- #1
Julio1
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Suppose that $u$ is the solution of the Laplace equation
$u_{xx}+u_{yy}=0$ in $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$
$u(x,y)=x$ for all $(x,y)\in \mathbb{R}^2$ such that $x^2+y^2=1.$
Find the value of $u$ in $(0,0).$ Use the property of median value.
$u_{xx}+u_{yy}=0$ in $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$
$u(x,y)=x$ for all $(x,y)\in \mathbb{R}^2$ such that $x^2+y^2=1.$
Find the value of $u$ in $(0,0).$ Use the property of median value.
Hello. The median value is $u(x)=\dfrac{\displaystyle\int_{\partial B(x,r)} u(y)dS(y)}{\displaystyle\int_{\partial B(x,r)} \, dS(y)}$. But how can apply for this case?