arXiv subjects

Joel Nathe

Publications and source records attributed to Joel Nathe.

4 recordsLinked to original sources

arXiv2026

Recovery of a Null Form in the Wave Equation from Scattering Data

We use highly oscillatory geometric optics solutions to solve the inverse problem for the system $$ \square \begin{bmatrix} u^{(1)}\\ u^{(2)}\\ \vdots\\ u^{(n)}\end{bmatrix} = \sum_{\substack{k,l=1\\k\geq l}}^n \left(Q_0(u^{(k)},u^{(l)}) \begin{bmatrix} q_{1kl}\\q_{2kl}\\\vdots\\q_{nkl} \end{bmatrix} + \sum_{\substack{i,j=1\\i>j}}^d Q_{ij}(u^{(k)},u^{(l)})\begin{bmatrix} p_{1ijkl} \\ p_{2ijkl} \\ \vdots \\ p_{nijkl} \end{bmatrix}\right), $$ where $Q_0$ and $Q_{ij}$ are the symmetric and anti-symmetric bilinear null forms. We present solutions in both the linear and weakly nonlinear regimes. In the linear regime, we show that the coefficients of order $h^3$ determine an injective light-ray transform of a vector field which depends on the coefficients $q_{rkl}, p_{rijkl}$. In the weakly nonlinear regime, we see that the coefficients of order $h$ determine the non-abelian light ray transform for matrices associated with the coefficients. While we do not have an injectivity result for this case, we do have one if we assume the coefficients do not depend on the time variable $x_0$, as our coefficients instead determine an injective non-abelian X-ray transform.

math.AP↗

arXiv2026

The Recovery of Semilinear Potentials Satisfying Null Conditions From Scattering Data

We construct oscillatory solutions of fully semilinear wave equations in Minkowski space satisfying a null condition of the form $$\square u:=(-\partial_{x_0}^2 +\sum_{j=1}^n \partial_{x_j}^2 )u= q(x,u)((\partial_{x_0}u)^2-|\nabla_{x'}u|^2),$$ $$x=(x_0,x'), \;\ x'=(x_1,\ldots, x_n) \text{ and } x_0=t \text{ is the time variable,}$$ on an interval $x_0\in [-T,T]$, $T<\infty$ arbitrary, which consist of the superposition of a non-oscillatory background solution and a single phase train of highly oscillatory waves of wave length $h\ll1$ and amplitudes given by powers of $h$; the waves interact with the nonlinearity and we measure the response $u(x_0,x')|_{x_0=T'}$ at a fixed time $x_0=T'<T$. We show that the coefficient of amplitude $h$ of the oscillatory part of the nonlinear geometric optics expansion of the solution determines the light-ray transform of a vector field associated with $q(x,u)$, which determines $q(x,u)$ uniquely in the maximal region determined by the data. Our methods also work for systems of semilinear wave equations satisfying null conditions, but in this paper we focus on the scalar case.

math.AP↗

arXiv2024

Subharmonic Kernels and Energy Minimizing Measures, with Applications to the Flat Torus

We study the minimization of the energy integral $I_K(\mu) = \int_{\Omega} \int_{\Omega} K(x,y) d\mu(x) d\mu(y)$ over all Borel probability measures $\mu$, where $(\Omega,\rho)$ is a compact connected metric space and $K:\Omega^2 \to [0,\infty]$ is continuous in the extended sense. We focus on kernels $K$ which are subharmonic, which we define so that the potential $U_K^\mu(x) = \int_{\Omega} K(x,y) d\mu(y)$ satisfies a maximum principle on $\Omega\setminus{\rm supp}{\mu}$. This extends the classical electrostatics minimization problem for logarithmic energy $\int_{\Omega}\int_{\Omega}\log\left(\frac{1}{||x-y||}\right)$, which is used heavily as a tool in approximation theory. Using properties of minimizing measures, we show that if the singularities of the subharmonic kernel $K$ are such that $K$ is regular, then $K$ is positive definite, and $\mu$ is a minimizing measure if and only if its potential is constant (outside of a small exceptional set).We then apply this result to group invariant kernels on compact homogeneous manifolds. In this case, the uniform measure $\sigma$ has constant potential, so subharmonicity implies that this is the minimizing measure. Finally, we look at the case of the $d$-dimensional flat torus $T^d$. We use our results to see that the Riesz kernel $K_s(x,y) = {\rm sign}(s)\rho(x,y)^{-s}$ is minimized by $\sigma$ (and thus positive definite) when $d > s \geq d-2$. Additionally, the positive definiteness gives us a condition which implies that the multivariate Fourier series of a function $f:[0,\pi]^d \to [0,\infty]$ has nonnegative coefficients.

math.CA↗

arXiv2024

Geodesic Distance Riesz Energy on Projective Spaces

We study probability measures that minimize the Riesz energy with respect to the geodesic distance $\vartheta (x,y)$ on projective spaces $\mathbb{FP}^d$ (such energies arise from the 1959 conjecture of Fejes Tóth about sums of non-obtuse angles), i.e. the integral \begin{equation} \frac{1}{s} \int_{\mathbb{FP}^d} \int_{\mathbb{FP}^d} \big( \vartheta (x,y) \big)^{-s} dμ(x) dμ(y) \,\,\, \text{ for } \,\,\, s<d \end{equation} and find ranges of the parameter $s$ for which the energy is minimized by the uniform measure $σ$ on $\mathbb{FP}^d$. To this end, we use various methods of harmonic analysis, such as Cesàro averages of Jacobi expansions and $A_1$ inequalities, and establish a rather general theorem guaranteeing that certain energies with singular kernels are minimized by $σ$. In addition, we obtain further results and present numerical evidence, which uncover a peculiar effect that minimizers this energy undergo numerous phase transitions, in sharp contrast with many analogous known examples (even the seemingly similar geodesic Riesz energy on the sphere), which usually have only one transition (between uniform and discrete minimizers).

math.CA↗