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Register a new userOn symmetry and uniqueness of ground states for linear and nonlinear elliptic PDEs
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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--Szego 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.
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We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.
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