<p>We establish continuity and uniqueness results for the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(BV(\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> minimisers of multidimensional scalar variational problems formulated on a bounded open set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. The integrand is assumed to be convex (but not necessarily strictly convex), with linear growth from below (but not necessarily from above), while the domain <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> and the boundary condition must satisfy suitable geometric and regularity conditions, such as convexity or Lipschitz continuity.</p>