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We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from…
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in [9],…
We study the long time statistics of a class of semi--linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase…
We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,\beta}$ regularity on the either side of the thin manifold, the optimal growth away from the free…
Attempts to construct a low-temperature version of the fluid/gravity correspondence have faced obstacles manifested in the form of logarithmic terms in the frequency, $\log(\omega)$, leading to non-local in time constitutive relations for…
Given that a solution to the 3D incompressible Euler equations on a bounded domain blows up at a time $T_\ast$ and that $T_\ast$ is the first such time, we provide pointwise-in-time lower bounds on $\|D^k\omega\|_{L^\infty(\Omega)}$ for $k…
Divergences often play important roles for study in information science so that it is indispensable to investigate their fundamental properties. There is also a mathematical significance of such results. In this paper, we introduce some…
We reveal strong and weak inequalities relating two fundamental macroscopic quantum geometric quantities, the quantum distance and Berry phase, for closed paths in the Hilbert space of wavefunctions. We recount the role of quantum geometry…
We study and measure the transmission coefficient of counter-propagating shallow-water waves produced by a wave generator and scattered by an obstacle. To precisely compare theoretical predictions and experimental data, we consider $\sim…
In this paper, we study the nonlinear Vlasov-Fokker-Planck equation with fixed collision frequency. We establish the global-in-time existence of weak solutions to the equation with large initial data. Moreover, we show that our solution…
The reflection, transmission and absorption of a symmetric electromagnetic pulse, which carrying frequency is close to the frequency of an interband transition in a QW (QW), are obtained. The energy levels of a QW are assumed discrete, one…
This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schr\"odinger equations with subcritical exponent. For some smooth bounded domain $\Omega\subset \mathbf R^n$, our boundary condition is given…
The paper studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations. We introduce a method to solve inverse problems for non-linear equations using interaction of three waves, that…
We study the stability issue for the inverse problem of determining a coefficient appearing in a Schr\"odinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of…
The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…
We consider the inverse problem of reconstructing the scattering and absorption coefficients using boundary measurements for a time dependent radiative transfer equation (RTE). As the measurement is mostly polluted by errors, both…
Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential…
Since the first theoretical proposal by Berezinskii, an odd-frequency superconductivity has encountered the fundamental problems on its thermodynamic stability and rigidity of a homogenous state accompanied by unphysical Meissner effect.…
We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves…
We consider the inverse scattering problem of retrieving the structural parameters of a stratified medium consisting of dispersive materials, given knowledge of the complex reflection coefficient in a finite frequency range. It is shown…