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A Gagliardo-Nirenberg inequality, with application to duality-based a posteriori estimation in the $$L^1$$ norm. (English) Zbl 1199.46086
Author’s abstract: We prove the Gagliardo-Nirenberg-type multiplicative interpolation inequality $\| v\| _{L^1(\mathbb R^n)}\leq C\| v\| ^{1/2}_{\operatorname{Lip}'(\mathbb R^n)}\| v\| ^{1/2}_{BV(\mathbb R^n)} \quad\forall v\in\operatorname{Lip}'(\mathbb R^n)\cap BV(\mathbb R^n),$ where $$C$$ is a positive constant, independent of $$v$$. Here, $$\| \cdot\| _{\operatorname{Lip}'(\mathbb R^n)}$$ is the norm of the dual to the Lipschitz space $$\operatorname{Lip}_0(\mathbb R^n)=C_0^{0,1}(\mathbb R^n)=C^{0,1}(\mathbb R^n)\cap C_0(\mathbb R^n)$$ and $$\| \cdot\| _{BV(\mathbb R^n)}$$ signifies the norm in the space $$BV(\mathbb R^n)$$ consisting of functions of bounded variation on $$\mathbb R^n$$. We then use a local version of this inequality to derive an a posteriori error bound in the $$L^1({\Omega}')$$ norm, with $$\bar\Omega'=(0,1)^n$$, 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 $$BV(\Omega)$$.
##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems