Suppose that $(v_n)$ is a sequence of solutions of $$\begin{cases}-\Delta v_n = f_n&\text{in }\Omega\\ v_n = 0&\text{on }\partial \Omega,\end{cases}$$ where $\Omega \subset \mathbb{R}^2$ is a bounded domain and $f_n$ is a sequence of measurable functions converging to $0$ in measure. Can one infer that $v_n$ converges uniformly to $0$?
Convergence of a sequence of solutions
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real-analysis
pde
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0Why 0? Wouldn't you except rather that they converge to the solution of $-\Delta v= f$ with $v=0$ on $\partial\Omega$? – 2012-09-05
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0The question is rather vague. How do you define $v_n$ to be a solution? I guess you need some elliptic regularity, but you give no starting point. Something more should be known about $f_n$. – 2012-09-05
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1For a sequence of solutions, I mean a sequence $(v_n)\subset L^1_{loc}(\Omega)$ such that $-\int_{\Omega}{v_n\Delta \varphi} = \int{f_n \varphi}$ for all $\varphi \in C^{\infty}_0(\Omega)$ – 2012-09-05
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0And I wrote $f$ instead of $0$... Now I've corrected it – 2012-09-05