## 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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