Related papers: Implicit solutions to some Lorentz invariant non-l…
We study the Cauchy problem of higher dimensional Einstein-Maxwell-Higgs system in the framework of Bondi coordinates. As a first step, the problem is reduced to a single first-order integro-differential equation by defining a generalized…
The universal field equations introduced by the author and his collaborators, which admit infinitely many inequivalent Lagrangian formulations are shown to arise as consistency conditions for the existence of non-trivial solutions to the…
We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…
We study the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions for the nonlinearity and the initial data, we obtain the global solution which satisfies weighted $L^1$ and $L^\infty$ estimates. Furthermore, we…
We consider rotational initial data for the two-dimensional incompressible Euler equations on an annulus. Using the convex integration framework, we show that there exist infinitely many admissible weak solutions (i.e. such with…
Consider the scattering of time-harmonic point sources by an infinite locally rough interface with bounded obstacles embedded in the lower half-space. The model problem is first reduced to an equivalent integral equation formulation defined…
A solution of linearized Einstein field equations in vacuum is given and discussed. First it is shown that, computing from our particular metric the linearized connections, the linearized Riemann tensor and the linearized Ricci tensor, the…
We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions…
This is the second and last paper of a series aimed at solving the local Cauchy problem for polarized $\mathbb U(1)$ symmetric solutions to the Einstein vacuum equations featuring the nonlinear interaction of three small amplitude impulsive…
We introduce a (variation of quadrics) ansatz for constructing explicit, real-valued solutions to broad classes of complex Hessian equations on domains in $\mathbb{C}^{n+1}$ and real Hessian equations on domains in $\mathbb{R}^{n+1}$. In…
A new class of exact solutions to the axisymmetric and stationary vacuum Einstein equations containing n arbitrary complex parameters and one arbitrary real solution of the axisymmetric three-dimensional Laplace equation is presented. The…
We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The…
A vacuum solution of the Einstein gravitational field equation is given that follows from a general ansatz but fails to follow from it if a certain symmetric matrix is assumed to be in diagonal form from the beginning.
It is shown that the Cotton tensor can describe the effects of gravity beyond general relativity. Any solution of the Einstein equations with or without the cosmological constant satisfies the field equations described by the Cotton tensor.…
This paper develops a characterisation of when solutions of forced second order linear differential equations converge to the zero solution of the asymptotically stable and unforced second order equation, or when the solution is bounded,…
The existence of stationary solutions of the Einstein-Vlasov-Maxwell system which are axially symmetric but not spherically symmetric is proven by means of the implicit function theorem on Banach spaces. The proof generalises the methods of…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
Assuming that a formal approximation of multiple waves has been obtained by matched asymptotic methods, we derive a {\em Spatial Shadowing lemma} to construct exact solutions near the formal approximation. In Part I, we consider a general…
The problem of neutral fermions subject to an inversely linear potential is revisited. It is shown that an infinite set of bound-state solutions can be found on the condition that the fermion is embedded in an additional uniform background…
In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and $L^1$ data. In the same setting we study sharp (local…