Narrow Results

If Ω ⊂ ℝⁿ is a bounded domain, the existence of solutions of div u = f for f ∈ L²(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution , where u is the velocity and p the pressure.

It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains.

In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal.

For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution for some r < 2 depending on the power of the cusp.

Let $X$ be a separable Banach space endowed with a nondegenerate centered Gaussian measure $\mu$ and let $w$ be a positive function on $X$ such that $w\in W^{1,s}(X,\mu )$ and $logw\in W^{1,t}(X,\mu )$ for some $s>1$ and $t>s^{\prime }$. In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure $\nu :=w\mu$. We obtain results regarding the divergence operator (i.e. the adjoint in $L^2$ of the gradient operator along the Cameron–Martin space) and the trace of Sobolev functions on hypersurfaces $\left\{x\in X|G(x)=0\right\}$, where $G$ is a suitable version of a Sobolev function.

We study the solutions $u$ to the equation