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The closed span of Rademacher functions is investigated in Nakano spaces on equipped with the Lebesgue measure. The main result of this paper states that under some conditions on distribution of the exponent function the Rademacher functions form in a basic sequence equivalent to the unit vector basis in .
We study the question of existence of time-periodic, spatially periodic solutions for dispersive evolution equations, and in particular, we introduce a framework for demonstrating the nonexistence of such solutions. We formulate the problem so that doubly periodic solutions correspond to fixed points of a certain operator. We prove that this operator is locally contracting, for almost every temporal period, if the Duhamel integral associated to the evolution exhibits a weak smoothing property. This implies the nonexistence of nontrivial, small-amplitude time-periodic solutions for almost every period if the smoothing property holds. This can be viewed as a partial analogue of scattering for dispersive equations on periodic intervals, since scattering in free space implies the nonexistence of small coherent structures. We use a normal form to demonstrate the smoothing property on specific examples, so that it can be seen that there are indeed equations for which the hypotheses of the general theorem hold. The nonexistence result is thus established through the novel combination of small-divisor estimates and dispersive smoothing estimates. The examples treated include the Korteweg–de Vries equation and the Kawahara equation.
We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderón, for conductivities that are degenerate, that is, they may not be bounded from above or below. Elliptic equations with such coefficient functions are essential for physical models used in transformation optics and the study of metamaterials, e.g., the zero permittivity materials. In particular, we show that the elliptic inverse problems can be solved in a class of conductivities which is larger than . Also, we give new counterexamples for the uniqueness of the inverse conductivity problem.
We say that a conductivity is visible if the inverse problem is solvable so that the conductivity inside of the domain can be uniquely determined, up to a change of coordinates, using the boundary measurements. The original counterexamples for the inverse problem are related to the invisibility cloaking. This means that there are conductivities for which a part of the domain is shielded from detection via boundary measurements and even the existence of the shielded domain is hidden. Such conductivities are called invisibility cloaks.
In the present paper, we identify the borderline of the visible conductivities and the borderline of invisibility cloaking conductivities. Surprisingly, these borderlines are not the same. We show that between the visible and the cloaking conductivities, there are the electric holograms. These are conductivities which create an illusion of a nonexisting body. Such conductivities give counterexamples for the uniqueness of the inverse problem which are less degenerate than the previously known ones. These examples are constructed using transformation optics and the inverse maps of the optimal blow-up maps. The proofs of the uniqueness results for inverse problems are based on the complex geometrical optics and the Orlicz space techniques.
Garnett, Killip, and Schul have exhibited a doubling measure with support equal to that is -rectifiable, meaning there are countably many curves of finite length for which . In this note, we characterize when a doubling measure with support equal to a connected metric space has a -rectifiable subset of positive measure and show this set coincides up to a set of -measure zero with the set of for which .
We construct Hadamard states for Klein–Gordon fields in a spacetime equal to the interior of the future lightcone from a base point in a globally hyperbolic spacetime .
Under some regularity conditions at the future infinity of , we identify a boundary symplectic space of functions on , which allows us to construct states for Klein–Gordon quantum fields in from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on , ensuring that the bulk state it induces in satisfies the Hadamard condition.
Using pseudodifferential calculus on the cone , we construct a large class of Hadamard states on the boundary with pseudodifferential covariances and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in .
We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schrödinger equation on a class of nontrapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices, including the endpoint, in both the homogeneous and inhomogeneous cases. This result improves on the results by Tao, Wunsch and the first author and by Mizutani, which are local in time, as well as results of the second author, which are global in time but with a loss of angular derivatives. In addition, the endpoint inhomogeneous estimate is a strengthened version of the uniform Sobolev estimate recently proved by Guillarmou and the first author. The second author has proved similar results for the wave equation.
We study the limiting probability distribution of the homogenization error for second order elliptic equations in divergence form with highly oscillatory periodic conductivity coefficients and highly oscillatory stochastic potential. The effective conductivity coefficients are the same as those of the standard periodic homogenization, and the effective potential is given by the mean. We show that the limiting distribution of the random part of the homogenization error, as random elements in proper Hilbert spaces, is Gaussian and can be characterized by the homogenized Green’s function, the homogenized solution and the statistics of the random potential. This generalizes previous results in the setting with slowly varying diffusion coefficients, and the current setting with fast oscillations in the differential operator requires new methods to prove compactness of the probability distributions of the random fluctuation.
We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this Lyapunov functional, we derive the blow-up rate and the blow-up limit of the solution. We also classify all asymptotic behaviors of the solution at the singularity and give precise blow-up profiles corresponding to these behaviors. Finally, we attain the blow-up profile numerically, thanks to a new mesh-refinement algorithm inspired by the rescaling method of Berger and Kohn. Note that our method is applicable to more general equations, in particular those with no scaling invariance.