Geometric Approach to Nonvariational Singular Elliptic Equations

详细信息    查看全文

• 作者：Dami?o Araújo (1)
  Eduardo V. Teixeira (1)
  
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2013
• 出版时间：September 2013
• 年：2013
• 卷：209
• 期：3
• 页码：1019-1054
• 全文大小：386KB
• 参考文献：1. Alt H.M., Caffarelli L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105-44 (1981)
  2. Alt H.M., Phillips D.: A free boundary problem for semilinear elliptic equations. J. Reine Angew. Math. 368, 63-07 (1986)
  3. Barles G.: A weak Bernstein method for fully nonlinear elliptic equations. Differ. Integral Equ. 4(2), 241-62 (1991)
  4. Barles G., Chasseigne E., Imbert C.: H?lder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 13(1), 1-6 (2011) CrossRef
  5. Caffarelli L.A.: The regularity of free boundaries in higher dimensions. Acta Math. 139(3-), 155-84 (1977) CrossRef
  6. Caffarelli L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189-13 (1989) CrossRef
  7. Caffarelli, L.A., Cabré, X.: / Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, Vol. 43. American Mathematical Society, Providence, 1995
  8. Caffarelli L., Karp L., Shahgholian H.: Regularity of a free boundary with application to the Pompeiu problem. Ann. Math. 151, 269-92 (2000) CrossRef
  9. Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1-7 (1992) CrossRef
  10. Crandall M.G., Rabinowitz P.H., Tartar L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2(2), 193-22 (1977) CrossRef
  11. Felmer, P., Quaas, A., Sirakov, B.: Existence and regularity results for fully nonlinear equations with singularities. / Math. Ann. 1-4. doi:10.1007/s00208-011-0741-5
  12. Giaquinta M., Giusti E.: Differentiability of minima of non-differentiable functionals. Invent. Math. 72, 285-98 (1983) CrossRef
  13. Giaquinta M., Giusti E.: Sharp estimates for the derivatives of local minima of variational integrals. Bollettino U.M.I. 6(3-A), 239-48 (1984)
  14. Imbert C., Silvestre L.: / C ^1,α regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233, 196-06 (2013) CrossRef
  15. Ishii H., Lions P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ Equ. 83(1), 26-8 (1990) CrossRef
  16. Krylov, N.V., Safonov, M.V.: An estimate of the probability that a diffusion process hits a set of positive measure. / Dokl. Akad. Nauk. SSSR 245, 235-55 (1979) [English translation in Sov. Math. Dokl. 20, 235-55 (1979)]
  17. Lee K.-A., Shahgholian H.: Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 54(1), 43-6 (2001) CrossRef
  18. Nadirashvili N., Vladut S.: Nonclassical solutions of fully nonlinear elliptic equations II. Hessian equations and octonions. Geom. Funct. Anal. 21, 483-98 (2011) CrossRef
  19. Phillips D.: A minimization problem and the regularity of solutions in the presence of a free boundary. Indiana Univ. Math. J. 32, 1-7 (1983) CrossRef
  20. Phillips D.: Hausdorff measure estimates of a free boundary for a minimum problem. Commun. Partial Differ. Equ. 8, 1409-454 (1983) CrossRef
  21. Ricarte G., Teixeira E.: Fully nonlinear singularly perturbed equations and asymptotic free boundaries. J. Funct Anal. 261(6), 1624-673 (2011) CrossRef
  22. Silvestre, L., Teixeira, E.: / Asymptotic regularity estimates for fully nonlinear equations. Preprint
  23. Teixeira E.V.: A variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(4), 633-58 (2008) CrossRef
  24. Teixeira, E.V.: / Universal moduli of continuity for solutions to fully nonlinear elliptic equations. Preprint. http://arxiv.org/abs/1111.2728
  25. Weiss G.: Partial regularity for weak solutions of an elliptic free boundary problem. Commun. Partial Differ. Equ. 23(3-), 439-55 (1998) CrossRef
  
• 作者单位：Dami?o Araújo (1)
  Eduardo V. Teixeira (1)
  
  1. Universidade Federal do Ceará, Campus of Pici, Bloco 914, Fortaleza, Ceará, 60.455-760, Brazil
  
• ISSN：1432-0673

文摘

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms, as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter (γ - 1), for 0 < γ < 1, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases.We establish existence as well as sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and we obtain fine geometric-measure properties of the free boundary ${\mathfrak{F} = \partial{u > 0}}$ . In particular, we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region {u > 0} and the ${\mathcal{H}^{n-1}}$ almost-everywhere weak differentiability property of ${\mathfrak{F}}$ .