arXiv subjects

Lars Bugiera

Publications and source records attributed to Lars Bugiera.

2 recordsLinked to original sources

arXiv2019

On symmetry and uniqueness of ground states for linear and nonlinear elliptic PDEs

We study ground state solutions for linear and nonlinear elliptic PDEs in $\mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.\,g. higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szeg\"o inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy-Littlewood majorant problem in $\mathbb{R}^n$ recently obtained by the last two authors of the present paper.

math.AP↗

arXiv2019

On symmetry of traveling solitary waves for dispersion generalized NLS

We consider dispersion generalized nonlinear Schr\"odinger equations (NLS) of the form $i \partial_t u = P(D) u - |u|^{2 \sigma} u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers $\sigma \in \mathbb{N}$. The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and half-wave equations.

math.AP↗