Sobolev Bound for solution to Dirichlet Problem from knowledge of boundary estimates

Question

Take some bounded domain, $\Omega \in \mathbb{R}^n$, $n \geq 3$. Denote by $\partial \Omega$ the boundary of $\Omega$, which we take to be Lipschitz. Let $L$ be an elliptic operator that satisfies ellipticity and boundness conditions. Let $u$ be the solution to the following Dirichlet problem: \begin{equation} \begin{cases} Lu = 0 \qquad & \mbox{in} \quad \Omega \\ u = -v &\mbox{on} \quad \partial \Omega \end{cases} \end{equation} Assume that we know the following is true about $v$ \begin{equation} \sup_{\partial \Omega} (|v| + |\nabla v|) \leq C_1 \end{equation} where $C_1$ is some positive constant. I am wondering, as a consequence of the inequality above, can we say \begin{equation} || u ||_{H^{1}(\Omega)} \leq C_2 \end{equation} where $C_2$ is a positive constant, possibly distinct from $C_1$?

Answer 1

There might be subtleties depending on the exact assumptions that you make, but the general reasoning is as follows.

When $u \in H^1(\Omega)$, its trace on $\partial \Omega$ belongs to $H^{1/2}(\partial\Omega)$. Conversely, if you take a $v \in H^{1/2}(\partial\Omega)$, you can find a $\bar{v} \in H^1(\Omega)$ such that $\bar v = v$ on $\partial \Omega$ and $$ \| \bar{v} \|_{H^1(\Omega)} \leq C \| v \|_{H^{1/2}(\partial\Omega)}. $$ Then you look for $u$ under the form $u = u' + \bar{v}$ so that $u'$ is a solution to $Lu' = f$ with $f = - L\bar{v}$ and $u' = 0$ on $\partial \Omega$. For this problem, you might know that $$ \| u' \|_{H^1(\Omega)} \leq C \| f \|_{H^{-1}(\Omega)} $$ and here you have $$ \|f\|_{H^{-1}(\Omega)} \leq C \| \bar{v} \|_{H^1(\Omega)}. $$ Putting all of this together leads you to the estimate $$ \| u \|_{H^1(\Omega)} \leq C \| v \|_{H^{1/2}(\Omega)}. $$ Now the $H^{1/2}=W^{1/2,2}$ fractional Sobolev norm is bounded above by the $W^{1,\infty}$ norm which you use in your assumption. So, essentially, what you are looking for is indeed valid.

As mentioned above, there are subtleties depending on the exact regularity of $\partial\Omega$ and on the coefficients of $L$.