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Background

Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\flat$ its dual 1-form (under musical isomorphism). Then let $\text{vol}_\partial=\iota_\nu \text{vol}$ orient $\partial M$ (standard). The Riemannian metric induces an $L^2$ structure on $\Omega^k$ (this is the dot product for differential forms, which differs from the dot product for tensor fields by a factor of $\frac{1}{k!}$).

Let $\jmath:\partial M \to M$ be the inclusion map. Then we can define the tangential and normal part of a differential form near the boundary, following Günter Schwarz $$ \forall w \in \Omega^k(M): \mathbf{t}w = \iota_\nu(\nu^\flat\wedge w), \mathbf{n}w = w - \mathbf{t}w = \nu^\flat\wedge\iota_\nu w $$ Obviously $\mathbf{n}f=0 \; \forall f\in \Omega^0(M)=C^\infty(M)$ and $\star$ (Hodge star) is $L^2$ isomorphism with $\star \mathbf{n} = \mathbf{t} \star,\; \mathbf{n} \star= \star \mathbf{t} $.

Now define $\Omega^k_{\text{hom}N}(M)=\{w\in \Omega^k: \mathbf{n}w=0,\mathbf{n}dw=0\}$ (relative Neumann boundary condition), and $\mathcal{H}^k_N =\{w\in \Omega^k: \mathbf{n}w=0,dw=0,\delta w=0\}$ (Neumann harmonic fields), where $\delta$ is the codifferential. Let ${(\mathcal{H}^k_N)}^{\perp}$ be the $L^2$ orthogonal complement in $\Omega^k$.

Let $\Delta=-(d\delta +\delta d)$ be the usual Hodge Laplacian (with negative spectrum). Then if we restrict its domain, we get the Hodge-Neumann Laplacian $\Delta_N : \Omega^k_{\text{hom}N}(M) \cap {(\mathcal{H}^k_N)}^{\perp}\to {(\mathcal{H}^k_N)}^{\perp} $ which is $C^\infty$ homeomorphism and essentially $L^2$-self-adjoint. The inverse is called the Neumann potential, which is a compact operator on ${(\mathcal{H}^k_N)}^{\perp}$ (subspace of $L^2$), so there is an orthonormal basis consisting of eigenvectors (spectral theorem for compact operators). See Günter Schwarz for more details.

The formulas for integration by parts are:

$$ \ll du,v \gg_M = \ll u,\delta v \gg_M + \ll \jmath^* u, \jmath^* \iota_\nu v \gg_{\partial M} $$ $$ \mathcal{D}(u,v)= \ll -\Delta u,v \gg_M + \ll \jmath^* \iota_\nu du, \jmath^* v \gg_{\partial M} - \ll \jmath^* \delta u, \jmath^* \iota_\nu v \gg_{\partial M} $$ where $\ll \cdot, \cdot \gg_M$ is the $L^2$ product on $M$ and $\mathcal{D}(u,v) = \ll du, dv \gg_M + \ll \delta u, \delta v \gg_M $ is called the Dirichlet integral. On $D(\Delta_N)$, we also have the Poincare-Hodge inequality (consider $0 \in \mathbb{N}$)

$$ ||w||_{W^{s+1,p}} \sim ||dw||_{W^{s,p}} + ||\delta w||_{W^{s,p}} \; \forall s\in \mathbb{N}, \forall p \in [2,+\infty) $$

Question

Let $w \in \Omega^k$ and $w \in D(\Delta_N^K)\; \forall K \in \mathbb{N}$. Can we show there exists $C_{k,K,M}\geq 0$ such that $$ \int_M |w|^{2K} \Delta(|w|^2) \leq C_{k,K,M} \int_M |w|^{2K+2} \; \forall K \in \mathbb{N} $$

where $|.|$ is the fiber norm induced by the Riemannian metric (again, the differential forms version differs from the tensor version by a factor of $\frac{1}{k!}$). If this is true, it would have non-trivial implications for index theory, Hodge theory and fluid dynamics (being a form of dissipativity). It is certainly true (in fact $C_{k,K,M}=0$) when $w$ is a scalar function ($w=f$,$\partial_\nu f =0$) or $\partial M = \emptyset$, or that the metric is flat. So already most possible cases for PDE are covered. I want to know if this is true for general M. If that fails, how about $$ \ll |w|^{2K}w, \Delta w \gg_M \leq C_{k,K,M} \int_M |w|^{2K+2} \; \forall K \in \mathbb{N} $$ (which is a weaker statement, following from the previous by Bochner's formula)

