# How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/332041/using-weak-maximum-principle-to-prove-continuous-dependence-of-the-boundary-data

Maybe someone here can help me? :/

I am currently looking at the following Dirichlet problem:

\begin{align} \begin{cases} \text{div} (\sigma |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\Omega \end{cases} \end{align}

so my initial conditions are the right side $$f \in L^2(\Omega)$$ and the boundary condition $$g \in W^{1,p}(\Omega)$$. I already proved that a unique solution $$u \in W^{1,p}_g$$ (Subspace of Sobolev spave $$W^{1,p}$$ which is $$g$$ on $$\partial\Omega$$). Furthermore I have an a priori estimate

\begin{align} ||u||_{W^{1,p}(\Omega)} \leq C \left ( ||f||^{\frac{1}{p-1}}_{L^2(\Omega)} + ||g||_{W^{1,p}(\Omega)} \right) \end{align}

But how do I prove "continous dependence of initial conditions"? How is this property even defined? Do exactly do I have to show? Intuitivly I thought of something like...

Let $$u$$ be the solution of the problem with the initial conditions $$g$$,$$f$$. Let $$f_k \rightarrow f$$, $$g_k \rightarrow g$$ and $$u_k$$ shall be the solution of the problem with initial conditions $$f_k$$, $$g_k$$. Then $$\lim_{k \rightarrow \infty} ||u - u_k||_{W^{1,p}(\Omega)} \rightarrow 0$$?

I would guess, that I need some terms like $$||u - u_k||_{W^{1,p}(\Omega)} \leq ... ||f - f_k|| ... ||g - g_k||$$?

But how do I get there? I am not very experienced with such estimations...

What I know about the solutions $$u$$ and $$u_k$$ is, that they are the minimizer of the strict convex functional

\begin{align} J(v) = \frac{1}{p}\int_{\Omega} \sigma |\nabla v|^p~dx - \int_{\Omega} fv~dx \end{align}

Therefore we have

\begin{align} J(u) \leq J(u - u_k) \end{align}

and

\begin{align} J_k(u_k) \leq J_k(u - u_k) = \frac{1}{p}\int_{\Omega} \sigma |\nabla (u - u_k)|^p~dx - \int_{\Omega} f_k (u - u_k)~dx \end{align}

but still I don't know how to show the continuous dependence... Maybe someone more experienced can help me?

EDIT: Ok, I think I got the continuous dependence on the "f":

Let $$u_1$$, $$u_2$$ be the solutions for $$f_1$$, $$f_2$$ on the right side of the equation. Since they have the same values on the boundary we have $$u_1 - u_2 \in W^{1,p}_0$$. Because they minimize the functional, for the frechet derivates we have

\begin{align} \int_{\Omega} \sigma|\nabla u_1|^{p-2} \nabla u_1 \nabla \phi~d x = \int_{\Omega} f_1 \phi ~dx\\ \int_{\Omega} \sigma|\nabla u_2|^{p-2} \nabla u_2 \nabla \phi~d x = \int_{\Omega} f_2 \phi ~dx \end{align}

This is true for all $$\phi \in W^{1,p}_0$$. There it is also true for $$u_1 - u_2 \in W^{1,p}_0$$:

\begin{align} \int_{\Omega} \sigma|\nabla u_1|^{p-2} \nabla u_1 \nabla (u_1 - u_2)~d x = \int_{\Omega} f_1 (u_1 - u_2) ~dx\\ \int_{\Omega} \sigma|\nabla u_2|^{p-2} \nabla u_2 \nabla (u_1 - u_2)~d x = \int_{\Omega} f_2 (u_1 - u_2) ~dx \end{align}

Combined we have

\begin{align} \int_{\Omega} (\sigma|\nabla u_1|^{p-2} \nabla u_1 - \sigma|\nabla u_2|^{p-2} \nabla u_2)\nabla (u_1 - u_2)~d x = \int_{\Omega} (f_1 - f_2) (u_1 - u_2) ~dx \end{align}

For the left side we have the estimation \begin{align} \int_{\Omega} \alpha |\nabla u_1 - \nabla u_2|^p ~dx \leq \int_{\Omega} (\sigma|\nabla u_1|^{p-2} \nabla u_1 - \sigma|\nabla u_2|^{p-2} \nabla u_2)\nabla (u_1 - u_2)~d x \end{align}

with an $$\alpha > 0$$ (this estimation is not obvious, but valid). For the right side we have

\begin{align} \alpha || u_1 - u_2 ||^p_{W^{1,p}_0}=\int_{\Omega} (f_1 - f_2) (u_1 - u_2) ~dx \leq ||f_1 - f_2||_{L^2}~||u_1 - u_2||_{W^{1,p}_0} \end{align}

Therefore \begin{align} || u_1 - u_2 ||_{W^{1,p}_0} \leq C ||f_1 - f_2||^{\frac{1}{p}}_{L^2} \end{align}

meaning that $$u_1 \rightarrow u_2$$ for $$f_1 \rightarrow f_2$$... Is this correct? And does anyone know how to handle the dependence of the boundary condition?

EDIT2: My idea for the continuous dependence of the boundary values: Let $$g_k \rightarrow g$$. Then $$\lim_{k \rightarrow \infty} (u_k - u)|_{\partial\Omega} = 0$$. So we have (the same equation as above but with $$f_1 = f_2$$ this time)

\begin{align} \lim_{k \rightarrow \infty} \int_{\Omega} (\sigma|\nabla u|^{p-2} \nabla u - \sigma|\nabla u_k|^{p-2} \nabla u_k)\nabla (u - u_k)~d x = 0 ~dx \end{align}

so \begin{align} \lim_{k \rightarrow \infty} \alpha || u_1 - u_2 ||^p_{W^{1,p}_0} = 0 \end{align}

At first sight it looks correct for me... or did I overlook something crucial?