Absence of the Lavrentiev phenomenon for degenerate parabolic double phase problems

TL;DR

This paper proves that for certain degenerate parabolic double phase problems, finite-energy solutions can be smoothly approximated without energy gaps, under specific conditions, ensuring better understanding of solution regularity.

Contribution

It establishes the absence of the Lavrentiev phenomenon for these problems and provides explicit gap bounds and improved estimates under additional assumptions.

Findings

Finite-energy functions can be approximated smoothly with energy convergence.

Explicit gap bounds are derived for the approximation.

Improved bounds are obtained under boundedness or stronger regularity assumptions.

Abstract

We establish the absence of the Lavrentiev phenomenon for degenerate parabolic double phase problems. Any finite-energy function in the natural parabolic class admits smooth approximations with convergence in the parabolic Sobolev space and convergence of the corresponding energy. We provide explicit gap bound conditions and derive improved bounds under additional assumptions such as boundedness or stronger time regularity.