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 local quasiminimizers of the Dirichlet energy under generalized growth conditions. Special cases include standard, variable exponent and double phase growths. We show that the gradient of a ...local quasiminimizer has local higher integrability.
We study the Hardy-Littlewood maximal operator M on Lp(·) (X) when X is an unbounded (quasi) metric measure space, and p may be unbounded. We consider both the doubling and general measure case, and ...use two versions of the log-Hölder condition. As a special case we obtain the criterion for a boundedness of M on Lp(·) (Rn, μ) for arbitrary, possibly non-doubling, Radon measures.
We study minimizers of non-autonomous functionals
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when $ \varphi $ has generalized Orlicz growth. We consider the case where the upper growth rate of $ \varphi $ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $ \varphi $ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.
Capacities in Generalized Orlicz Spaces Baruah, Debangana; Harjulehto, Petteri; Hästö, Peter
Journal of function spaces,
01/2018, Letnik:
2018
Journal Article
Recenzirano
Odprti dostop
In this paper basic properties of both Sobolev and relative capacities are studied in generalized Orlicz spaces. The capacities are compared with each other and the Hausdorff measure. As an ...application, the existence of quasicontinuous representative of generalized Orlicz functions is proved.
Double phase image restoration Harjulehto, Petteri; Hästö, Peter
Journal of mathematical analysis and applications,
09/2021, Letnik:
501, Številka:
1
Journal Article
Recenzirano
Odprti dostop
In this paper we explore the potential of the double phase functional in an image processing context. To this end, we study minimizers of the double phase energy for functions with bounded variation ...and show that this energy can be obtained by Γ-convergence or relaxation of regularized functionals. A central tool is a capped fractional maximal function of the derivative of BV functions.
We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content H∞δ. In particular, if Ω is a bounded John domain in Rn, n≥2, and 0<δ≤n, we prove that the ...corresponding (δp/(δ−p),p)-Poincaré-Sobolev inequalities hold for all continuously differentiable functions defined on Ω whenever δ/n<p<δ. We prove also that the (p,p)-Poincaré inequality is valid for all p>δ/n.
We study minimizers of the Dirichlet φ‐energy integral with generalized Orlicz growth. We prove the Kellogg property, the set of irregular points has zero capacity, and give characterizations of ...semiregular boundary points. The results are new ever for the special cases double phase and Orlicz growth.
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