July  2013, 12(4): 1569-1585. doi: 10.3934/cpaa.2013.12.1569

Uniqueness for elliptic problems with Hölder--type dependence on the solution

1. 

Dipartimento di Matematica, Università di Roma 1, Piazza A. Moro 2, 00185 Roma

2. 

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scienti ca 1, 00133 Roma

Received  May 2011 Revised  June 2012 Published  November 2012

We prove uniqueness of weak (or entropy) solutions for nonmonotone elliptic equations of the type \begin{eqnarray} -div (a(x,u)\nabla u)=f \end{eqnarray} in a bounded set $\Omega\subset R^N$ with Dirichlet boundary conditions. The novelty of our results consists in the possibility to deal with cases when $a(x,u)$ is only Hölder continuous with respect to $u$.
Citation: Lucio Boccardo, Alessio Porretta. Uniqueness for elliptic problems with Hölder--type dependence on the solution. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1569-1585. doi: 10.3934/cpaa.2013.12.1569
References:
[1]

M. Artola, Sur une classe de problèmes paraboliques quasi-linéaires, Boll. U.M.I. B., 5 (1986), 51-70.

[2]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vàzquez, An $L^1$ theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 240-273.

[3]

D. Blanchard, F. Désir and O. Guibé, Quasi-linear degenerate elliptic problems with $L^1$ data, Nonlinear Anal., 60 (2005), 557-587. doi: 10.1016/S0362-546X(04)00395-5.

[4]

L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form, Progress in elliptic and parabolic partial differential equations (Capri, 1994), 43-57.

[5]

L. Boccardo, Uniqueness of solutions for some nonlinear Dirichlet problems, dedicated to M. Artola, unpublished.

[6]

L. Boccardo, A remark on some nonlinear elliptic problems, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electron. J. Diff. Eqns. Conf. 08, (2002), 47-52.

[7]

L. Boccardo and B. Dacorogna, Monotonicity of certain differential operators in divergence form, Manuscripta Math., 64 (1989), 253-260. doi: 10.1007/BF01160123.

[8]

L. Boccardo, I. Diaz, D. Giachetti and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation, in "Recent Advances in Nonlinear Elliptic and Parabolic Problems" (Nancy, 1988), Pitman Res. Notes Math. Ser. 208, 229-246, Longman, 1989.

[9]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. P.D.E., 17 (1992), 641-655. doi: 10.1080/03605309208820857.

[10]

L. Boccardo, T. Gallouët and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires, C. R. Acad. Sc. Paris, 315 (1992), 1159-1164.

[11]

J. Carrillo and M. Chipot, On some elliptic equations involving derivatives of the nonlinearity, Proc. Roy. Soc. Edinburgh, 100 (1985), 281-294. doi: 10.1017/S0308210500013822.

[12]

J. Casado Diaz, F. Murat and A. Porretta, Uniqueness results for pseudomonotone problems with $p>2$, C. R. Math. Acad. Sci. Paris, 344 (2007), 487-492. doi: 10.1016/j.crma.2007.02.007.

[13]

M. Chipot and G. Michaille, Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Sc. Norm. Sup. Pisa, 16 (1989), 137-166.

[14]

A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), 207-240. doi: 10.1007/BF01758989.

[15]

O. Guibé, Uniqueness of the solution to quasilinear elliptic equations under a local condition on the diffusion matrix, Adv. Math. Sci. Appl., 17 (2007), 357-368.

[16]

O. Guibé, Uniqueness of the renormalized solution to a class of nonlinear elliptic equations, in "On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments," Quaderni di Matematica 23, 459-497. Department of Mathematics, Seconda Universit\`a di Napoli, Caserta, 2008.

[17]

A. G. Kartsatos and I. V. Skrypnik, The index of a critical point for nonlinear elliptic operators with strong coefficient growth, J. Math. Soc. Japan, 52 (2000), 109-137. doi: 10.2969/jmsj/05210109.

[18]

C. Leone and A. Porretta, Entropy solutions for nonlinear elliptic equations in $ L^1$, Nonlinear Anal., 32 (1998), 325-334. doi: 10.1016/S0362-546X(96)00323-9.

[19]

M. Marcus and V. J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal., 33 (1979), 217-229. doi: 10.1016/0022-1236(79)90113-7.

[20]

A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915-936.

[21]

A. Porretta, Uniqueness of solutions for some nonlinear Dirichlet problems, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 407-430. doi: 10.1007/s00030-004-0031-y.

[22]

A. Porretta, Some remarks on the regularity of solutions for a class of elliptic equations with measure data, Houston J. Math., 26 (2000), 183-213.

