Scalar minimizers with maximal singular sets and lack of Meyers property

TL;DR

This paper introduces a method to construct convex scalar variational problems with minimizers that have maximal singular sets and lack higher integrability, challenging existing assumptions about regularity.

Contribution

It provides a general procedure for creating examples of variational problems with highly singular minimizers and demonstrates the failure of Meyers property in these cases.

Findings

Maximal dimension of singular sets in constructed minimizers

Existence of convex scalar problems with non-integrable minimizers

Failure of Meyers property in these examples

Abstract

We present a general procedure to construct examples of convex scalar variational problems which admit a minimizers with large singular sets. The dimension of the set of singularities is maximal and the minimizer has no higher integrability property (failure of Meyers property).