Superharmonic functions are locally renormalized solutions
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 6, pp. 775-795.

We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.

DOI : 10.1016/j.anihpc.2011.03.004
Classification : 35J92, 35A01
@article{AIHPC_2011__28_6_775_0,
     author = {Kilpel\"ainen, Tero and Kuusi, Tuomo and Tuhola-Kujanp\"a\"a, Anna},
     title = {Superharmonic functions are locally renormalized solutions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {775--795},
     publisher = {Elsevier},
     volume = {28},
     number = {6},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.03.004},
     mrnumber = {2859927},
     zbl = {1234.35121},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.004/}
}
TY  - JOUR
AU  - Kilpeläinen, Tero
AU  - Kuusi, Tuomo
AU  - Tuhola-Kujanpää, Anna
TI  - Superharmonic functions are locally renormalized solutions
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 775
EP  - 795
VL  - 28
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.004/
DO  - 10.1016/j.anihpc.2011.03.004
LA  - en
ID  - AIHPC_2011__28_6_775_0
ER  - 
%0 Journal Article
%A Kilpeläinen, Tero
%A Kuusi, Tuomo
%A Tuhola-Kujanpää, Anna
%T Superharmonic functions are locally renormalized solutions
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 775-795
%V 28
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.004/
%R 10.1016/j.anihpc.2011.03.004
%G en
%F AIHPC_2011__28_6_775_0
Kilpeläinen, Tero; Kuusi, Tuomo; Tuhola-Kujanpää, Anna. Superharmonic functions are locally renormalized solutions. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 6, pp. 775-795. doi : 10.1016/j.anihpc.2011.03.004. http://www.numdam.org/articles/10.1016/j.anihpc.2011.03.004/

[1] H. Abdel Hamid, M.F. Bidaut-Véron, On the connection between two quasilinear elliptic problems with a source terms of order 0 or 1, http://arxiv.org/abs/0811.3292v1 | MR | Zbl

[2] B. Abdellaoui, A. DallʼAglio, I. Peral, Some remarks on elliptic problems with critical growth in the gradient, J. Differential Equations 222 no. 1 (2006), 21-62

[3] B. Abdellaoui, A. DallʼAglio, I. Peral, Corrigendum to “Some remarks on elliptic problems with critical growth in the gradient”, J. Differential Equations 246 no. 7 (2009), 2988-2990 | MR | Zbl

[4] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J.L. Vázquez, An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 241-273 | EuDML | Numdam | MR | Zbl

[5] M.F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation with absorption or source term and measure data, Adv. Nonlinear Stud. 3 no. 1 (2003), 25-63 | MR | Zbl

[6] L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 no. 1 (1989), 149-169 | MR | Zbl

[7] A. DallʼAglio, Approximated solutions of equations with L 1 data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl. (4) 170 (1996), 207-240 | MR

[8] A. DallʼAglio, D. Giachetti, J.-P. Puel, Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl. (4) 181 no. 4 (2002), 407-426 | MR | Zbl

[9] G. Dal Maso, A. Malusa, Some properties of reachable solutions of nonlinear elliptic equations with measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 375-396 | EuDML | Numdam | MR | Zbl

[10] E. Dibenedetto, C 1+α Local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. Ser. A: Theory Methods 7 no. 8 (1983), 827-850 | MR | Zbl

[11] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 741-808 | EuDML | Numdam | MR | Zbl

[12] V. Ferone, B. Messano, Comparison and existence results for classes of nonlinear elliptic equations with general growth in the gradient, Adv. Nonlinear Stud. 7 no. 1 (2007), 31-46 | MR | Zbl

[13] V. Ferone, E. Giarrusso, B. Messano, M.R. Posteraro, Estimates for blow-up solutions to nonlinear elliptic equations with p-growth in the gradient, Z. Anal. Anwend. 29 no. 2 (2010), 219-234 | MR | Zbl

[14] V. Ferone, F. Murat, Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal. Ser. A: Theory Methods 42 no. 7 (2000), 1309-1326 | MR | Zbl

[15] N. Grenon, Existence and comparison results for quasilinear elliptic equations with critical growth in the gradient, J. Differential Equations 171 no. 1 (2001), 1-23 | MR | Zbl

[16] N. Grenon, C. Trombetti, Existence results for a class of nonlinear elliptic problems with p-growth in the gradient, Nonlinear Anal. 52 no. 3 (2003), 931-942 | MR | Zbl

[17] K. Hansson, V.G. MazʼYa, I.E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat. 37 no. 1 (1999), 87-120 | MR

[18] L.I. Hedberg, Th.H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 no. 4 (1983), 161-187 | EuDML | Numdam | MR | Zbl

[19] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, NY (2006) | MR | Zbl

[20] B.J. Jaye, I.E. Verbitsky, The fundamental solution of nonlinear equations with natural growth terms, http://arxiv.org/abs/1002.4664

[21] T. Kilpeläinen, J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 591-613 | EuDML | Numdam | MR | Zbl

[22] T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161 | MR | Zbl

[23] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure Appl. Math. vol. 88, Academic Press, New York, London (1980) | MR | Zbl

[24] R. Korte, T. Kuusi, A note on the Wolff potential estimate for solutions to elliptic equations involving measures, Adv. Calc. Var. 3 (2010), 99-113 | MR | Zbl

[25] P.-L. Lions, F. Murat, Sur les solutions renormalisées dʼéquations elliptiques non linéaires (informal communication).

[26] O. Martio, Quasilinear Riccati type equations and quasiminimizers, preprint 509, Department of Mathematics and Statistics, University of Helsinki, 2010. | MR

[27] B. Messano, Symmetrization results for classes of nonlinear elliptic equations with q-growth in the gradient, Nonlinear Anal. 64 no. 12 (2006), 2688-2703 | MR | Zbl

[28] P. Mikkonen, On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996) | MR | Zbl

[29] G. Mingione, Gradient estimates below the duality exponent, Math. Ann. 346 (2010), 571-627 | MR | Zbl

[30] G. Mingione, The Calderon–Zygmund theory for elliptic problems with measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 6 (2007), 195-261 | EuDML | Numdam | MR | Zbl

[31] G. Mingione, Nonlinear aspects of Calderon–Zygmund theory, Jahresber. Dtsch. Math.-Ver. 112 (2010), 159-191 | MR | Zbl

[32] G. Mingione, Gradient potential estimates, J. Europ. Math. Soc. 13 (2011), 459-486 | EuDML | MR | Zbl

[33] N.C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations 35 (2010), 1958-1981 | MR | Zbl

[34] A. Porretta, S. Segura De León, Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl. 85 (2006), 465-492 | MR | Zbl

[35] J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 385-387 | EuDML | Numdam | MR | Zbl

[36] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 no. 3–4 (1977), 219-240 | MR | Zbl

[37] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 no. 1 (1984), 126-150 | MR | Zbl

[38] A. Tuhola-Kujanpää, A potential theory approach to the equation -Δu=|u| 2 , Ann. Acad. Sci. Fenn. Math. 35 (2010), 633-640 | MR | Zbl

[39] N.S. Trudinger, X.J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), 369-410 | MR | Zbl

[40] N.S. Trudinger, X.J. Wang, Quasilinear elliptic equations with signed measure data, Disc. Cont. Dyn. Systems 124 (2002), 369-410 | MR

[41] W.P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Grad. Texts in Math. vol. 120, Springer-Verlag, New York (1989) | MR | Zbl

Cité par Sources :