4,278 research outputs found

### Phase-space geometry of the generalized Langevin equation

The generalized Langevin equation is widely used to model the influence of a
heat bath upon a reactive system. This equation will here be studied from a
geometric point of view. A dynamical phase space that represents all possible
states of the system will be constructed, the generalized Langevin equation
will be formally rewritten as a pair of coupled ordinary differential
equations, and the fundamental geometric structures in phase space will be
described. It will be shown that the phase space itself and its geometric
structure depend critically on the preparation of the system: A system that is
assumed to have been in existence for ever has a larger phase space with a
simpler structure than a system that is prepared at a finite time. These
differences persist even in the long-time limit, where one might expect the
details of preparation to become irrelevant

### A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent

Let $(M,\textit{g},\sigma)$ be a compact Riemannian spin manifold of
dimension $m\geq2$, let $\mathbb{S}(M)$ denote the spinor bundle on $M$, and
let $D$ be the Atiyah-Singer Dirac operator acting on spinors
$\psi:M\to\mathbb{S}(M)$. We study the existence of solutions of the nonlinear
Dirac equation with critical exponent $D\psi = \lambda\psi + f(|\psi|)\psi + |\psi|^{\frac2{m-1}}\psi \tag{NLD}$
where $\lambda\in\mathbb{R}$ and $f(|\psi|)\psi$ is a subcritical nonlinearity
in the sense that $f(s)=o\big(s^{\frac2{m-1}}\big)$ as $s\to\infty$. A model
nonlinearity is $f(s)=\alpha s^{p-2}$ with $2<p<\frac{2m}{m-1}$,
$\alpha\in\mathbb{R}$. In particular we study the nonlinear Dirac equation $D\psi=\lambda\psi+|\psi|^{\frac2{m-1}}\psi, \quad \lambda\in\mathbb{R}.
\tag{BND}$ This equation is a spinorial analogue of the Brezis-Nirenberg
problem. As corollary of our main results we obtain the existence of least
energy solutions $(\lambda,\psi)$ of (BND) and (NLD) for every $\lambda>0$,
even if $\lambda$ is an eigenvalue of $D$. For some classes of nonlinearities
$f$ we also obtain solutions of (NLD) for every $\lambda\in\mathbb{R}$, except
for non-positive eigenvalues. If $m\not\equiv3$ (mod 4) we obtain solutions of
(NLD) for every $\lambda\in\mathbb{R}$, except for a finite number of
non-positive eigenvalues. In certain parameter ranges we obtain multiple
solutions of (NLD) and (BND), some near the trivial branch, others away from
it.
The proofs of our results are based on variational methods using the strongly
indefinite energy functional associated to (NLD).Comment: 42 page

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