[23]

M. M. Porzio, A uniqueness result for monotone elliptic problems, C. R. Math. Acad. Sci. Paris, 337 (2003), 313-316. doi: 10.1016/S1631-073X(03)00347-9.

[24]

N. Trudinger, On the comparison principle for quasilinear divergence structure equations, Arch. for Rat. Mech. Anal., 57 (1975), 128-133. doi: 10.1007/BF00248414.

show all references

References:
[1]

M. Artola, Sur une classe de problèmes paraboliques quasi-linéaires, Boll. U.M.I. B., 5 (1986), 51-70.

[2]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vàzquez, An $L^1$ theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 240-273.

[3]

D. Blanchard, F. Désir and O. Guibé, Quasi-linear degenerate elliptic problems with $L^1$ data, Nonlinear Anal., 60 (2005), 557-587. doi: 10.1016/S0362-546X(04)00395-5.

[4]

L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form, Progress in elliptic and parabolic partial differential equations (Capri, 1994), 43-57.

[5]

L. Boccardo, Uniqueness of solutions for some nonlinear Dirichlet problems, dedicated to M. Artola, unpublished.

[6]

L. Boccardo, A remark on some nonlinear elliptic problems, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electron. J. Diff. Eqns. Conf. 08, (2002), 47-52.

[7]

L. Boccardo and B. Dacorogna, Monotonicity of certain differential operators in divergence form, Manuscripta Math., 64 (1989), 253-260. doi: 10.1007/BF01160123.

[8]

L. Boccardo, I. Diaz, D. Giachetti and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation, in "Recent Advances in Nonlinear Elliptic and Parabolic Problems" (Nancy, 1988), Pitman Res. Notes Math. Ser. 208, 229-246, Longman, 1989.

[9]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. P.D.E., 17 (1992), 641-655. doi: 10.1080/03605309208820857.

[10]

L. Boccardo, T. Gallouët and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires, C. R. Acad. Sc. Paris, 315 (1992), 1159-1164.

[11]

J. Carrillo and M. Chipot, On some elliptic equations involving derivatives of the nonlinearity, Proc. Roy. Soc. Edinburgh, 100 (1985), 281-294. doi: 10.1017/S0308210500013822.

[12]

J. Casado Diaz, F. Murat and A. Porretta, Uniqueness results for pseudomonotone problems with $p>2$, C. R. Math. Acad. Sci. Paris, 344 (2007), 487-492. doi: 10.1016/j.crma.2007.02.007.

[13]

M. Chipot and G. Michaille, Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Sc. Norm. Sup. Pisa, 16 (1989), 137-166.

[14]

A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), 207-240. doi: 10.1007/BF01758989.

[15]

O. Guibé, Uniqueness of the solution to quasilinear elliptic equations under a local condition on the diffusion matrix, Adv. Math. Sci. Appl., 17 (2007), 357-368.

[16]

O. Guibé, Uniqueness of the renormalized solution to a class of nonlinear elliptic equations, in "On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments," Quaderni di Matematica 23, 459-497. Department of Mathematics, Seconda Universit\`a di Napoli, Caserta, 2008.

[17]

A. G. Kartsatos and I. V. Skrypnik, The index of a critical point for nonlinear elliptic operators with strong coefficient growth, J. Math. Soc. Japan, 52 (2000), 109-137. doi: 10.2969/jmsj/05210109.

[18]

C. Leone and A. Porretta, Entropy solutions for nonlinear elliptic equations in $ L^1$, Nonlinear Anal., 32 (1998), 325-334. doi: 10.1016/S0362-546X(96)00323-9.

[19]

M. Marcus and V. J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal., 33 (1979), 217-229. doi: 10.1016/0022-1236(79)90113-7.

[20]

A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915-936.

[21]

A. Porretta, Uniqueness of solutions for some nonlinear Dirichlet problems, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 407-430. doi: 10.1007/s00030-004-0031-y.

[22]

A. Porretta, Some remarks on the regularity of solutions for a class of elliptic equations with measure data, Houston J. Math., 26 (2000), 183-213.

[23]

M. M. Porzio, A uniqueness result for monotone elliptic problems, C. R. Math. Acad. Sci. Paris, 337 (2003), 313-316. doi: 10.1016/S1631-073X(03)00347-9.

[24]

N. Trudinger, On the comparison principle for quasilinear divergence structure equations, Arch. for Rat. Mech. Anal., 57 (1975), 128-133. doi: 10.1007/BF00248414.

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