Monotonicity formulas for obstacle problems with Lipschitz coefficients

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• 作者：M. Focardi ; M. S. Gelli ; E. Spadaro
• 关键词：35R35 ; 49N60
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：54
• 期：2
• 页码：1547-1573
• 全文大小：640 KB
• 参考文献：1.Alt, H.W., Caffarelli, L.A., Friedman, A.: Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282(2), 431-61 (1984)MathSciNet CrossRef MATH
  2.Athanasopoulos, I., Caffarelli, L.A.: A theorem of real analysis and its application to free boundary problems. Commun. Pure Appl. Math. 38(5), 499-02 (1985)MathSciNet CrossRef
  3.Blank, I.: Sharp results for the regularity and stability of the free boundary in the obstacle problem. Indiana Univ. Math. J. 50(3), 1077-112 (2001)MathSciNet CrossRef MATH
  4.Caffarelli, L.A.: The regularity of free boundaries in higher dimensions. Acta Math. 139(3-), 155-84 (1977)MathSciNet CrossRef
  5.Caffarelli, L.A.: Compactness methods in free boundary problems. Commun. Partial Differ. Equ. 5(4), 427-48 (1980)MathSciNet CrossRef MATH
  6.Caffarelli, L.A.: The obstacle problem revisited. Lezioni Fermiane [Fermi Lectures]. Accademia Nazionale dei Lincei, Rome; Scuola Normale Superiore, Pisa, ii+54 pp (1998)
  7.Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4(4-), 383-02 (1998)MathSciNet CrossRef MATH
  8.Caffarelli, L.A., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30(4), 621-40 (1981)MathSciNet CrossRef MATH
  9.Caffarelli, L.A., Salsa, S.: A geometric approach to free boundary problems. Graduate Studies in Mathematics, vol. 68. American Mathematical Society, Providence, x+270 pp (2005)
  10.Caffarelli, L.A., Karp, L., Shahgholian, H.: Regularity of a free boundary with application to the Pompeiu problem. Ann. Math. (2) 151(1), 269-92 (2000)
  11.Cerutti, M.C., Ferrari, F., Salsa, S.: Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are \(C^{1,\gamma }\) . Arch. Ration. Mech. Anal. 171(3), 329-48 (2004)MathSciNet CrossRef MATH
  12.Ferrari, F., Salsa, S.: Regularity of the free boundary in two-phase problems for linear elliptic operators. Adv. Math. 214(1), 288-22 (2007)MathSciNet CrossRef MATH
  13.Ferrari, F., Salsa, S.: Regularity of the solutions for parabolic two-phase free boundary problems. Commun. Partial Differ. Equ. 35(6), 1095-129 (2010)MathSciNet CrossRef MATH
  14.Friedman, A.: Variational Principles and Free Boundary Problems, 2nd edn. Robert E. Krieger Publishing Co., Inc, Malabar, x+710 pp (1988)
  15.Giusti, E.: Equazioni ellittiche del secondo ordine, Quaderni Unione Matematica Italiana, vol. 6. Pitagora Editrice, Bologna, v+213 pp (1978)
  16.Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, xiv+517 pp (2001)
  17.Jerison, D., Kenig, C.: Boundary behaviour of harmonic functions in nontangentially accessible domains. Adv. Math. 46(1), 80-47 (1982)MathSciNet CrossRef MATH
  18.Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Pure and Applied Mathematics, vol. 88. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, xiv+313 pp (1980)
  19.Kukavica, I.: Quantitative uniqueness for second-order elliptic operators. Duke Math. J. 91(2), 225-40 (1998)MathSciNet CrossRef MATH
  20.Lin, F.: On regularity and singularity of free boundaries in obstacle problems. Chin. Ann. Math. Ser. B 30(5), 645-52 (2009)MathSciNet CrossRef MATH
  21.Matevosyan, N., Petrosyan, A.: Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients. Commun. Pure Appl. Math. 64(2), 271-11 (2011)MathSciNet CrossRef MATH
  22.Monneau, R.: On the number of singularities for the obstacle problem in two dimensions. J. Geom. Anal. 13(2), 359-89 (2003)MathSciNet CrossRef
  23.Petrosyan, A., Shahgholian, H.: Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem. Am. J. Math. 129(6), 1659-688 (2007)MathSciNet CrossRef MATH
  24.Petrosyan, A., Shahgholian, H., Uraltseva, N.: Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, vol. 136. American Mathematical Society, Providence, x+221 pp (2012)
  25.Rodrigues, J.F.: Obstacle problems in mathematical physics. North-Holland Mathematics Studies, vol. 134. Notas de Matemtica [Mathematical Notes], vol. 114. North-Holland Publishing Co., Amsterdam, xvi+352 pp (1987)
  26.Wang, P.Y.: Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. I. Lipschitz free boundaries are \(C^{1,\alpha }\) . Commun. Pure Appl. Math. 53(7), 799-10 (2000)CrossRef MATH
  27.Wang, P.Y.: Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. II. Flat free boundaries are Lipschitz. Commun. Partial Differ. Equ. 27(7-), 1497-514 (2002)CrossRef MATH
  28.Weiss, G.S.: A homogeneity improvement approach to the obstacle problem
• 作者单位：M. Focardi (1)
  M. S. Gelli (2)
  E. Spadaro (3)
  
  1. DiMaI “U. Dini- Università di Firenze, V.le Morgagni 67/A, 50134, Florence, Italy
  2. Dipartimento di Matematica, Università di Pisa, L.go Pontecorvo 5, 56127, Pisa, Italy
  3. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22, 04103, Leipzig, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a H?lder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli (J Fourier Anal Appl 4(4-), 383-02, 1998), Monneau (J Geom Anal 13(2), 359-89, 2003), and Weiss (Invent Math 138(1), 23-0, 1999). Mathematics Subject Classification 35R35 49N60