On some coercive inequalities for linear partial differential operators (Q1288300)

Let \(X\) be a Banach space. The symbols \(C\), \(L_p\), and \(M_p\) stand for the Banach spaces of \(X\)-valued functions endowed with the norms \[ \| u\| _{C}=\sup_{x\in {\mathbb R}^n}\| u(x)\| _{X},\;\| u\| _{L_p}^p=\int_{{\mathbb R}^n}\| u(x)\| _{X}^p dx,\;\| u\| _{M_p}^p= \sup_{x\in {\mathbb R}^n}\int_{K(x)}\| u(y)\| _{X}^p dy, \] with \(K(x)\) the unit cube centered at \(x\in {\mathbb R}^n\). If \(F\) is one of these spaces then the space \(W^m(F)\) consists of functions whose derivatives up to order \(m\) belong to \(F\). Let \(L(X,X)\) denote the space of linear bounded operators from \(X\) into \(X\). A linear operator \[ P= \sum_{| \alpha| \leq m} A_{\alpha}(x)D^{\alpha}:W^m(F)\to F \quad \bigl(A_{\alpha}\in C({\mathbb R}^n;L(X,X))\bigr) \] is called coercive with respect to a seminorm \(\langle\cdot\rangle\) if, for every \(\varepsilon >0\), there exists a constant \(c(\varepsilon)\) such that \(\| u\| _{W^m(F)}\leq \varepsilon \langle u\rangle + c(\varepsilon)\| Pu\| _F\). The author studies this notion using one of the following seminorms or their sums: \[ \begin{aligned} \langle u\rangle _1 &=\sum_{| \alpha| =m}\sup_{x,y\in {\mathbb R}^n} \frac{\| D^{\alpha}u(x)-D^{\alpha}u(y)\| _{X}}{| x-y| ^{\gamma}} \quad \bigl(\gamma \in (0,1)\bigr), \\ \langle u\rangle _2^p &=\sum_{| \alpha| =m}\sup_{x,y\in {\mathbb R}^n} \int_{K(x)}\int_{K(y)}\frac{\| D^{\alpha}u(\xi)-D^{\alpha}u(\eta)\| _{X}^p}{| \xi-\eta| ^{n+\gamma p}} d\xi d\eta \quad \bigl(\gamma \in (0,1)\bigr), \\ \langle u\rangle _3^p &=\sum_{| \alpha| \leq m} \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \frac{\| D^{\alpha}u(x)-D^{\alpha}u(y)\| _{X}^p}{| x-y| ^{n+\gamma p}} dxdy \quad \bigl(\gamma \in (0,1)\bigr). \end{aligned} \] In particular, the author describes connections between coerciveness and bounded invertibility of the operator \(P\). The results are applied to the study of solvability of elliptic systems in \({\mathbb R}^n\).