Local boundedness for vectorial minimizers of non-uniform variational integrals

Abstract

We establish the local boundedness of vectorial local minimizers for a specific class of integral functionals with rank-one convex integrands under appropriate structural assumptions. Our method adapts the renowned De Giorgi‘s iteration technique and employs a suitable Caccioppoli-type inequality. Our findings are applicable to polyconvex integrals$\underset{\Omega }{\int }\left\{\sum _{\alpha =1}^N\lambda \right(x\left)\right|D{u}^{\alpha }{|}^p+\mu \left(x\right)\left|Du{|}^r\right\}dx$ with suitable $\lambda \left(x\right),\mu \left(x\right)>0$ and $p,r>1$.