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Assume that $A=A(r,1)={x: r<||x||<1} \subset R^n$ is an annulus. Whether is known the constant of Poincare inequality for A or some its estimation (w.r.t. $L^2$): the constant $C$ in the inequality $||f||_ {L^2(A)}\le C ||\nabla f||_{L^2(A)},$ where $ f\in W^{1,2}_0(A) $ |
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The constant $C^{-2}$, that is, the infimum of the Raleigh quotient $$\min _ { f \in W^{1,2} _0(A) } \frac {\int_A|\nabla f|^2 dx}{ \int_A|f|^2 dx } $$ is the first eigenvalue of the Laplacian on $A$ with Dirichlet boundary conditions. As a general fact, a positive solution of $-\Delta f=\lambda f$ with Dirichlet boundary conditions is necessarily the first eigenfunction. Here, this allow to find it just by solving the corresponding ODE, where radial simmetry is assumed. The subject is of course classic. However, I do not have access to MatSciNet in this moment, but I'm pretty sure that a search on "first eigenvalue of the Laplacian on an annulus" should give you useful results; you may also like to write and solve the ODE by yourself, and to compute the corresponding Raleigh quotient. |
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If you apply Poincare Inequality on the ball $B\supset A$, thus $|\vert f-\frac{1}{|B|}\int_A f|\vert_{L^2(B)}\le C |\vert\nabla f|\vert_{L^{2}(B)}$ and as we know $f=0$ outside $A$, thus $||\frac{1}{|B|}\int f||\le\frac{|A|}{|B|}||f||=\alpha ||f||$ By triangular inequality, $|\vert f-\frac{1}{|B|}\int f|\vert_{L^2}\ge (1-\alpha)||f||_{L^2}$. |
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