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Lower semicontinuity for an integral functional in BV


We prove a lower semicontinuity result for a functional of linear growth initially defined by

$$\begin{aligned} \int _{\Omega }F\left( \frac{dDu}{d\mu }\right) \,d\mu \end{aligned}$$

for \(u\in \mathrm {BV}(\Omega ;{\mathbb R}^N)\) with \(Du\ll \mu \). The positive Radon measure \(\mu \) is only assumed to satisfy \(\mathcal L^n\ll \mu \).

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This research was done while P. L. was visiting the Oxford Centre for Nonlinear Partial Differential Equations (OxPDE) from September 2014 to July 2016. During this time, P.L. was supported by Aalto University as well as the Finnish Cultural Foundation.

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Correspondence to Panu Lahti.

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Communicated by L. Ambrosio.

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Kristensen, J., Lahti, P. Lower semicontinuity for an integral functional in BV. Calc. Var. 55, 70 (2016).

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Mathematics Subject Classification

  • 49J45
  • 26B30
  • 52A99