Abstract
Abstract In this paper we prove local Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands. The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the Hölder continuity. In the final section, we provide some non-trivial applications of our results.
Highlights
In this paper we establish Hölder regularity for vector-valued minimizers of a class of integral functionals of the Calculus of Variations
In this paper we prove local Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands
The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the Hölder continuity
Summary
In this paper we establish Hölder regularity for vector-valued minimizers of a class of integral functionals of the Calculus of Variations. We shall apply such results to minimizers of quasiconvex integrands, satisfying the natural condition to ensure existence in the vectorial setting. For equations and scalar integrals, such a topic is strictly related to the celebrated De Giorgi result in [1]. Several generalizations in the scalar case have been given, let us mention the contribution of Giaquinta-. The question whether the previous theory and results extend to systems and vectorial integrals was solved in [3] by De Giorgi himself constructing an example of a second order linear elliptic system with solution x |x|γ.
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