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Given a domain $\Omega\subset\mathbb{R}^d$, suppose that the boundary is sufficiently regular. Then by Gröger's regularity theory we know that the following operator \begin{equation} \nabla\cdot\nabla …

I have also posted the question here. Let me explain what difficulties I have. In fact, one may write \begin{equation} \partial_1(f-\partial_1 u)=0 \end{equation} in $\Omega$. Then one may have the fo …

Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain $\Omega\subset\mathbb{R}^n$ with certain regular bondary and a function …

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that $u|_{\partial\O …

I think the reason that you can hardly find reference about the vector valued functions is that you can always reduce your situation to scalar ones, since you can always use test function such that it …