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.
Double phase image restoration Harjulehto, Petteri; Hästö, Peter
Journal of mathematical analysis and applications,
09/2021, Letnik:
501, Številka:
1
Journal Article
Recenzirano
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 establish maximal local regularity results of weak solutions or local minimizers of
div
A
(
x
,
D
u
)
=
0
and
min
u
∫
Ω
F
(
x
,
D
u
)
d
x
,
providing new ellipticity and continuity assumptions on
...A
or
F
with general (
p
,
q
)-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity
F
satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as
t
p
,
φ
(
t
)
,
t
p
(
x
)
,
t
p
+
a
(
x
)
t
q
, and not only
F
but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to, special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using
A
or
F
only, which entails that the function is quasi-isotropic, i.e. it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on
A
or
F
without specific structure and without direct restriction on the ratio
q
p
of the parameters from the (
p
,
q
)-growth condition. We establish local
C
1
,
α
-regularity for some
α
∈
(
0
,
1
)
and
C
α
-regularity for any
α
∈
(
0
,
1
)
of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.
We establish the W2,φ(⋅)-solvability of the linear elliptic equations in non-divergence form under a suitable, essentially minimal, condition of the generalized Orlicz function φ(⋅)=φ(x,t), by ...deriving Calderón–Zygmund type estimates. The class of generalized Orlicz spaces we consider here contains as special cases classical Lebesgue and Orlicz spaces, as well as non-standard growth cases like 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 prove higher integrability of the spatial gradient of weak solutions to parabolic systems with φ-growth, where φ=φ(t) is a general Orlicz function. The parabolic systems need be neither degenerate ...nor singular. Our result is a generalized version of the one of Kinnunen and Lewis (2000) 34 for the parabolic p-Laplace systems.
We study unbounded weak supersolutions of elliptic partial differential equations with generalized Orlicz (Musielak–Orlicz) growth. We show that they satisfy the weak Harnack inequality with optimal ...exponent provided that they belong to a suitable Lebesgue or Sobolev space. Furthermore, we establish the sharpness of our central assumptions.
In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove independence of the choice of basis functions, as well as several other basic properties. We ...also give Sobolev-type embeddings, and show that our scale contains variable order Hölder–Zygmund spaces as special cases. We provide an alternative characterization of the Besov space using approximations by analytic functions.
We study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea–Mihăilescu (Orlicz case) and Eleuteri–Prinari (variable exponent case) ...and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.
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