Maximal regularity for non-autonomous evolution equations

Haak, Bernhard Hermann; Ouhabaz, El Maati

doi:10.1007/s00208-015-1199-7

Cited by 40 publications

(68 citation statements)

References 14 publications

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“…For q = 1 this is the condition used by Ouhabaz and Haak [5]. We also see that any of the conditions above is implied by α-Hölder continuity for an α > 1 2 .…”

Section: Comparison To Earlier Resultsmentioning

confidence: 61%

On non-autonomous maximal regularity for elliptic operators in divergence form

Auscher

Egert

2016

Arch. Math.

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We consider the Cauchy problem for non-autonomous forms inducing elliptic operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open subset Ω ⊆ R n . We obtain maximal regularity in L 2 (Ω) if the coefficients are bounded, uniformly elliptic, and satisfy a scale invariant bound on their fractional time-derivative of order one-half. Previous results even for such forms required control on a time-derivative of order larger than one-half.Mathematics Subject Classification. Primary 35K15; Secondary 47A07, 26A33.

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“…For q = 1 this is the condition used by Ouhabaz and Haak [5]. We also see that any of the conditions above is implied by α-Hölder continuity for an α > 1 2 .…”

Section: Comparison To Earlier Resultsmentioning

confidence: 61%

On non-autonomous maximal regularity for elliptic operators in divergence form

Auscher

Egert

2016

Arch. Math.

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“…Step 2: We show that

t \mapsto \int_{0}^{t} A (t) e^{- (t - s) A (t)} f (s) d s \in L^{2} (0, T; H)

. The proof here is much more elementary than the one of [, Lemma 2.5] thanks to our stronger condition on the form

fraktura

. By A (0) satisfies maximal regularity and

t \mapsto \int_{0}^{t} A (0) e^{- (t - s) A (0)} f (s) d s \in L^{2} (0, T; H) .

Thus it suffices to show that

ϕ : t \mapsto \int_{0}^{t} A (t) e^{- (t - s) A (t)} f (s) d s - \int_{0}^{t} A (0) e^{- (t - s) A (0)} f (s) d s \in L^{2} (0, T; H) .

As before we have

ϕ (t) = \int_{0}^{t} \frac{1}{2 π i} \int_{Γ} λ e^{- (t - s) λ} {(λ Id - A (t))}^{- 1} (scriptA (t) - scriptA (0)) {(λ Id - A (0))}^{- 1} f (s) d λ d s .

...…”

Section: Maximal Regularity In Hmentioning

confidence: 96%

“…For that we use the decomposition and show that

A (\cdot) u_{j} (\cdot) \in L^{2} (0, T; H)

for

j = 1, 2, 3

where

u_{1} (t) = e^{- t A (t)} u_{0}, u_{2} (t) = \int_{0}^{t} e^{- (t - s) A (t)} f (s) d s,

and u_{3} (t) = \int_{0}^{t} e^{- (t - s) A (t)} (scriptA (t) - scriptA (s)) u (s) d s .

Remark that implies for all

t, s \in [0, T]

A (t) - A (s) \in L (V, V_{γ}^{'}) and ∥ scriptA (t) - A (s) ∥_{L (V, V_{γ}^{'})} \leq c ω (| t - s |) .

We divide the proof into three steps to treat each

u_{j}

j = 1, 2, 3

separately. Step 1: We adapt the proof of [, Lemma 2.7] to our situation. Since

…”

Section: Maximal Regularity In Hmentioning

confidence: 99%

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Maximal regularity for non-autonomous Robin boundary conditions

Arendt

Monniaux

2016

Math. Nachr.

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We consider a non‐autonomous Cauchy problem u̇(t)+A(t)u(t)=f(t),u(0)=u0where A(t) is associated with the form a(t;.,.):V×V→C, where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in H1(0,T;H) (if u0∈V and f∈L2(0,T;H)) under a new regularity condition on the form fraktura with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.

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“…If the domains of the operators A(t) do not depend on time, we mention [42], [3], [5], and in the case of time-depending domains, we refer to [30], [31], [40], [6] and [27].…”

Section: Maximal Regularity Of Abstract Cauchy Problemsmentioning

confidence: 99%

Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces

Chill

Fiorenza

2014

J. Evol. Equ.

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To cite this version:Ralph Chill, Alberto Fiorenza. Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces. Journal of Evolution Equations, Springer Verlag, 2014, 14 (4), pp.795-828. 10.1007/s00028-014-0239-1. hal-01275671 SINGULAR INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS, AND EXTRAPOLATION OF MAXIMAL REGULARITY INTO REARRANGEMENT INVARIANT BANACH FUNCTION SPACESRALPH CHILL AND ALBERTO FIORENZA Abstract. We prove two extrapolation results for singular integral operators with operator-valued kernels and we apply these results in order to obtain the following extrapolation of L p -maximal regularity: if an autonomous Cauchy problem on a Banach space has L p -maximal regularity for some p ∈ (1, ∞), then it has E w -maximal regularity for every rearrangement-invariant Banach function space E with Boyd indices 1 < p E ≤ q E < ∞ and every Muckenhoupt weight w ∈ A p E . We prove a similar result for non-autonomous Cauchy problems on the line.

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Maximal regularity for non-autonomous evolution equations

Cited by 40 publications

References 14 publications

On non-autonomous maximal regularity for elliptic operators in divergence form

On non-autonomous maximal regularity for elliptic operators in divergence form

Maximal regularity for non-autonomous Robin boundary conditions

Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces

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