Almost minimizers for a singular system with free boundary. (English) Zbl 1496.35461
Summary: In this paper we study vector-valued almost minimizers of the energy functional
\[
\int_D (| \nabla \mathbf{u} |^2 + 2 | \mathbf{u} |) d x.
\]
We establish the regularity for both minimizers and the “regular” part of the free boundary. The analysis of the free boundary is based on Weiss-type monotonicity formula and the epiperimetric inequality for the energy minimizers.
MSC:
| 35R35 | Free boundary problems for PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
Keywords:
almost minimizers; singular system; regular set; Weiss-type monotonocity formula; epiperimetric inequalityReferences:
| [1] | Andersson, J.; Shahgholian, H.; Uraltseva, N.; Weiss, G., Equilibrium points of a singular cooperative system with free boundary, Adv. Math., 280, 743-771 (2015) · Zbl 1319.35306 |
| [2] | Anzellotti, G., On the \(C^{1 , \alpha}\)-regularity of ω-minima of quadratic functionals, Boll. Un. Mat. Ital. C (6), 2, 1, 195-212 (1983) · Zbl 0522.49005 |
| [3] | David, G.; Engelstein, M.; Toro, T., Free boundary regularity for almost-minimizers, Adv. Math., 350, 1109-1192 (2019) · Zbl 1455.35303 |
| [4] | David, G.; Toro, T., Regularity of almost minimizers with free boundary, Calc. Var. Partial Differ. Equ., 54, 1, 455-524 (2015) · Zbl 1378.35337 |
| [5] | De Silva, D.; Savin, O., Thin one-phase almost minimizers, Nonlinear Anal., 193, Article 111507 pp. (2020), 23 pp. · Zbl 1437.49051 |
| [6] | De Silva, D.; Savin, O., Almost minimizers of the one-phase free boundary problem, Commun. Partial Differ. Equ., 45, 8, 913-930 (2020) · Zbl 1452.35006 |
| [7] | Han, Q.; Lin, F., Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1 (2011), Courant Institute of Mathematical Sciences: Courant Institute of Mathematical Sciences New York, American Mathematical Society, Providence, RI, x+147 pp. · Zbl 1210.35031 |
| [8] | Jeon, S.; Petrosyan, A., Almost minimizers for certain fractional variational problems, Algebra Anal.. Algebra Anal., St. Petersburg Math. J., 32, 4, 729-751 (2021), reprinted in · Zbl 1475.49045 |
| [9] | Jeon, S.; Petrosyan, A., Almost minimizers for the thin obstacle problem, Calc. Var. Partial Differ. Equ., 60, Article 124 pp. (2021), 59 · Zbl 1468.49041 |
| [10] | S. Jeon, A. Petrosyan, M. Smit Vega Garcia, Almost minimizers for the thin obstacle problem with variable coefficients, Preprint. · Zbl 07902357 |
| [11] | Weiss, G. S., A homogeneity improvement approach to the obstacle problem, Invent. Math., 138, 1, 23-50 (1999) · Zbl 0940.35102 |
| [12] | Weiss, G. S., An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary, Interfaces Free Bound., 3, 2, 121-128 (2001) · Zbl 0986.35139 |
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