While doing Lp estimates, is the constant C monotonically increasing with respect to the parameters it depends on?

Question

For example, consider the third boundary value problem: \begin{align} &\frac{\partial u}{\partial t}-\sum_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n b_i(x,t) \frac{\partial u}{\partial x_i} + c(x,t) u = f(x,t), \quad (x,t) \in Q_T \\ &\frac{\partial u}{\partial \mathbf{n}} + b(x,t) u = g(x,t), \quad x \in \partial \Omega, \ 0<t \leq T, \\ &u(x,0)=u_0(x), \quad x \in \Omega, \end{align} where $Q_T=\Omega \times (0,T]$, $0<T<+\infty$, $\Omega ⊂ \mathbb{R}^n$ is open and bounded, and $\partial \Omega \in C^2$. Suppose $a_{ij} \in C(\overline{Q_T})$, $a_{ij}=a_{ji}$ $(i,j=1,\dots,n)$, and there exist positive constants $\lambda$ and $\Lambda$, s.t. \begin{equation} \lambda |\xi|^2 \leq \sum_{i,j=1}^n a_{ij}(x,t) \xi_i \xi_j \leq \Lambda |\xi|^2, \quad \forall (x,t) \in \overline{Q_T}, \ \xi \in \mathbb{R}^n. \end{equation} Suppose also that $b_i, c \in L^\infty(Q_T)$ $(i=1,\dots,n)$, $f \in L^p(Q_T)$ $(1<p<+\infty)$, $b \in C^{1,1/2}(\partial \Omega \times [0,T])$ and is nonnegative, $g \in W^{2,1}_p(Q_T)$, $u_0 \in W^2_p(\Omega)$, and the compatibility condition \begin{equation} \frac{\partial u_0(x)}{\partial \mathbf{n}}+b(x,0) u_0(x)=g(x,0), \quad x \in \partial \Omega \end{equation} is satisfied. It is asserted in the lecture note that the problem admits a unique solution $u \in W^{2,1}_p(Q_T)$, and the following $L^p$ estimate holds: \begin{equation} \|u\|_{W^{2,1}_p(Q_T)} \leq C (\|f\|_{L^p(Q_T)} + \|g\|_{W^{2,1}_p(Q_T)} + \|u_0\|_{W^2_p(\Omega)}), \end{equation} where $C$ is a positive constant which only depends on $n,\lambda,\Lambda,\Omega,T,\|b_i\|_{L^\infty(Q_T)},\|c\|_{L^\infty(Q_T)},\|b\|_{C^{1,1/2}(\partial \Omega \times [0,T])}$ and the continuous modulus of $a_{ij}$.

The question is, is $C$ monotonically increasing with respect to $n,\lambda,\Lambda,T,\|b_i\|_{L^\infty(Q_T)}$, etc.? The lecture note I referred omits the proof and I haven't found the proof elsewhere, therefore I have no idea how the constant $C$ depends on those parameters exactly. Can anyone give me some hints or tell me where I can find the proof? Thank you very much.

Answer 1

In general, the best way to determine this is to work through the proof and explicitly keep track of the constants involved. I am not aware of any references that do this in explicit detail as it is often just a lengthy computation that doesn't offer much additional insight, however it is a great exercise to make sure you understand the full details.

This typically involves establishing the estimate uniformly assuming that, for instance assume $\lVert b_i \rVert_{L^{\infty}(Q_T)} \leq M$ and show that you can establish the estimate where $C$ depends on $M$ instead of the $\lVert b_i \rVert_{L^{\infty}(Q_T)}.$ Can you see why this would imply that $C$ can be chosen to depend monotonically on $\lVert b_i \rVert_{L^{\infty}(Q_T)}$?

There are a few exceptions where this dependence generally does not hold. In your setting these are the following:

• The ellipticity constant $\lambda,$ as the equation degenerates as $\lambda \to 0$ we expect it to increase monotonically in $1/\lambda$ instead.
• The constant $p,$ as similarly we expect a degeneration when $p \to 1$ and $p \to \infty.$ However the constant can be chosen to increase monotonically in $p$ and $p' = \frac{p}{p-1}.$

Other than these, you should be able to show that the constant $C$ can be chosen to increase monotonically in the remaining numerical parameters.