A Gagliardo-Nirenberg inequality, with application to duality-based a posteriori estimation in the L^1 norm

Endre Suli

We prove the Gagliardo--Nirenberg-type multiplicative interpolation inequality \[ \|v\|_{L^1({\mathbb R}^n)} eq C \|v\|^{1/2}_{{\rm Lip}'({\mathbb R}^n)} \|v\|^{1/2}_{{\rm BV}({\mathbb R}^n)}\qquad \forall v \in {\rm Lip}'(\Rn)\cap {\rm BV}({\mathbb R}^n), \] where $C$ is a positive constant, independent of $v$. Here $\|\cdot\|_{{\rm Lip}'({\mathbb R}^n)}$ is the norm of the dual to the Lipschitz space ${\rm Lip}_{\,0}({\mathbb R}^n) := {\rm C}^{0,1}_0({\mathbb R}^n)={\rm C}^{0,1}(\Rn) \cap {\rm C}_0(\Rn)$ and $\|\cdot\|_{{\rm BV}({\mathbb R}^n)}$ signifies the norm in the space ${\rm BV}({\mathbb R}^n)$ consisting of functions of bounded variation on $\Rn$. We then use a local version of this inequality to derive an {\em a posteriori} error bound in the ${\rm L}^1(\Omega')$ norm, with $\bar\Omega' \subset\Omega=(0,1)^n$, for a finite element approximation to a boundary-value problem for a first-order linear hyperbolic equation, under the limited regularity requirement that the solution to the problem belongs to ${\rm BV}(\Omega)$.}