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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
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