Optimal regularity for the parabolic no-sign obstacle type problem

Andersson, John; Lindgren, Erik; Shahgholian, Henrik

doi:10.4171/ifb/311

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…In [28], the elliptic part of the operator is allowed to be fully nonlinear. A slightly more general free boundary problem of parabolic type is studied in [6], [12], [11] and [1].…”

Section: Known Resultsmentioning

confidence: 99%

Optimal regularity for the obstacle problem for the p-Laplacian

Andersson

Lindgren

Shahgholian

2015

Journal of Differential Equations

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Abstract. In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such thatwith φ ∈ C 1,1 (B 1 ) and given boundary datum on ∂B 1 . We prove that the solution is uniformly C 1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L ∞ . When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C 1,

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“…In [28], the elliptic part of the operator is allowed to be fully nonlinear. A slightly more general free boundary problem of parabolic type is studied in [6], [12], [11] and [1].…”

Section: Known Resultsmentioning

confidence: 99%

Optimal regularity for the obstacle problem for the p-Laplacian

Andersson

Lindgren

Shahgholian

2015

Journal of Differential Equations

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“…In the elliptic case, it has been recently studied by the authors in [10]. We also refer to several other articles concerning similar type of problems: For elliptic case see [5], [1], and for parabolic case see [6], [2]. One may find applications and relevant discussions about these kinds of problems in these articles.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A general class of free boundary problems for fully nonlinear parabolic equations

Figalli

Shahgholian

2014

Annali di Matematica

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In this paper we consider the fully nonlinear parabolic free boundary problemwhere K > 0 is a positive constant, and Ω is an (unknown) open set.Our main result is the optimal regularity for solutions to this problem: namely, we prove that W 2,n x ∩ W 1,n t solutions are locally C 1,1x ∩ C 0,1 t inside Q1. A key starting point for this result is a new BMO-type estimate which extends to the parabolic setting the main result in [4].Once optimal regularity for u is obtained, we also show regularity for the free boundary ∂Ω ∩ Q1 under the extra condition that Ω ⊃ {u = 0}, and a uniform thickness assumption on the coincidence set {u = 0}, A.

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“…In [28], the elliptic part of the operator is allowed to be fully nonlinear. A slightly more general free boundary problem of parabolic type is studied in [6], [12], [11] and [1].…”

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confidence: 99%

Optimal regularity for the obstacle problem for the $p$-Laplacian

Andersson¹,

Lindgren²,

Shahgholian³

2014

Preprint

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In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such that) and given boundary datum on ∂B 1 . We prove that the solution is uniformly C 1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L ∞ . When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C 1, 1 p−1 in the spatial variables.

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Optimal regularity for the parabolic no-sign obstacle type problem

Cited by 4 publications

References 8 publications

Optimal regularity for the obstacle problem for the p-Laplacian

Optimal regularity for the obstacle problem for the p-Laplacian

A general class of free boundary problems for fully nonlinear parabolic equations

Optimal regularity for the obstacle problem for the $p$-Laplacian

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