H\"{o}lder regularity for logarithmic double phase problems

We investigate boundedness and regularity properties of weak solutions to a class of generalized logarithmic double phase equations with variable exponents.

The considered operators arise from Musielak-Orlicz type energies of logarithmic double phase type and exhibit nonstandard growth features.

Under general structural assumptions, we derive a priori boundedness estimates in the subcritical setting and establish boundedness of weak solutions also in the presence of critical growth terms.

In addition, we prove global Hölder continuity up to the boundary by means of the De Giorgi iteration scheme, localization arguments, and the frozen functional technique.

The obtained results extend several existing regularity results for double phase and related nonstandard growth problems.