Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


IMPACT FACTOR 2017: 1.029
5-year IMPACT FACTOR: 1.147

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 1.588
Source Normalized Impact per Paper (SNIP) 2017: 0.971

Mathematical Citation Quotient (MCQ) 2017: 1.03

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 18, Issue 2

Issues

On a Class of Quasilinear Elliptic Equations with Degenerate Coerciveness and Measure Data

Flavia Smarrazzo
Published Online: 2017-09-27 | DOI: https://doi.org/10.1515/ans-2017-6032

Abstract

We study the existence of measure-valued solutions for a class of degenerate elliptic equations with measure data. The notion of solution is natural, since it is obtained by a regularization procedure which also relies on a standard approximation of the datum μ. We provide partial uniqueness results and qualitative properties of the constructed solutions concerning, in particular, the structure of their diffuse part with respect to the harmonic-capacity.

Keywords: Measure-Valued Solutions; Degenerate Elliptic Equations; Radon Measures

MSC 2010: 35J15; 35D30; 28A33

References

  • [1]

    A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl. (4) 182 (2003), no. 1, 53–79. CrossrefGoogle Scholar

  • [2]

    J. M. Ball, A version of the fundamental theorem for Young measures, PDEs and Continuum Models of Phase Transitions (Nice 1988), Lecture Notes in Phys. 344, Springer, Berlin (1989), 207–215. Google Scholar

  • [3]

    P. Baras and M. Pierre, Singularités éliminables pour des équations semilinéaires, Ann. Inst. Fourier 34 (1984), 185–206. CrossrefGoogle Scholar

  • [4]

    D. Bartolucci, F. Leoni, L. Orsina and A. C. Ponce, Semilinear equations with exponential nonlinearity and measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, 799–815. CrossrefGoogle Scholar

  • [5]

    P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 22 (1995), no. 2, 241–273. Google Scholar

  • [6]

    P. Bénilan and H. Brézis, Nonlinear problems related to the Thomas–Fermi equation, J. Evol. Equ. 3 (2004), 673–770. Google Scholar

  • [7]

    P. Bénilan, H. Brézis and M. Crandall, A semilinear equation in L1(N), Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 2 (1975), 523–555. Google Scholar

  • [8]

    L. Boccardo, On the regularizing effect of strongly increasing lower order terms, J. Evol. Equ. 3 (2003), no. 2, 225–236. CrossrefGoogle Scholar

  • [9]

    L. Boccardo and H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. 6 (2003), 521–530. Google Scholar

  • [10]

    L. Boccardo, G. Croce and L. Orsina, A semilinear problem with a W01,1 solution, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 23 (2012), no. 2, 97–103. Google Scholar

  • [11]

    L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with W01,1(Ω) solutions, Manuscripta Math. 137 (2012), no. 3–4, 419–439. Web of ScienceGoogle Scholar

  • [12]

    L. Boccardo, A. Dall’Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 51–81, Google Scholar

  • [13]

    L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 241–273. Google Scholar

  • [14]

    L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side a measure, Comm. Partial Differential Equations 17 (1992), 641–655. Google Scholar

  • [15]

    L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 5, 539–551. CrossrefGoogle Scholar

  • [16]

    H. Brézis, Some variational problems of the Thomas–Fermi type, Variational Inequalities and Complementarity Problems (Erice 1978), Wiley, Chichester (1980), 53–73. Google Scholar

  • [17]

    H. Brézis and F. E. Browder, Strongly nonlinear elliptic boundary value problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 5 (1978), 587–603. Google Scholar

  • [18]

    H. Brézis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited, Mathematical Aspects of Nonlinear Dispersive Equations, Ann. of Math. Stud. 163, Princeton University Press, Princeton (2007), 55–109. Google Scholar

  • [19]

    H. Brézis and A. C. Ponce, Reduced measures for obstacle problems, Adv. Differential Equations 10 (2005), 1201–1234. Google Scholar

  • [20]

    H. Brézis and W. A. Strauss, Semilinear second-order elliptic equations in L1, J. Math. Soc. Japan 25 (1973), 565–590. Google Scholar

  • [21]

    H. Brézis and L. Véron, Removable singularities for some nonlinear elliptic equations, Arch. Ration. Mech. Anal. 75 (1980), 1–6. Google Scholar

  • [22]

    G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1999), no. 4, 741–808.Google Scholar

  • [23]

    J. Droniou, Global and local estimates for nonlinear noncoercive elliptic equations with measure data, Comm. Partial Differential Equations 28 (2003), no. 1–2, 129–153. CrossrefGoogle Scholar

  • [24]

    L. Dupaigne, A. C. Ponce and A. Porretta, Elliptic equations with vertical asymptotes in the nonlinear term, J. Anal. Math. 98 (2006), 349–396. CrossrefGoogle Scholar

  • [25]

    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. Google Scholar

  • [26]

    M. Fukushima, K.-I. Sato and S. Taniguchi, On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures, Osaka J. Math. 28 (1991), no. 3, 517–535. Google Scholar

  • [27]

    T. Gallouet and J.-M. Morel, Resolution of a semilinear equation in L1, Proc. Roy. Soc. Edinburgh Sect. A 96 (1984), 275–288. Google Scholar

  • [28]

    J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty–Browder, Bull. Soc. Math. France 93 (1965), 97–107. Google Scholar

  • [29]

    E. H. Lieb and B. Simon, The Thomas–Fermi theory of atoms, molecules and solids, Adv. Math. 23 (1977), no. 1, 22–116. CrossrefGoogle Scholar

  • [30]

    L. Orsina and A. C. Ponce, Semilinear elliptic equations and systems with diffuse measures, J. Evol. Equ. 8 (2008), 781–812. CrossrefWeb of ScienceGoogle Scholar

  • [31]

    A. Porretta, Uniqueness of solutions for some nonlinear Dirichlet problems, NoDEA Nonlinear Differential Equations Appl. 11 (2004), no. 4, 407–430. CrossrefGoogle Scholar

  • [32]

    M. M. Porzio and F. Smarrazzo, Radon measure-valued solutions for some quasilinear degenerate elliptic equations, Ann. Mat. Pura Appl. (4) 194 (2015), no. 2, 495–532. CrossrefWeb of ScienceGoogle Scholar

  • [33]

    M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Ration. Mech. Anal. 210 (2013), no. 3, 713–772. CrossrefWeb of ScienceGoogle Scholar

  • [34]

    J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Sup. Pisa (3) 18 (1964), 385–387. Google Scholar

  • [35]

    G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. 1, 189–258. CrossrefGoogle Scholar

  • [36]

    M. Valadier, A course on Young measures, Rend. Istit. Mat. Univ. Trieste 26 (1994), 349–394. Google Scholar

  • [37]

    J. L. Vázquez, On a semilinear equation in 𝐑2 involving bounded measures, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), no. 3–4, 181–202. Google Scholar

  • [38]

    L. Véron, Elliptic equations involving measures, Stationary Partial Differential Equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam (2004), 593–712. Google Scholar

About the article


Received: 2017-06-02

Revised: 2017-08-27

Accepted: 2017-08-29

Published Online: 2017-09-27

Published in Print: 2018-04-01


Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 2, Pages 361–392, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2017-6032.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in