Linear and Non-Linear Waves
http://hdl.handle.net/20.500.11824/4
Sat, 30 Sep 2023 12:43:15 GMT2023-09-30T12:43:15ZOn the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols
http://hdl.handle.net/20.500.11824/1553
On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols
Covi, G.; García-Ferrero, M.A.; Rüland, A.
In this article we consider direct and inverse problems for α-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective “domain of dependence of the operator” and in several, adapted function spaces. This formulation allows one to avoid natural “gauges” which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only “one-sided operators” new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.
Sun, 25 Dec 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15532022-12-25T00:00:00ZSharp local smoothing estimates for Fourier integral operators
http://hdl.handle.net/20.500.11824/1513
Sharp local smoothing estimates for Fourier integral operators
Beltran D.; Hickman J.; Sogge C.D.
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory
integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions
arising from the structure of the Fourier integrals.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15132019-01-01T00:00:00ZENERGY CONSERVATION FOR 2D EULER WITH VORTICITY IN L(log L)α*
http://hdl.handle.net/20.500.11824/1486
ENERGY CONSERVATION FOR 2D EULER WITH VORTICITY IN L(log L)α*
Ciampa, G.
In these notes we discuss the conservation of the energy for weak solutions of the twodimensional incompressible Euler equations. Weak solutions with vorticity in (Formula presented) with p > 3/2 are always conservative, while for less integrable vorticity the conservation of the energy may depend on the approximation method used to construct the solution. Here we prove that the canonical approximations introduced by DiPerna and Majda provide conservative solutions when the initial vorticity is in the class L(logL)α with α > 1/2.
Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14862022-01-01T00:00:00ZOn the Hausdorff dimension of Riemann's non-differentiable function
http://hdl.handle.net/20.500.11824/1454
On the Hausdorff dimension of Riemann's non-differentiable function
Eceizabarrena, D.
Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14542021-01-01T00:00:00Z