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On Gårding’s inequality. (English) Zbl 0635.35011

Gårding examined a linear differential operator of order m, (1) \(q=\sum_{| \beta | \leq m}q_{\beta}(x)D_{\beta}\), where \(q_{\beta}(x)\) are assumed for simplicity to be infinitely differentiable in an open subset, S, of real n-dimensional space. Restricting the right-hand side of (1) to be derivatives of order exactly equal to m, gives an operator called the principal part of q. The principal part of q together with its associated polynomial, \(q_ m(x,\xi)=\sum_{| \beta | =m}q_{\beta}(x)\xi_{\beta}\), is elliptic iff \(q_ m(x,\xi)\neq 0\), \(x\in S\) and \(\xi =0.\)
This paper replaces (1) with (2) \(L(\cdot,D)=\sum_{\sigma,\tau \in \Gamma}D^{\sigma}(a_{\sigma \tau}(\cdot)D^{\tau})\) where the functions \(a_{\sigma \tau}\) are bounded and sufficiently smooth in an open bounded set, \(G\subset {\mathbb{R}}^ n\). \(\Gamma\) is a finite and non- empty subset of \(N^ n_ 0=\{(\alpha_ 1,...,\alpha_ n)|\alpha_ i\) positive integers, \(1\leq i\leq n\}\). The length of \(\alpha\) is defined to be \(| \alpha | =\sum^{n}_{i=1}\alpha_ i\). The space \(C_ 0^{\infty}(G)\) is equipped with a scalar product \((\phi,\psi)_{\Gamma}=\sum_{\sigma \in \Gamma}(D^{\sigma}\phi,D^{\sigma}\psi)_ 0+(\phi,\psi)_ 0\), where \((\phi,\psi)_ 0=\int_{G}\overline{\phi (x)}\psi (x)dx\) is the \(L^ 2(G)\) scalar product.
The set \(\Gamma\) is examined and if it admits a decomposition \(\Gamma =\Gamma^*\cup \Gamma '\), \(\Gamma^*\cap \Gamma '=\emptyset\) several generalized results follow. It is noted that \(\Gamma^*\) plays the role of \(\{\alpha \in N^ n_ 0| | \alpha | =m\}\). From the operator (2) a sesquilinear form, \(B(.,.)\) on \(C_ 0^{\infty}(\sigma)\times C_ 0^{\infty}(\sigma)\) is defined by \[ B(\phi,\psi)=\sum_{\sigma,\tau \in \Gamma}\int_{G}\overline{a_{\sigma\tau}(x) D^{\tau}\phi(x)} D^{\sigma}\psi(x) dx \] and Gårding’s inequality becomes Real \(B(\phi,\psi) \geq C_ 0 \| \phi \|^ 2_{\Gamma^*}-C_ 1\| \phi \|^ 2_ 0\) for all \(\phi \in C_ 0^{\infty}(b)\) under appropriate conditions. A form of a compact imbedding theorem in a Hilbert space is proven and the paper concludes with a wonderful example illustrating the results with the implementation of the operator \[ L(D)=-\frac{\partial^ 6}{\partial x^ 2_ 1 \partial x^ 4_ 2}- \frac{\partial^ 6}{\partial x^ 4_ 1 \partial x^ 2_ 2}- \frac{\partial^ 4}{\partial x^ 2_ 1 \partial x^ 2_ 2}. \]
Reviewer: J.Schmeelk

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35G15 Boundary value problems for linear higher-order PDEs
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:

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