The Dirichlet problem for second order parabolic operators

Nyström, Kaj

doi:10.1512/iumj.1997.46.1277

Cited by 46 publications

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“…We assume that u vanishes continuously on a part of the boundary and we study the type of geometric assumptions we need to pose on the boundary in order to prove the Carleson estimate for (2), initially proved for equations of p-parabolic type by the author together with Ugo Gianazza and Sandro Salsa for cylindrical Lipschitz domains, in [1]. In the linear ( p = 2) case, the Carleson seems to be very useful, see for example [9][10][11][12][13][14][15]18,20], thus it seems meaningful to establish it for p > 2.…”

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confidence: 99%

On time dependent domains for the degenerate $$p$$ p -parabolic equation: Carleson estimate and Hölder continuity

Avelin

2015

Math. Ann.

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In this paper we propose a definition of "parabolic NTA" for solutions to the degenerate p-parabolic equation. Given this definition we prove the Carleson estimate, originally proved for this equation in Avelin et al. (J Eur Math Soc, 2015) for cylindrical domains. Moreover we study a non-optimal, stronger "outer corkscrew" condition, such that we obtain Hölder continuity up to the boundary, for non-negative solutions vanishing on a part of the boundary.Mathematics Subject Classification Primary 35K20; Secondary 35K65 · 35K92 · 35B65

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confidence: 99%

On time dependent domains for the degenerate $$p$$ p -parabolic equation: Carleson estimate and Hölder continuity

Avelin

2015

Math. Ann.

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“…We remark that, while these authors work in Lipschitz cylinders, one can easily see that their proofs can be generalized to the setting of bounded NTA-cylinders. While the works Fabes, Safonov and Yuan completed, for linear uniformly parabolic equations, the line of research considered in this paper, contributions by other researchers are contained in [17], [23], [25], [33], [16], [21], [44]. For the elliptic versions of Theorem 1.1-Theorem 1.3 we refer to [18], [19], [20], and we emphasize that in the elliptic case the assumption λ ∈ A 2 (R n ) on the weight is sufficient for the validity of the corresponding versions of Theorem 1.1-Theorem 1.3.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…The proof of the corresponding lemma in [44] can easily be adapted to prove Lemma 5.3. We omit further details.…”

Section: Remark 52mentioning

confidence: 99%

Boundary estimates for solutions to linear degenerate parabolic equations

Nyström

Persson

Sande

2015

Journal of Differential Equations

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Let Ω ⊂ R n be a bounded NTA-domain and let Ω T = Ω × (0, T ) for some T > 0. We study the boundary behaviour of non-negative solutions to the equationWe assume that A(x, t) = {a ij (x, t)} is measurable, real, symmetric and thatfor some constant β ≥ 1 and for some non-negative and real-valued function λ = λ(x) belonging to the Muckenhoupt class A 1+2/n (R n ). Our main results include the doubling property of the associated parabolic measure and the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes, Kenig, Jerison, Serapioni, see [18], [19], [20], to a parabolic setting.

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“…Apart from these references many of the relevant ideas used in the proofs can also be found in [5], [6] and [19]. In particular, in [19] all relevant estimates are stated and proved, in Lip(1, 1/2) domains, in the general setting of second order parabolic equations in divergence form.…”

Section: Basic Estimatesmentioning

confidence: 99%

Regularity below the continuous threshold in a two-phase parabolic free boundary problem

Nyström¹

2006

MATH. SCAND.

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In this paper we study free boundary regularity in a parabolic two-phase problem below the continuous threshold. We consider unbounded domains ⊂ R n+1 assuming that ∂ separates R n+1 into two connected components 1 = and 2 = R n+1 \ . We furthermore assume that both 1 and 2 are parabolic NTA-domains, that ∂ is Ahlfors regular and for i ∈ {1, 2} we define ω i (X i ,t i , ·) to be the caloric measure at (X i ,t i ) ∈ i defined with respect to i . In the paper we make the additional assumption that ω i (X i ,t i , ·), for i ∈ {1, 2}, is absolutely continuous with respect to an appropriate surface measure σ on ∂ and that the Poisson kernels k i (X i ,t i , ·) = dω i (X i ,t i , ·)/dσ are such that log k i (X i ,t i , ·) ∈ VMO(dσ ).Our main result (Theorem 1) states that, under these assumptions, C r (X, t) ∩ ∂ is Reifenberg flat with vanishing constant whenever (X, t) ∈ ∂ and min{t 1 ,t 2 } > t + 4r 2 . This result has an important consequence (Theorem 3) stating that if the two-phase condition on the Poisson kernels is fulfilled, 1 and 2 are parabolic NTA-domains and ∂ is Ahlfors regular then if is close to being a chord arc domain with vanishing constant we can in fact conclude that is a chord arc domain with vanishing constant.

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The Dirichlet problem for second order parabolic operators

Cited by 46 publications

References 10 publications

On time dependent domains for the degenerate $$p$$ p -parabolic equation: Carleson estimate and Hölder continuity

On time dependent domains for the degenerate $$p$$ p -parabolic equation: Carleson estimate and Hölder continuity

Boundary estimates for solutions to linear degenerate parabolic equations

Regularity below the continuous threshold in a two-phase parabolic free boundary problem

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