My attempt

$\mathbf{n}w=0$ and $\mathbf{n}dw=0$ are equivalent to $\iota_\nu w = 0$ and $\iota_\nu dw = 0$. Integration by parts immediately yields

$$ \begin{align} \ll \Delta(|w|^2), |w|^{2K} \gg_M & = - \ll d(|w|^2), d(|w|^{2K}) \gg_M + \ll \partial_\nu(|w|^2), |w|^{2K} \gg_{\partial M} \\ & \leq \; \ll \partial_\nu(|w|^2), |w|^{2K} \gg_{\partial M} = 2 \ll \nabla_\nu w, |w|^{2K} w \gg_{\partial M} \\ & = \frac{2}{K+1} \int_{\partial M} \partial_\nu(|w|^{2(K+1)}) = \frac{2}{K+1} \int_M \Delta(|w|^{2(K+1)}) \end{align}$$

The last 2 equalities look nice (for future applications), but I'm not sure if they are useful here.

We shall use Penrose's abstract index notation (also confer Wald's General Relativity). Write $a_1,..,a_{n-1}$ for abstract indices representing coordinates on $\partial M$ and $n$ representing the outwards normal direction. Let $\nabla$ be the Levi-Civita connection, and $\partial$ be the ordinary derivative (in local coordinates) then

$$ \begin{align} 0 =(dw)_{na_1...a_k} & = \nabla_n w_{a_1...a_k} + \sum_i (\pm 1) \nabla_{a_i} w_{na_1...\widehat{a_i}...a_k} \\ &= \partial_n w_{a_1...a_k} + \sum_i (\pm 1) \partial_{a_i} w_{na_1...\widehat{a_i}...a_k} = \partial_n w_{a_1...a_k} \end{align} $$ since $ w_{na_1...\widehat{a_i}...a_k} = 0$. But I am not certain whether $\nabla_\nu w = 0$ or $\partial_\nu (|w|^2)=0$ (unless it's a flat metric).

Plugging this in, we get $$ \begin{align} k! \ll \nabla_\nu w, |w|^{2K} w \gg_{\partial M} & =\int_{\partial M} \nabla_n w_{a_1...a_k} |w|^{2K}w^{a_1...a_k} \\ & = \sum_i (\pm 1) \int_{\partial M} \nabla_{a_i}w_{na_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \end{align} $$

But the catch is that $\nabla_{a_i}w_{na_1...\widehat{a_i}...a_k} \neq \nabla_{a_i}(\iota_\nu w)_{a_1...\widehat{a_i}...a_k} $, so I'm stuck.

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1 Answer 1

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I think I've found the answer (maybe writing this down has made me think harder about it).

The application I have in mind uses interpolation, so $K\in \mathbb{N}$ was really just for simplicity's sake, and I can assume $K$ to be odd, $K+1=2L$, $L\in \mathbb{N}$.

Continuing from the end of my attempt: $$ \begin{align} &\int_{\partial M} \nabla_{a_i}w_{na_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \\ &= \int_{\partial M} \nabla_{a_i}{(\iota_\nu w)}_{a_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} - \int_{\partial M} {(\iota_{\nabla_{a_i}\nu} w)}_{a_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \\ \end{align} $$ Here the first term vanishes as $\iota_\nu w=0$, while the second term spits out some Christoffel symbols. Since the manifold is compact, we get $O( \int_{\partial M} |w|^{2K+2})$.

We can control this error by the negative quantity (which I previously threw away, prematurely) $$ - \ll d(|w|^2), d(|w|^{2K}) \gg_M = -K \int_M \left|d(|w|^2)\right|^2 |w|^{2(K-1)} $$ Let $f=|w|^2$, then we want $$C\int_{\partial M} f^{K+1} - \int_M |df|^2 f^{K-1} \lesssim \int_M f^{K+1}$$ where $C$ is some constant (depending on $k,K,M$). As $K+1=2L$, set $F=f^L$, and our problem, after simplification, becomes $$C\int_{\partial M} F^2 - \int_M |dF|^2 \lesssim \int_M F^2$$

This is just Ehrling's inequality. As $\tau:H^1(M)\to L^2(\partial M)$ compact and $H^1(M)\hookrightarrow L^2(M)$ continuous, we have $$ \forall \epsilon >0, \exists C_\epsilon > 0: ||\tau F||_{L^2(\partial M)} \leq \epsilon ||F||_{H^1} + C_\epsilon ||F||_{L^2} $$

I am very excited that this turns out to be true. This further confirms my belief that the above setting is, in a sense, universal.

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