The regularity of some vector-valued variational inequalities with gradient constraints

TL;DR

This paper establishes optimal regularity results for certain vector-valued variational inequalities with gradient constraints, also providing new proofs and equivalence to obstacle problems in scalar and vector cases.

Contribution

It introduces new proofs for regularity and demonstrates the equivalence of gradient-constrained variational inequalities to obstacle problems.

Findings

Proved optimal regularity for vector-valued variational inequalities.

Provided a new proof for scalar variational inequalities.

Established equivalence to obstacle problems.

Abstract

We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar and vector-valued case.