In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity permeability on domains a perfectly conducting boundary. This leads to an initial boundary value problem for first-order hyperbolic system characteristic We prove priori estimates solutions in H m . Moreover, show the existence of unique -solution if coefficients data are sufficiently regula...

Consider a smooth bounded domain Ω ⊆ R3, a time interval [0, T ), 0 < T ≤ ∞, and a weak solution u of the Navier-Stokes system. Our aim is to develop several new sufficient conditions on u yielding uniqueness and/or regularity. Based on semigroup properties of the Stokes operator we obtain that the local left-hand Serrin condition for each t ∈ (0, T ) is sufficient for the regularity of u. Some...

We study the pathwise regularity of the map φ 7→ I(φ) = ∫ T 0 〈φ(Xt), dXt〉 where φ is a vector function on R belonging to some Banach space V , X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A stochastic current is a continuous version of this map, seen as a random element of the topological dual of V . We give sufficient conditio...

Abstract. The regularity property of a norm-sequence ρ(n) = ‖Tn‖ (n ∈ N) ensures that the operator T can be intertwined with an isometry V , which relation can be exploited to obtain a lot of information for T itself, as it was shown in [1] and [2]. In [3] general sufficient conditions of regularity were provided. In the present note a necessary and sufficient condition of regularity is given. ...

We derive estimates on the magnitude of the interaction between a wide class of analytic partial differential equations and a high-frequency quasiperiodic oscillator. Assuming high regularity of initial conditions, the equations are transformed to an uncoupled system of an infinite dimensional dynamical system and a linear quasiperiodic flow on a torus; up to coupling terms which are exponentia...

This paper is devoted to the analysis of a special kind of regularity of multifunction which we call metric pseudo-(sub)regularity, when the multifunction is not metrically (sub)regular at a given point but is metrically (sub)regular at certain points in a neighborhood with moduli possibly tending to infinity with a certain order. By using advanced techniques of generalized differentiation we d...

In this article, we numerically study the regularity loss of the solutions of non-parametric minimal surfaces with non-zero boundary conditions. Parts of the boundaries have non-positive mean curvature. As expected from theoretical results in such geometry, we find that the solutions may or may not satisfy the boundary conditions depending upon the data. Firstly, we validate the numerical study...