We consider the functional∫Ωg(∇u+X∗) dL2nwheregis convex andX∗(x,y)=2(−y,x)and we study the minimizers in BV(Ω) of the associated Dirichlet problem. We prove that,under the bounded slope condition on the boundary datum, and suitable conditions ong,there exists a unique minimizer which is also Lipschitz continuous. The assumptions ongallow to consider both the case with superlinear growth and the one with linear growth.Moreover neither uniform ellipticity nor smoothness ofgare assumed.
Lipschitz minimizers for a class of integral functionals under the bounded slope condition
Giulia Treu
2022
Abstract
We consider the functional∫Ωg(∇u+X∗) dL2nwheregis convex andX∗(x,y)=2(−y,x)and we study the minimizers in BV(Ω) of the associated Dirichlet problem. We prove that,under the bounded slope condition on the boundary datum, and suitable conditions ong,there exists a unique minimizer which is also Lipschitz continuous. The assumptions ongallow to consider both the case with superlinear growth and the one with linear growth.Moreover neither uniform ellipticity nor smoothness ofgare assumed.
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