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For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$ $$ \Delta u=0, \text{ on } D, \\ \left. u\right|_{\partial D}=f \in C\left( \partial D\right). $$ With a fix $x$ in $D$, an application of the Riesz representation theorem gives $$ u(x)=\int_{\partial D} f(t) d\mu_x(t), $$ for some measure $\mu_x$.

What is the operator on which Riesz theorem is applied?
Which Riesz theorem is used?

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1 Answer 1

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This assumes that the Dirichlet problem is solvable for all continuous $f$ (which means that each boundary point admits a barrier). Now, if you fix a point $x\in\Omega$, and look at the value $u(x)$, then clearly $P_x:f\mapsto u(x)$ is linear, and by the maximum principle it is positive, and $$ |u(x)| \leq \|f\|_{C(\partial\Omega)}, $$ so $P_x$ is a bounded linear functional on the space of continuous functions on the boundary. Finally, by the Riesz representation theorem, there exists a measure $\mu_x$ that represents $P_x$.

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