We prove the optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.
Mots-clés : Minimal immersion, Thin obstacle problem, Free boundary, 2-Valued functions
@article{AIHPC_2020__37_4_1017_0, author = {Focardi, Matteo and Spadaro, Emanuele}, title = {How a minimal surface leaves a thin obstacle}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1017--1046}, publisher = {Elsevier}, volume = {37}, number = {4}, year = {2020}, doi = {10.1016/j.anihpc.2020.02.005}, mrnumber = {4104833}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.02.005/} }
TY - JOUR AU - Focardi, Matteo AU - Spadaro, Emanuele TI - How a minimal surface leaves a thin obstacle JO - Annales de l'I.H.P. Analyse non linéaire PY - 2020 SP - 1017 EP - 1046 VL - 37 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2020.02.005/ DO - 10.1016/j.anihpc.2020.02.005 LA - en ID - AIHPC_2020__37_4_1017_0 ER -
%0 Journal Article %A Focardi, Matteo %A Spadaro, Emanuele %T How a minimal surface leaves a thin obstacle %J Annales de l'I.H.P. Analyse non linéaire %D 2020 %P 1017-1046 %V 37 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2020.02.005/ %R 10.1016/j.anihpc.2020.02.005 %G en %F AIHPC_2020__37_4_1017_0
Focardi, Matteo; Spadaro, Emanuele. How a minimal surface leaves a thin obstacle. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 1017-1046. doi : 10.1016/j.anihpc.2020.02.005. http://www.numdam.org/articles/10.1016/j.anihpc.2020.02.005/
[1] Optimal regularity of lower dimensional obstacle problems, Zap. Nauč. Semin. POMI, Volume 310 (2004) no. 3, pp. 49–66 226; translation in J. Math. Sci. (N.Y.), 132, 2006, 274–284 | MR | Zbl
[2] The structure of the free boundary for lower dimensional obstacle problems, Am. J. Math., Volume 130 (2008) no. 2, pp. 485–498 | DOI | MR | Zbl
[3] Characterization of -rectifiability in terms of Jones' square function: part II, Geom. Funct. Anal., Volume 25 (2015) no. 5, pp. 1371–1412 | DOI | MR
[4] Una maggiorazione a priori relativa alle ipersuperficie minimali non parametriche, Arch. Ration. Mech. Anal., Volume 32 (1969), pp. 255–267 | DOI | MR | Zbl
[5] Further regularity in the Signorini problem, Commun. Partial Differ. Equ., Volume 4 (1979), pp. 1067–1076 | DOI | MR | Zbl
[6] Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., Volume 171 (2008) no. 2, pp. 425–461 | DOI | MR | Zbl
[7] Regolarità di frontere minimali con ostacoli sottili, Rend. Semin. Mat. Univ. Padova, Volume 61 (1979), pp. 133–144 | Numdam | MR | Zbl
[8] Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. (3), Volume 3 (1957), pp. 25–43 | MR | Zbl
[9] Problemi di superfici minime con ostacoli: forma non cartesiana, Boll. Unione Mat. Ital. (4), Volume 8 (1973) no. 2, pp. 80–88 | MR | Zbl
[10] Frontiere orientate di misura minima e questioni collegate, Scuola Normale Superiore, Pisa, 1972 (177 pp) | MR | Zbl
[11] Rectifiability and upper Minkowski bounds for singularities of harmonic -valued maps, Comment. Math. Helv., Volume 93 (2018) no. 4, pp. 737–779 | DOI | MR
[12] -valued functions revisited, Mem. Am. Math. Soc., Volume 211 (2011) no. 991 (vi+79 pp) | MR | Zbl
[13] Multiple valued functions and integral currents, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 14 (2015) no. 4, pp. 1239–1269 | MR
[14] Regularity of area minimizing currents III: blow-up, Ann. Math. (2), Volume 183 (2016) no. 2, pp. 577–617 | MR
[15] estimates for the fully nonlinear Signorini problem, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 4 | DOI | MR
[16] Regularity of minimal surfaces with lower dimensional obstacles | arXiv
[17] Monotonicity formulas for obstacle problems with Lipschitz coefficients, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 2, pp. 1547–1573 | DOI | MR
[18] Quasi-monotonicity formulas for classical obstacle problems with Sobolev coefficients and applications, J. Optim. Theory Appl., Volume 184 (2020) no. 1, pp. 125–138 | DOI | MR
[19] Improved estimate of the singular set of Dir-minimizing -valued functions via an abstract regularity result, J. Funct. Anal., Volume 268 (2015) no. 11, pp. 3290–3325 | DOI | MR
[20] An epiperimetric inequality for the fractional obstacle problem, Adv. Differ. Equ., Volume 21 (2016) no. 1–2, pp. 153–200 | MR
[21] On the measure and the structure of the free boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal., Volume 230 (2018) no. 1, pp. 125–184 | DOI | MR
