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- SIGMUND SELBERG
- 2006

We prove global well-posedness below the charge norm (i.e., the L 2 norm of the Dirac spinor) for the Dirac-Klein-Gordon system of equations (DKG) in one space dimension. Adapting a method due to Bourgain, we split off the high frequency part of the initial data for the spinor, and exploit nonlinear smoothing effects to control the evolution of the high… (More)

- SIGMUND SELBERG
- 2008

We first review the L 2 bilinear generalizations of the L 4 estimate of Strichartz for solutions of the homogeneous 3D wave equation, and give a short proof based solely on an estimate for the volume of intersection of two thickened spheres. We then go on to prove a number of new results, the main theme being how additional, anisotropic Fourier restrictions… (More)

- Sigmund Selberg
- 2005

We prove a quadrilinear integral estimate in space-time for solutions of the homogeneous wave equation on R 1+2. This estimate is a generalization of a previously known bilinear L 2 estimate, and it arises naturally in the study of the local regularity properties of a hyperbolic model equation connected with wave maps from Minkowski space R 1+2 into a… (More)

- Sigmund Selberg
- 2001

We prove estimates for solutions of the Cauchy problem for the in-homogeneous wave equation on R 1+n in a class of Banach spaces whose norms only depend on the size of the space-time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier… (More)

- Sigmund Selberg
- 2002

We prove that the Maxwell-Klein-Gordon system on R 1þ4 relative to the Coulomb gauge is locally well-posed for initial data in H 1þ" for all " > 0. This builds on previous work by Klainerman and Machedon [6] who proved the corresponding result, with the additional restriction of small-norm data, for a model problem obtained by ignoring the elliptic features… (More)

- Sigmund Selberg
- 2007

We prove that the Maxwell-Klein-Gordon equations on R 1+4 relative to the Coulomb gauge are locally well-posed for initial data in H 1+ε for all ε > 0. This builds on previous work by Klainerman and Machedon [3] who proved the corresponding result for a model problem derived from the Maxwell-Klein-Gordon system by ignoring the elliptic features of the… (More)

We prove that in the nonrelativistic limit c → ∞, where c is the speed of light, solutions of the Klein-Gordon-Maxwell system on R 1+3 converge in the energy space C([0, T ]; H 1) to solutions of a Schrödinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two… (More)

- SIGMUND SELBERG, ACHENEF TESFAHUN
- 2006

We extend recent results of S. Machihara and H. Pecher on low regularity well-posedness of the Dirac-Klein-Gordon (DKG) system in one dimension. Our proof, like that of Pecher, relies on the null structure of DKG, recently completed by D'Ancona, Foschi and Selberg, but we show that in 1d the argument can be simplified by modifying the choice of projections… (More)

We prove that in the nonrelativistic limit c → ∞, where c is the speed of light, solutions of the Klein-Gordon-Maxwell system on R 1+3 converge in the energy space C([0, T]; H 1) to solutions of a Schrödinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two… (More)

- SIGMUND SELBERG
- 2007

Last week we proved existence of weak solutions of the Dirichlet problem for a class of elliptic operators (in particular, for the Laplace operator), by using the Riesz representation theorem. This method, however, is limited to linear PDEs. It is therefore of interest to study more robust methods, which can be applied also to nonlinear PDEs. One such… (More)