We prove Harnack's inequality for local (quasi)minimizers in generalized Orlicz spaces without polynomial growth or coercivity conditions. As a consequence, we obtain the local Hölder continuity of local (quasi)minimizers. The results include as special cases standard, variable exponent and double phase growth.
We study the balayage related to the supersolutions of the variable exponent p(·)-Laplace equation. We prove the fundamental convergence theorem for the balayage and apply it for proving the Kellogg property, boundary regularity results for the balayage, and a removability theorem for p(·)-solutions.
We prove the strong minimum principle for non-negative quasisuperminimizers of the variable exponent Dirichlet energy integral under the assumption that the exponent has modulus of continuity slightly more general than Lipschitz. The proof is based on a new version of the weak Harnack estimate.
RésuméNous prouvons le fort principe du minimum pour des quasisuperminimizeurs non-négatifs de problème de Dirichlet de l'exposant variable en supposant que l'exposant a le module de continuité un peu plus général que Lipschitz. La démonstration est fondée sur une nouvelle version de la faible inégalité de Harnack.
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