Lorentzian Calderón problem under curvature bounds

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of arbitrary, smooth perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calderón problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior of the double null cone emanating from a point. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.

math.AP↗
arXiv2020

Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors

We show a stability result for the Schwarzschild singularity (inside the black hole region) for the Einstein vacuum equations. The result is proven in the class of polarized axial symmetry, under perturbations of the Schwarzschild data induced on a hypersurface $\{r=\e\}$, $\e<<2M$. Our result is only partly a stability result, in that we show that while a (space-like) singularity persists under perturbations as above, the behaviour of the metric approaching the singularity is much more involved than for the Schwarzschild solution. Indeed, we find that the solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity. These Kasner-type asymptotics are very far from isotropic, since (as in Schwarzschild) there are two contracting directions and one expanding one. Our proof relies on energy methods and on a new approach to the EVE in axial symmetry, which we believe has wider applicability: In this symmetry class and under a suitable geodesic gauge, the EVE can be studied as a free wave coupled to (nonlinear) ODEs, which couple the geometry of the projected, 2+1 space-time to the free wave. The fact that the nonlinear part of the Einstein equations is described by ODEs lies at the heart of how one can overcome a certain linear instability exhibited by the singularity.

math.AP↗
arXiv2018

Determining a Riemannian Metric from Minimal Areas

We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth Riemannian metric $g$, and mean-convex boundary $\partial M$ then knowledge of least areas circumscribed by simple closed curves $γ\subset \partial M$ uniquely determines the metric $g$, under some additional geometric assumptions. These are that $g$ is either a) $C^3$-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. %sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves $γ\subset \partial M$. We also prove a corresponding local result: assuming only that $(M,g)$ has strictly mean convex boundary at a point $p\in\partial M$, we prove that knowledge of the least areas circumscribed by any simple closed curve $γ$ in a neighbourhood $U\subset \partial M$ of $p$ uniquely determines the metric near $p$. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations "from all directions". The proofs rely on finding the metric along a continuous sweep-out of $M$ by area-minimizing surfaces; they bring together ideas from the 2D-Calderón inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.

math.DG↗
arXiv2017

On the profile of energy concentration at blow-up points for sub-conformal focusing nonlinear waves

We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the $H^1$-norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this $H^1$-energy can be distributed inside such cones. Our proof relies on new localized estimates obtained using Carleman- type inequalities for such nonlinear waves. These bound the $L^{p+1}$-norm in the interior of timelike cones by their $H^1$-norm near the boundary of the cones. Such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations, in the interior of time cones.

math.AP↗
arXiv2016

Bounds on the Bondi Energy by a Flux of Curvature

We consider smooth null cones in a vacuum spacetime that extend to future null infinity. For such cones that are perturbations of shear-free outgoing null cones in Schwarzschild spacetimes, we prove bounds for the Bondi energy, momentum, and rate of energy loss. The bounds depend on the closeness between the given cone and a corresponding cone in a Schwarzschild spacetime, measured purely in terms of the differences between certain weighted $L^2$-norms of the space-time curvature on the cones, and of the geometries of the spheres from which they emanate. A key step in this paper is the construction of a family of asymptotically round cuts of our cone, relative to which the Bondi energy is measured.

math.AP↗
arXiv2015

Global uniqueness theorems for linear and nonlinear waves

We prove a unique continuation from infinity theorem for regular waves of the form $[ \Box + \mathcal{V} (t, x) ]ϕ=0$. Under the assumption of no incoming and no outgoing radiation on specific halves of past and future null infinities, we show that the solution must vanish everywhere. The "no radiation" assumption is captured in a specific, finite rate of decay which in general depends on the $L^\infty$-profile of the potential $\mathcal{V}$. We show that the result is optimal in many regards. These results are then extended to certain power-law type nonlinear wave equations, where the order of decay one must assume is independent of the size of the nonlinear term. These results are obtained using a new family of global Carleman-type estimates on the exterior of a null cone. A companion paper to this one explores further applications of these new estimates to such nonlinear waves.

math.AP↗
arXiv2015

The Penrose inequality on perturbations of the Schwarzschild exterior

We prove a version the Penrose inequality for black hole space-times which are perturbations of the Schwarzschild exterior in a slab around a null hypersurface $\underline{\mathcal{N}}_0$. $\underline{\mathcal{N}}_0$ terminates at past null infinity $\mathcal{I}^-$ and $\mathcal{S}_0:=\partial\underline{\mathcal{N}}_0$ is chosen to be a marginally outer trapped sphere. We show that the area of $\mathcal{S}_0$ yields a lower bound for the Bondi energy of sections of past null infinity, thus also for the total ADM energy. Our argument is perturbative, and rests on suitably deforming the initial null hypersurface $\underline{\mathcal{N}}_0$ to one for which the natural "luminosity" foliation originally introduced by Hawking yields a monotonically increasing Hawking mass, and for which the leaves of this foliation become asymptotically round. It is to ensure the latter (essential) property that we perform the deformation of the initial nullhypersurface $\underline{\mathcal{N}}_0$.

gr-qc↗
arXiv2015

Non-existence of time-periodic vacuum spacetimes

We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be periodic in time, are in fact stationary in a neighborhood of infinity. Our result applies under physically relevant regularity assumptions purely at the level of the initial data. In particular, our work removes the assumption of analyticity up to null infinity in [Bicak, Scholtz, and Tod; 2010]. The proof relies on extending a suitably constructed "candidate" Killing vector field from null infinity, via Carleman-type estimates obtained in [Alexakis, Schlue, Shao; 2013].

gr-qc↗
arXiv2015

Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence.

math.DG↗