Weighted $L^p$-theory for Poisson, biharmonic and Stokes problems on periodic unbounded strips of ${{\mathbb {R}}}^n$

Abstract

This paper establishes isomorphisms for Laplace, biharmonic and Stokes operators in weighted Sobolev spaces. The $W^{m,p}_{\alpha }({{\mathbb {R}}}^n)$-spaces are similar to standard Sobolev spaces $W^{m,p}_{}({\mathbb {R}}^n)$, but they are endowed with weights $(1+|x|^2)^{\alpha /2}$ prescribing functions’ growth or decay at infinity. Although well established in ${{\mathbb {R}}}^n$ [3], these weighted results do not apply in the specific hypothesis of periodicity. This kind of problem appears when studying singularly perturbed domains (roughness, sieves, porous media, etc): when zooming on a single perturbation pattern, one often ends with a periodic problem set on an infinite strip. We present a unified framework that enables a systematic treatment of such problems in the context of periodic strips. We provide existence and uniqueness of solutions in our weighted Sobolev spaces. This gives a refined description of solution’s behavior at infinity which is of importance in the multi-scale context. The isomorphisms are valid for any relative integer m, any p in $(1,\infty )$, and any real $\alpha$ out of a countable set of critical values for the Stokes, the biharmonic and the Laplace operators.

Compactly supported distributions, \({\mathcal E}'(\hat{G})\)

Theorem 10

If $u\in {\mathcal E}'(\hat{G})$ of order p, then for any test function $\psi \in {\mathcal E}(\hat{G})$ s.t. $D^j_{\xi ^j} \psi (\eta ) = 0$ for all points $\eta$ in the support of u and all $j \in \{0,\dots ,p\}$, one has $\langle u,\psi \rangle =0$, where the brackets denote the duality in ${\mathcal E}'(\hat{G}),{\mathcal E}(\hat{G})$.

Proof

As the support of u is compact in $\hat{G}$ there exists a finite collection of integers $J \subset {\mathbb {N}}$ and finite sequence of compact intervals $(K(j))_{j\in J}$ s.t. $\hbox {supp}u \in \{({\mathbf k}_j,K(j))_{j\in J}\}$, we denote the set of all $S:=\{ ({\mathbf k}_j)_{j\in J}\}$, a finite subset of ${\varPi }_{\ell =1}^{n-1} (2\pi$ $\mathbb {Z}/L_\ell )$. Let $K_\epsilon (j):=\{ \zeta \in {{\mathbb {R}}}\text { s.t. }\mathrm{d}(\zeta ,K(j))\le \epsilon \}$. We define the smooth cut-off function

$$\begin{aligned} \chi _{\epsilon }({\mathbf k},\xi ) := \frac{1}{\epsilon }{\left\{ \begin{array}{ll} \int _{K_{2\epsilon }(j)} \varphi ((\xi -\zeta )/\epsilon ) \mathrm{d}\zeta &{} \text { if } {\mathbf k}\in S, \\ 0&{} \text { otherwise}, \end{array}\right. } \end{aligned}$$

where $\varphi \in C^\infty _0({{\mathbb {R}}})$ s.t. $\varphi \ge 0$, $\int _{{{\mathbb {R}}}} \varphi (\xi )\mathrm{d}\xi =1$ and $\hbox {supp}\varphi \subset B(0,1)$. Then it is clear that $\chi _{\epsilon }({\mathbf k}_j,\cdot ) = 1$ in $K_\epsilon (j)$ and that $\hbox {supp}\chi _{\epsilon }({\mathbf k}_j,\cdot ) \subset K_{3\epsilon }(j)$. One has then $\langle u,\psi \rangle =\langle u,\psi \chi _\epsilon \rangle$ since $\psi (1-\chi _\epsilon )=0$ in a neighborhood of $\hbox {supp}u$. Hence

$$\begin{aligned} |\langle u,\psi \rangle | \le C \sup _{\eta \in \hat{G},j \le p} \left| D^j_{\xi ^j} ( \psi \chi _\epsilon ) \right| \end{aligned}$$

which by arguments similar to Theorem 1.5.4 [19] is bounded uniformly by $\epsilon$, and thus tends to 0, as $\epsilon \rightarrow 0$. Indeed, for any fixed ${\mathbf k}_j$ using

$$\begin{aligned} |\psi ({\mathbf k}_j,\xi ')| \le \frac{1}{(p+1)!} \sup _{t\in (0,1)} \left| D^{p+1}_{\xi ^{p+1}} \psi ( {\mathbf k}_j,\xi + t(\xi '-\xi )) \right| |\xi '-\xi |^{(p+1)}, \end{aligned}$$

for all $\xi \in {{\mathbb {R}}}$ and similar estimates for all derivatives of higher order, it implies that $\sup _{\xi ' \in K_{3 \epsilon }}$ $| D^q \psi ({\mathbf k}_j,\xi ') | \le c_{j,q} \epsilon ^{p+1-q}$. On the other hand $\sup _{\xi \in K_{3\epsilon }(j)} |D^{q'} \chi _\epsilon ({\mathbf k}_j,\xi ) | \le c'_{j,q'} \epsilon ^{-q'}$, which together with the Leibnitz formula gives that

$$\begin{aligned} |\langle u,\psi \rangle | \le c \sup _{j \in J} \sup _{ \xi \in K_{3 \epsilon }(j) } \begin{pmatrix} p \\ q \end{pmatrix} c_{j,q}\epsilon ^{p+1-q} c'_{j,p-q} \epsilon ^{-(p-q)} \le c \epsilon . \end{aligned}$$

Theorem 11

If $u\in {\mathcal E}'(\hat{G})$ s.t. $\hbox {supp}u \subset \{ 0 \}$ then for any $\varphi \in {\mathcal E}(\hat{G})$, there exists a $p\in {\mathbb {N}}$ s.t. $\langle u , \varphi \rangle = \sum _{r=0}^p c_{r} \partial _{\xi ^{r}}^{r} \varphi (0).$

Proof

Let p be the order of u. Set $\psi ({\mathbf k},\xi )= \varphi ({\mathbf k},\xi ) - \sum _{r=0}^p \partial _{\xi ^r}^r \varphi ({\mathbf k},0) \frac{\xi ^r}{r !}$ it is a smooth function s.t. any derivative in direction $\xi$ vanishes up to order p. Since u is compactly supported, one defines $\chi ({\mathbf k},\xi )={\mathbbm {1}}_{{\mathbf k}=0}$ which is in $C^\infty (\hat{G})$. Applying the previous result it comes $\langle u,\psi \rangle =\langle u,\chi \psi \rangle = 0$. Then using that u is linear, one recovers:

$$\begin{aligned} \langle u , \varphi \rangle = \left\langle u, \chi _{{\mathbf k}=0} \sum _{r=0}^p \partial _{\xi ^r}^r \varphi ({\mathbf k},0) \frac{\xi ^r}{r !}\right\rangle = \sum _{r=0}^p \left\langle u , \chi _{{\mathbf k}=0} \frac{\xi ^r}{r !}\right\rangle \partial _{\xi ^r}^r \varphi (0), \end{aligned}$$

which gives the definition of the constants $(c_r)_{r\in \{0,\dots ,p\}}$.