[22] Correction to: on the measure and the structure of the free boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal., Volume 230 (2018) no. 2, pp. 783–784 | DOI | MR
[23] The local structure of the free boundary in the fractional obstacle problem | arXiv
[24] Two-dimensional variational problems with thin obstacles, Math. Z., Volume 143 (1975), pp. 279–288 | DOI | MR | Zbl
[25] On Signorini's problem and variational problems with thin obstacles, Ann. Sc. Norm. Super. Pisa, Volume 4 (1977), pp. 343–362 | Numdam | MR | Zbl
[26] New monotonicity formulas and the optimal regularity in the Signorini problem with variable coefficients, Adv. Math., Volume 262 (2014), pp. 682–750 | DOI | MR | Zbl
[27] Global regularity for second order quasilinear elliptic equations in divergence form, J. Reine Angew. Math., Volume 351 (1984), pp. 55–65 | MR | Zbl
[28] Regolarità Lipschitziana per la soluzione di alcuni problemi di minimo con vincolo, Ann. Mat. Pura Appl., Volume 106 (1975) no. 4, pp. 95–117 | MR | Zbl
[29] Esistenza e regolarità per il problema dell'area minima con ostacolo in variabili, Ann. Sc. Norm. Super. Pisa, Volume 25 (1971) no. 3, pp. 481–507 | Numdam | MR | Zbl
[30] Superfici minime cartesiane con ostacoli discontinui, Arch. Ration. Mech. Anal., Volume 40 (1971), pp. 251–267 (in Italian) | DOI | MR | Zbl
[31] Non-parametric minimal surfaces with discontinuous and thin obstacles, Arch. Ration. Mech. Anal., Volume 49 (1972/1973), pp. 41–56 | DOI | MR | Zbl
[32] Geometric Measure Theory and Minimal Surfaces, Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Minimal surfaces with obstacles, Edizioni Cremonese, Rome (1973), pp. 119–153 (Varenna, 1972) | MR | Zbl
[33] Symposia Mathematica, vol. XIV, Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, Maggiorazioni a priori del gradiente, e regolarità delle superfici minime non-parametriche con ostacoli, Academic Press, London (1974), pp. 481–491 (INDAM, Rome, 1973) | MR | Zbl
[34] Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003 (viii+403 pp) | DOI | MR | Zbl
[35] Optimal regularity for the Signorini problem, Calc. Var. Partial Differ. Equ., Volume 36 (2009) no. 4, pp. 533–546 | DOI | MR | Zbl
[36] The regularity of minimal surfaces defined over slit domains, Pac. J. Math., Volume 37 (1971), pp. 109–117 | DOI | MR | Zbl
[37] The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl., Volume 60 (1981), pp. 193–212 | MR | Zbl
[38] The variable coefficient thin obstacle problem: optimal regularity and regularity of the regular free boundary, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 34 (2017) no. 4, pp. 845–897 | Numdam | MR
[39] Fine properties of branch point singularities: two-valued harmonic functions | arXiv
[40] On a refinement of Evans' law in potential theory, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat., Volume 48 (1970) no. 8, pp. 1–9 | MR | Zbl
[41] On the coincidence set in variational inequalities, J. Differ. Geom., Volume 6 (1972), pp. 497–501 | DOI | MR | Zbl
[42] Regularity for the nonlinear Signorini problem, Adv. Math., Volume 217 (2008) no. 3, pp. 1301–1312 | DOI | MR | Zbl
[43] On an isoperimetric-isodiametric inequality, Anal. PDE, Volume 10 (2017) no. 1, pp. 95–126 | DOI | MR
[44] Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps, Ann. Math. (2), Volume 185 (2017) no. 1, pp. 131–227 | DOI | MR
[45] Variational problems with inequalities as boundary conditions or how to fashion a cheap hat for Giacometti's brother, Arch. Ration. Mech. Anal., Volume 35 (1969), pp. 83–113 | MR | Zbl
[46] Variational problems with thin obstacles, The University of British Columbia, Canada, 1978 Thesis (Ph.D.) | MR
[47] The structure of the free boundary in the fully nonlinear thin obstacle problem, Adv. Math., Volume 316 (2017), pp. 710–747 | DOI | MR
[48] A frequency function and singular set bounds for branched minimal immersions, Commun. Pure Appl. Math., Volume 69 (2016) no. 7, pp. 1213–1258 | DOI | MR
[49] Hölder continuity of gradients of solutions of parabolic equations with boundary conditions of Signorini type, Dokl. Akad. Nauk SSSR, Volume 280 (1985), pp. 563–565 | MR | Zbl
[50] On the regularity of solutions of variational inequalities, Usp. Mat. Nauk, Volume 42 (1987) no. 6(258), pp. 151–174 (in Russian) 248 | MR | Zbl
[51] Estimation on the boundary of the domain of derivatives of solutions of variational inequalities, J. Sov. Math., Volume 45 (1989) no. 3, pp. 1181–1191 | DOI | Zbl
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