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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 \(BV(\Omega)\).
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