In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding.
Statement of the inequality Let Ω {\displaystyle \Omega } be a bounded, open domain in n {\displaystyle n} -dimensional Euclidean space and let H k ( Ω ) {\displaystyle H^{k}(\Omega )} denote the Sobolev space of k {\displaystyle k} -times weakly differentiable functions u : Ω → R {\displaystyle u\colon \Omega \rightarrow \mathbb {R} } with weak derivatives in L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} . Assume that Ω {\displaystyle \Omega } satisfies the k {\displaystyle k} -extension property, i.e., that there exists a bounded linear operator E : H k ( Ω ) → H k ( R n ) {\displaystyle E\colon H^{k}(\Omega )\rightarrow H^{k}(\mathbb {R} ^{n})} such that E u | Ω = u {\displaystyle Eu\vert _{\Omega }=u} for all u ∈ H k ( Ω ) {\displaystyle u\in H^{k}(\Omega )} . Let L be a linear partial differential operator of even order 2k, written in divergence form
( L u ) ( x ) = ∑ 0 ≤ | α | , | β | ≤ k ( − 1 ) | α | D α ( A α β ( x ) D β u ( x ) ) , {\displaystyle (Lu)(x)=\sum _{0\leq |\alpha |,|\beta |\leq k}(-1)^{|\alpha |}\mathrm {D} ^{\alpha }\left(A_{\alpha \beta }(x)\mathrm {D} ^{\beta }u(x)\right),}
and suppose that L is uniformly elliptic, i.e., there exists a constant θ > 0 such that
∑ | α | , | β | = k ξ α A α β ( x ) ξ β > θ | ξ | 2 k for all x ∈ Ω , ξ ∈ R n ∖ { 0 } . {\displaystyle \sum _{|\alpha |,|\beta |=k}\xi ^{\alpha }A_{\alpha \beta }(x)\xi ^{\beta }>\theta |\xi |^{2k}{\mbox{ for all }}x\in \Omega ,\xi \in \mathbb {R} ^{n}\setminus \{0\}.}
Finally, suppose that the coefficients Aαβ are bounded, continuous functions on the closure of Ω for |α| = |β| = k and that
A α β ∈ L ∞ ( Ω ) for all | α | , | β | ≤ k . {\displaystyle A_{\alpha \beta }\in L^{\infty }(\Omega ){\mbox{ for all }}|\alpha |,|\beta |\leq k.}
Then Gårding's inequality holds: there exist constants C > 0 and G ≥ 0
B [ u , u ] + G ‖ u ‖ L 2 ( Ω ) 2 ≥ C ‖ u ‖ H k ( Ω ) 2 for all u ∈ H 0 k ( Ω ) , {\displaystyle B[u,u]+G\|u\|_{L^{2}(\Omega )}^{2}\geq C\|u\|_{H^{k}(\Omega )}^{2}{\mbox{ for all }}u\in H_{0}^{k}(\Omega ),}
where
B [ v , u ] = ∑ 0 ≤ | α | , | β | ≤ k ∫ Ω A α β ( x ) D α u ( x ) D β v ( x ) d x {\displaystyle B[v,u]=\sum _{0\leq |\alpha |,|\beta |\leq k}\int _{\Omega }A_{\alpha \beta }(x)\mathrm {D} ^{\alpha }u(x)\mathrm {D} ^{\beta }v(x)\,\mathrm {d} x}
is the bilinear form associated to the operator L.
Application: the Laplace operator and the Poisson problem Be careful, in this application, Garding's Inequality seems useless here as the final result is a direct consequence of Poincaré's Inequality, or Friedrich Inequality. (See talk on the article). As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L2(Ω) the Poisson equation
{ − Δ u ( x ) = f ( x ) , x ∈ Ω ; u ( x ) = 0 , x ∈ ∂ Ω ; {\displaystyle {\begin{cases}-\Delta u(x)=f(x),&x\in \Omega ;\\u(x)=0,&x\in \partial \Omega ;\end{cases}}}
where Ω is a bounded Lipschitz domain in Rn. The corresponding weak form of the problem is to find u in the Sobolev space H01(Ω) such that
B [ u , v ] = ⟨ f , v ⟩ for all v ∈ H 0 1 ( Ω ) , {\displaystyle B[u,v]=\langle f,v\rangle {\mbox{ for all }}v\in H_{0}^{1}(\Omega ),}
where
B [ u , v ] = ∫ Ω ∇ u ( x ) ⋅ ∇ v ( x ) d x , {\displaystyle B[u,v]=\int _{\Omega }\nabla u(x)\cdot \nabla v(x)\,\mathrm {d} x,}
⟨ f , v ⟩ = ∫ Ω f ( x ) v ( x ) d x . {\displaystyle \langle f,v\rangle =\int _{\Omega }f(x)v(x)\,\mathrm {d} x.}
The Lax–Milgram lemma ensures that if the bilinear form B is both continuous and elliptic with respect to the norm on H01(Ω), then, for each f ∈ L2(Ω), a unique solution u must exist in H01(Ω). The hypotheses of Gårding's inequality are easy to verify for the Laplace operator Δ, so there exist constants C and G ≥ 0
B [ u , u ] ≥ C ‖ u ‖ H 1 ( Ω ) 2 − G ‖ u ‖ L 2 ( Ω ) 2 for all u ∈ H 0 1 ( Ω ) . {\displaystyle B[u,u]\geq C\|u\|_{H^{1}(\Omega )}^{2}-G\|u\|_{L^{2}(\Omega )}^{2}{\mbox{ for all }}u\in H_{0}^{1}(\Omega ).}
Applying the Poincaré inequality allows the two terms on the right-hand side to be combined, yielding a new constant K > 0 with
B [ u , u ] ≥ K ‖ u ‖ H 1 ( Ω ) 2 for all u ∈ H 0 1 ( Ω ) , {\displaystyle B[u,u]\geq K\|u\|_{H^{1}(\Omega )}^{2}{\mbox{ for all }}u\in H_{0}^{1}(\Omega ),}
which is precisely the statement that B is elliptic. The continuity of B is even easier to see: simply apply the Cauchy–Schwarz inequality and the fact that the Sobolev norm is controlled by the L2 norm of the gradient.
References Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. (Theorem 9.17)