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Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach
Locally Lipschitz perturbations of bisemigroups
| 1. | Department of Mathematics, University of Toledo, Toledo, Ohio 43606, United States |
$\cdot $ We show the existence and uniqueness of what we call local dichotomous mild solutions (DMS) that take the form
$x(t) = e^{(t-t_1)L}x_1 + \int_{t_1}^{t} e^{(t-s)L} f(\xi(s), s)ds$
$ y(t)= e^{-(t_2-t)R} y_2 - \int_{t}^{t_2} e^{-(s-t)R} g(\xi(s), s) ds$
$ t_1 < t_2, \qquad t_1\leq t \leq t_2$
for any sufficiently small time interval $[t_1, t_2]$
and any given $\xi :=(x_1, y_2)$ in a sufficiently small neighbourhood.
$\cdot $ We show that in the uniform $C^0$-norm DMSs vary continuously with $[t_1, t_2]$
and Lipschitz-continuously with $\xi $.
$\cdot $ We study the regularity of DMSs under various hypotheses.
$\cdot $ A simple example that leads to a bisemigroup is a semilinear
elliptic system that arises when
considering solitary waves in an infinite cylinder:
$u_{x x}+\Delta u = f(u), \quad u|_{\Gamma} = 0, \quad\Gamma= \mathbb{R}\times \partial\Omega, \quad (x, y, u)\in \mathbb{R}\times \Omega\times\mathbb{R}^m
where $\Omega$ is a bounded region in $ \mathbb{R}^n$ with $C^2$ boundary and $\Delta$ is the Laplacian in the variable $y\in \Omega$.
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