Boundary estimates for solutions to the two-phase parabolic obstacle problem

Apushkinskaya, D. E.; Uraltseva, Nina

doi:10.1007/s10958-009-9284-7

Cited by 9 publications

(17 citation statements)

References 2 publications

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“…From here on we will no longer, with the exception of Lemma 16, need any explicit calculations using the curl operator and we will therefore write R n instead of R 3 . Some of the ideas in this section to control the growth of blow up sequences comes from [2]. Proposition 2.…”

Section: Flatness Of the Free Boundarymentioning

confidence: 99%

“…But due to Korn's inequality both conditions are equivalent. If we denote Π = {x; x 3 = 0} and Λ u = {x ∈ Π; u 3 (x) = 0} then it is easy to see that the minimizers solves the following Euler-Lagrange equations (2) Lu ≡ ∆u + 2+λ 2 ∇div(u) = 0 in R 3…”

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confidence: 99%

“…Instead of maximum principle methods we will rely on the blow-up method, the Liouiville Theorem and linearization together with some simple geometric observations and a nice way to control blow-up sequences that we get from [2].…”

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confidence: 99%

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Optimal regularity for the Signorini problem and its free boundary

Andersson

2015

Invent. math.

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We will show optimal regularity for minimizers of the Signorini problem for the Lame system. InThen u ∈ C 1,1/2 (B + 1/2 ). Moreover the free boundary, given by Γu = ∂{x; u 3 (x) = 0, x 3 = 0} ∩ B 1 , will be a C 1,α graph close to points where u is not degenerate.Similar results have been know before for scalar partial differential equations (see for instance [4] and [5]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.

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Section: Flatness Of the Free Boundarymentioning

confidence: 99%

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confidence: 99%

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Optimal regularity for the Signorini problem and its free boundary

Andersson

2015

Invent. math.

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“…Remark 3.2. More detailed information about Caffarelli's monotonicity formula and its local rescaled version can be found in book [18] and in papers [14,19], respectively.…”

Section: Historical Reviewmentioning

confidence: 99%

Free boundaries in problems with hysteresis

Apushkinskaya

Uraltseva

2015

Phil. Trans. R. Soc. A.

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Here, we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes 'with memory' in which various substances interact according to hysteresis law. Our main objective is to discuss the structure of the free boundaries and the properties of the socalled 'strong solutions' belonging to the anisotropic Sobolev class W 2,1 q with sufficiently large q. Several open problems in this direction are proposed as well.

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“…x 2 1 2 is caloric in R n+1 ∩ {t ≤ 0} and has quadratic growth with respect to x and at most linear growth with respect to t. By the Liouville theorem (see Lemma 2.1 in [AUS1]), the function v 0 , and, consequently, the function u 0 , is a polynomial of degree 2, i.e., there exist constants a i ≥ 0 and c ≥ 0 such that the exact representation (2.10) is valid.…”

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confidence: 99%