Two-phase semilinear free boundary problem with a degenerate phase

Matevosyan, Norayr; Petrosyan, Arshak

doi:10.1007/s00526-010-0367-6

Cited by 3 publications

(2 citation statements)

References 7 publications

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“…Singular Lane-Emden equations like (1.1), and the properties of the nodal sets for their solutions, have been considered in the literature, mainly with a nonlinearity with opposite sign with respect to ours. One and two phases problems of type (1.1) for q ∈ (0, 1), but with λ ± ≤ 0, have been studied in [1,13,15,18,19] (see also [2] for a fully nonlinear analogue). In this case, nontrivial solutions may vanish on open sets, and no unique continuation principle holds.…”

Section: Introductionmentioning

confidence: 99%

The nodal set of solutions to some elliptic problems: Singular nonlinearities

Soave

Terracini

2019

Journal de Mathématiques Pures et Appliquées

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This paper deals with solutions to the equationis the unit ball in R N , N ≥ 2, and u + := max{u, 0}, u − := max{−u, 0} are the positive and the negative part of u, respectively. We extend to this class of singular equations the results recently obtained in [24] for sublinear and discontinuous equations, 1 ≤ q < 2, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most N − 2 (locally finite when N = 2). As an intermediate step, we establish the regularity of a class of not necessarily minimal solutions.The proofs are based on a priori bounds, monotonicity formulae for a 2-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogenous solutions.

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Section: Introductionmentioning

confidence: 99%

The nodal set of solutions to some elliptic problems: Singular nonlinearities

Soave

Terracini

2019

Journal de Mathématiques Pures et Appliquées

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“…Minimizers in this case are

C^{1, α}

for an optimal value α and this is essential in the fine analysis of the free boundary. Let us mention that some results for the two phase version of were obtained in and using a monotonicity formula due to Weiss . A related nonvariational problem was also considered in .…”

Section: Introductionmentioning

confidence: 99%

Almost minimizers for semilinear free boundary problems with variable coefficients

Queiroz

Tavares

2018

Mathematische Nachrichten

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We study regularity results for almost minimizers of the functionalwhere is a matrix with Hölder continuous coefficients. In the case 0 < ≤ 1 we show that an almost minimizer belongs to 1, , where the exponent is related with the competition between the Hölder continuity of the matrix , the parameter of almost minimization and . In some sense, this regularity is optimal. As far as the case = 0 is concerned, our results show that an almost minimizer is 1, locally in each phase { > 0} and { < 0}, improving in some sense a recent result of David & Toro. K E Y W O R D S almost minimizers, free boundary problems, regularity theory M S C ( 2 0 1 0 ) 35J20, 35J61, 35R35, 49J10 1486

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Regularity issues for semilinear PDE-s (a narrative approach)

Shahgholian

2016

St. Petersburg Math. J.

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Two-phase semilinear free boundary problem with a degenerate phase

Abstract: Abstract. We study minimizers of the energy functionalfor p ∈ (0, 1) without any sign restriction on the function u. The main result states that in dimension two the free boundaries Γ + = ∂{u > 0} ∩ D and Γ − = ∂{u < 0} ∩ D are C 1 -regular, provided 1 − 0 < p < 1.

Cited by 3 publications

References 7 publications

The nodal set of solutions to some elliptic problems: Singular nonlinearities

The nodal set of solutions to some elliptic problems: Singular nonlinearities

Almost minimizers for semilinear free boundary problems with variable coefficients

Regularity issues for semilinear PDE-s (a narrative approach)

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