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fixed a minor typo in the last bound

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edited Apr 4, 2022 at 16:41

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Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$, and let $\{\phi_j\}_{j=1}^{\infty}$ be an orthonormal basis of $L^2(M)$ consisting of the associated Dirihclet eigenfunctions. Consider the boundary value problem $$ -\Delta_g u = 0 \quad \text{on $M$} \quad \text{and}\quad u|_{\partial M}=f.$$ Given any $f \in C^{\infty}(\partial M)$, it is straightforward to see that $$ u = -\sum_{j=1}^{\infty} \lambda_j^{-1}\,(\int_{\partial M} f \partial_\nu \phi_j\,dV_g)\, \phi,$$ where the convergence is in the $L^2(M)$ topology. My confusion arises since the convergence can not possibly take place say in $H^1(M)$ (otherwise the trace of $u$ must vanish on $\partial M$) but on the other hand we have the bound $$ |\int_{\partial M} f \partial_\nu \phi_j\,dV_g| \leq C_k \|f\|_{H^k(\partial M)}\, \|\partial_\nu \phi_j\|_{H^{-k}(\partial M)} \leq C \|f\|_{H^k(\partial M)}\, \lambda_j^{-\frac{k+1}{2}},$$$$ |\int_{\partial M} f \partial_\nu \phi_j\,dV_g| \leq C_k \|f\|_{H^k(\partial M)}\, \|\partial_\nu \phi_j\|_{H^{-k}(\partial M)} \leq C \|f\|_{H^k(\partial M)}\, \lambda_j^{\frac{-k+1}{2}},$$ where the last inequality is using equation (2.2) in the paper "Hassell, Andrew; Tao, Terence, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions, Math. Res. Lett. 9 (2002), no. 2-3, 289–305". Thus taking $k$ sufficiently negativelarge (for instance $k\geq n+3$$k=n+3$), it seems to suggest that for the equation of $u$, the convergence of the series also happens in $H^1(M)$. Can someone help me see the flaw in this argument?

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$, and let $\{\phi_j\}_{j=1}^{\infty}$ be an orthonormal basis of $L^2(M)$ consisting of the associated Dirihclet eigenfunctions. Consider the boundary value problem $$ -\Delta_g u = 0 \quad \text{on $M$} \quad \text{and}\quad u|_{\partial M}=f.$$ Given any $f \in C^{\infty}(\partial M)$, it is straightforward to see that $$ u = -\sum_{j=1}^{\infty} \lambda_j^{-1}\,(\int_{\partial M} f \partial_\nu \phi_j\,dV_g)\, \phi,$$ where the convergence is in the $L^2(M)$ topology. My confusion arises since the convergence can not possibly take place say in $H^1(M)$ (otherwise the trace of $u$ must vanish on $\partial M$) but on the other hand we have the bound $$ |\int_{\partial M} f \partial_\nu \phi_j\,dV_g| \leq C_k \|f\|_{H^k(\partial M)}\, \|\partial_\nu \phi_j\|_{H^{-k}(\partial M)} \leq C \|f\|_{H^k(\partial M)}\, \lambda_j^{-\frac{k+1}{2}},$$ where the last inequality is using equation (2.2) in the paper "Hassell, Andrew; Tao, Terence, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions, Math. Res. Lett. 9 (2002), no. 2-3, 289–305". Thus taking $k$ sufficiently negative (for instance $k\geq n+3$), it seems to suggest that for the equation of $u$, the convergence of the series also happens in $H^1(M)$. Can someone help me see the flaw in this argument?

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$, and let $\{\phi_j\}_{j=1}^{\infty}$ be an orthonormal basis of $L^2(M)$ consisting of the associated Dirihclet eigenfunctions. Consider the boundary value problem $$ -\Delta_g u = 0 \quad \text{on $M$} \quad \text{and}\quad u|_{\partial M}=f.$$ Given any $f \in C^{\infty}(\partial M)$, it is straightforward to see that $$ u = -\sum_{j=1}^{\infty} \lambda_j^{-1}\,(\int_{\partial M} f \partial_\nu \phi_j\,dV_g)\, \phi,$$ where the convergence is in the $L^2(M)$ topology. My confusion arises since the convergence can not possibly take place say in $H^1(M)$ (otherwise the trace of $u$ must vanish on $\partial M$) but on the other hand we have the bound $$ |\int_{\partial M} f \partial_\nu \phi_j\,dV_g| \leq C_k \|f\|_{H^k(\partial M)}\, \|\partial_\nu \phi_j\|_{H^{-k}(\partial M)} \leq C \|f\|_{H^k(\partial M)}\, \lambda_j^{\frac{-k+1}{2}},$$ where the last inequality is using equation (2.2) in the paper "Hassell, Andrew; Tao, Terence, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions, Math. Res. Lett. 9 (2002), no. 2-3, 289–305". Thus taking $k$ sufficiently large (for instance $k=n+3$), it seems to suggest that for the equation of $u$, the convergence of the series also happens in $H^1(M)$. Can someone help me see the flaw in this argument?

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asked Apr 4, 2022 at 16:34

Ali

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On the convergence of the spectral decomposition of a harmonic function

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$, and let $\{\phi_j\}_{j=1}^{\infty}$ be an orthonormal basis of $L^2(M)$ consisting of the associated Dirihclet eigenfunctions. Consider the boundary value problem $$ -\Delta_g u = 0 \quad \text{on $M$} \quad \text{and}\quad u|_{\partial M}=f.$$ Given any $f \in C^{\infty}(\partial M)$, it is straightforward to see that $$ u = -\sum_{j=1}^{\infty} \lambda_j^{-1}\,(\int_{\partial M} f \partial_\nu \phi_j\,dV_g)\, \phi,$$ where the convergence is in the $L^2(M)$ topology. My confusion arises since the convergence can not possibly take place say in $H^1(M)$ (otherwise the trace of $u$ must vanish on $\partial M$) but on the other hand we have the bound $$ |\int_{\partial M} f \partial_\nu \phi_j\,dV_g| \leq C_k \|f\|_{H^k(\partial M)}\, \|\partial_\nu \phi_j\|_{H^{-k}(\partial M)} \leq C \|f\|_{H^k(\partial M)}\, \lambda_j^{-\frac{k+1}{2}},$$ where the last inequality is using equation (2.2) in the paper "Hassell, Andrew; Tao, Terence, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions, Math. Res. Lett. 9 (2002), no. 2-3, 289–305". Thus taking $k$ sufficiently negative (for instance $k\geq n+3$), it seems to suggest that for the equation of $u$, the convergence of the series also happens in $H^1(M)$. Can someone help me see the flaw in this argument?