Full regularity of the free boundary in a Bernoulli-type problem in two dimensions

Danielli, Donatella; Petrosyan, Arshak

doi:10.4310/mrl.2006.v13.n4.a14

Cited by 17 publications

(24 citation statements)

References 8 publications

(19 reference statements)

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“…For other references related to the free boundary problem under consideration in this paper we would like to refer the reader to [3], [4], [5], [9], [10], [11], [27], [28], [30], [31], [32], [34], [35] and the references therein. This list is by no means exhaustive.…”

Section: Introductionmentioning

confidence: 99%

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Lederman¹,

Wolanski²

2017

Interfaces Free Bound.

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Abstract. In this paper we study a one phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We prove that the free boundary of a weak solution is a C 1,α surface in a neighborhood of every "flat" free boundary point. We also obtain further regularity results on the free boundary, under further regularity assumptions on the data. We apply these results to limit functions of an inhomogeneous singular perturbation problem for the p(x)-Laplacian that we studied in [25]. IntroductionIn this paper we study the following inhomogeneous free boundary problem for the p(x)-Laplacian: u ≥ 0 andThe p(x)-Laplacian serves as a model for a stationary non-newtonian fluid with properties depending on the point in the region where it moves. For example, such a situation corresponds to an electrorheological fluid. These are fluids such that their properties depend on the magnitude of the electric field applied to it. In some cases, fluid and Maxwell's equations become uncoupled and a single equation for the p(x)-Laplacian appears (see [33]).The free boundary problem P (f, p, λ * ) appears, for instance, in the limit of a singular perturbation problem that may model high activation energy deflagration flames in a fluid with electromagnetic sensitivity (see [25]). When p(x) ≡ 2 (in which case the p(x)-Laplacian coincides with the Laplacian) this singular perturbation problem was introduced by Zeldovich and Frank-Kamenetski in order to model these kind of flames in [37]. In this latter case, the right hand side f may come from nonlocal effects as well as from external sources (see [23]).The free boundary problem considered in this paper also appears in an inhomogeneous minimization problem that we study in [26] where we prove that minimizers are weak solutions to P (f, p, λ * ).In the present article we prove that the free boundary ∂{u > 0} -with u a weak solution of P (f, p, λ * )-is a smooth hypersurface in a neighborhood of every "flat" free boundary point.Key words and phrases. Free boundary problem, variable exponent spaces, regularity of the free boundary, singular perturbation, inhomogeneous problem.

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Section: Introductionmentioning

confidence: 99%

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Lederman¹,

Wolanski²

2017

Interfaces Free Bound.

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“…Since v = |D 2 u 0 | 2 , we conclude that ∇u 0 is constant in each connected component of {u 0 > 0}. Therefore, by Lemma C.1 (6) and (8), we have…”

Section: By the Definition Of δ T We Havementioning

confidence: 69%

“…Indeed, in that case the whole free boundary is regular. Full regularity of the free boundary in dimension 2 was proved in [1] and [4] in the case of uniformly elliptic operators, in [6] for the p-laplacian with 2 − δ p < ∞ for a small δ > 0, and also in [12] for a penalization problem. In dimension 3 for p close to 2 a similar result was proved by A. Petrosyan (see [17]).…”

Section: Introductionmentioning

confidence: 99%

An optimization problem with volume constraint in Orlicz spaces

Martı́nez¹

2008

Journal of Mathematical Analysis and Applications

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We consider the optimization problem of minimizing Ω G(|∇u|) dx in the class of functions W 1,G (Ω), with a constraint on the volume of {u > 0}. The conditions on the function G allow for a different behavior at 0 and at ∞. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution u is locally Lipschitz continuous and that the free boundary, ∂{u > 0} ∩ Ω is smooth.

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“…As a corollary we have the following regularity result for the free boundary ∂{u > 0}. [6]) proved the full regularity of the free boundary of the minimizers of (1.2) if 2 − δ p < ∞ for a small δ > 0. Also, a similar result was proved by Petrosyan in dimension 3 for p close to 2 (see [15]).…”

Section: The Penalized Problemmentioning

confidence: 92%

“…if K {u 0 = 0}, then u k = 0 in K for big enough k, (5) if K {u 0 > 0} ∪ {u 0 = 0} • , then ∇u k → ∇u 0 uniformly in K, (6) there exists a constant 0 < λ < 1 such that |B R (y 0 ) ∩ {u 0 = 0}| |B R (y 0 )| λ, ∀R > 0, ∀y 0 ∈ ∂{u 0 > 0}, (7) ∇u k → ∇u 0 a.e. in Ω, (8) if x k ∈ ∂{u > 0}, then 0 ∈ ∂{u 0 > 0}.…”

Section: Appendix B Blow-up Limitsmentioning

confidence: 99%

An optimization problem with volume constraint for a degenerate quasilinear operator

Bonder¹,

Martı́nez²,

Wolanski³

2006

Journal of Differential Equations

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Full regularity of the free boundary in a Bernoulli-type problem in two dimensions

Abstract: Abstract. In this note we prove that in dimension n = 2 there are no singular points on the free boundary ∂{u > 0}∩Ω in the Bernoulli-type problem governed by the p-Laplace operatorfor p in the range 2 − ε 0 < p < ∞ for an absolute constant ε 0 > 0.

Cited by 17 publications

References 8 publications

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

An optimization problem with volume constraint in Orlicz spaces

An optimization problem with volume constraint for a degenerate quasilinear operator

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