Relaxation of gap conditions under additional solution bounds

Relax the gap conditions eqref{gap_1} or eqref{gap_2} for vector-valued local minimizers of the double-phase Orlicz functional mathcal{F}_{mathbb{A},varphi_1,varphi_2}(u;Omega) under a priori boundedness or higher-integrability assumptions on the solution, and establish partial regularity in the resulting broader setting.

Background

The paper establishes partial regularity for vector-valued local minimizers of a non-autonomous double-phase functional with Orlicz growth, subject to gap conditions controlling the modulus of continuity of the phase coefficient and the relative growth of the two N-functions. The first main theorem treats a vanishing gap quantity, while the borderline theorem permits a bounded gap quantity when the phase coefficient is Hölder continuous.

The authors identify a possible extension in which these gap restrictions are weakened, provided that the minimizers satisfy additional a priori boundedness or higher-integrability assumptions. Such an extension would broaden the class of vectorial double-phase Orlicz problems covered by the partial regularity theory developed in the paper.

References

On the other hand, inspired by the literatures , we leave for future consideration the relaxation of the gap conditions gap_1 or gap_2 to show the partial regularity for the vector-valued minimizers of functional main_funl under a-priori boundedness or higher integrability assumptions on the solution.

gap_1:

lim sup⁡r→0+ ωa(r) (φ2∘φ1−1)(r−n)r−n=0.\limsup_{r \to 0^+}\,\omega_a(r)\, \frac{\big({\varphi_2} \circ \varphi_1^{-1}\big)\left(r^{-n}\right)}{r^{-n}} =0.

gap_2:

lim sup⁡r→0+ ωa(r) (φ2∘φ1−1)(r−n)r−n<∞.\limsup_{r \to 0^+}\,\omega_a(r)\, \frac{\big({\varphi_2} \circ \varphi_1^{-1}\big)\left(r^{-n}\right)}{r^{-n}} < \infty.

main_funl:

FA,φ1,φ2(u;Ω):=∫Ω(φ1(∣Du∣Au)+a(x) φ2(∣Du∣Au)) dx,\mathcal{F}_{\mathbb{A},{\varphi_1},{\varphi_2}}(u ; \Omega): = \int_\Omega \Big( {\varphi_1}\big(|Du|_{\mathbb{A}^u}\big) + a(x)\,{\varphi_2} \big(|Du|_{\mathbb{A}^u}\big) \Big)\, dx,

— Partial regularity for local minimizers of functionals with double-phase Orlicz growth  (2609.31242 - Chang et al., 25 Sep 2026) in Remark following Theorem 3, Introduction