Interior gradient regularity for BV minimizers of singular variational problems
Nonlinear Anal., Theory Methods Appl.
We consider a class of vectorial integrals with linear growth, where, as a key feature, some degenerate//singular behavior is allowed. For generalized minimizers of these integrals in BV, we establish interior gradient regularity and --- as a consequence --- uniqueness up to constants. In particular, these results apply, for $1<p<2$, to the singular model integrals
\int_\Omega(1+\lvert\nabla w(x)\rvert^p)^\frac1p \,dx\,.
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