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December  2011, 31(4): 1053-1067. doi: 10.3934/dcds.2011.31.1053

Planar quasilinear elliptic equations with right-hand side in $L(\log L)^{\delta}$

1. 

Consiglio Nazionale delle Ricerche, Istituto per le Applicazioni del Calcolo “Mauro Picone” - Sezione di Napoli, Via Pietro Castellino 111, 80131, Napoli, Italy

2. 

Università degli Studi di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Via Cintia, 80126, Napoli, Italy, Italy

Received  November 2009 Revised  July 2010 Published  September 2011

For $\Omega\subset \mathbb{R}^2$ a bounded open set with $\mathcal{C}^1$ boundary, we study the regularity of the variational solution $v\in W_0^{1,2}(\Omega)$ to the quasilinear elliptic equation of Leray-Lions \begin{equation*} - \,\textrm{div}\, A(x, \nabla v) = f \end{equation*} when $f$ belongs to the Zygmund space $L(\log L)^{\delta}(\Omega)$, $\frac{1}{2} \leq \delta \leq 1$. We prove that $|\nabla v|$ belongs to the Lorentz space $L^{2, 1/\delta}(\Omega)$.
Citation: Angela Alberico, Teresa Alberico, Carlo Sbordone. Planar quasilinear elliptic equations with right-hand side in $L(\log L)^{\delta}$. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1053-1067. doi: 10.3934/dcds.2011.31.1053
References:
[1]

A. Alberico and V. Ferone, Regularity properties of solutions of elliptic equations in $\mathbbR^2$ in limit cases,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 6 (1995), 237.   Google Scholar

[2]

C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces,, Dissertationes Math. (Rozprawy Mat.), 175 (1980), 1.   Google Scholar

[3]

C. Bennett and R. Sharpley, "Interpolation of Operators,", Pure and Applied Mathematics, 129 (1988).   Google Scholar

[4]

L. Boccardo, Quelques problemes de Dirichlet avec donneées dans de grand espaces de Sobolev, (French) [Some Dirichlet problems with data in large Sobolev spaces],, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 1269.   Google Scholar

[5]

C. Capone and A. Fiorenza, On small Lebesgue spaces,, J. Funct. Spaces Appl., 3 (2005), 73.   Google Scholar

[6]

M. Carozza and C. Sbordone, The distance to $L^\infty$ in same function spaces and applications,, Differential Integral Equations, 10 (1997), 599.   Google Scholar

[7]

A. Cianchi, Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 575.   Google Scholar

[8]

M. Cwikel, The dual of Weak $L^p$,, Ann. Inst. Fourier (Grenoble), 25 (1975), 81.  doi: 10.5802/aif.556.  Google Scholar

[9]

G. di Blasio, F. Feo and M. R. Posteraro, Existence results for nonlinear elliptic equations related to Gauss measure in a limit case,, Commun. Pure Appl. Anal., 7 (2008), 1497.  doi: 10.3934/cpaa.2008.7.1497.  Google Scholar

[10]

G. Di Fratta and A. Fiorenza, A direct approach to the duality of grand and small Lebesgue spaces,, Nonlinear Anal., 70 (2009), 2582.  doi: 10.1016/j.na.2008.03.044.  Google Scholar

[11]

D. E. Edmunds and H. Triebel, "Function Spaces, Entropy Numbers, Differential Operators,", Cambridge Tracts in Mathematics, 120 (1996).   Google Scholar

[12]

D. E. Edmunds and H. Triebel, Logarithmic Sobolev spaces and their applications to spectral theory,, Proc. London Math. Soc. (3), 71 (1995), 333.  doi: 10.1112/plms/s3-71.2.333.  Google Scholar

[13]

D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces,, Indiana Univ. Math. J., 44 (1995), 19.  doi: 10.1512/iumj.1995.44.1977.  Google Scholar

[14]

A. Fiorenza, Duality and reflexivity in grand Lebesgue spaces,, Collect. Math., 51 (2000), 131.   Google Scholar

[15]

A. Fiorenza and G. E. Karadzhov, Grand and small Lebesgue spaces and their analogs,, Z. Anal. Anwendungen, 23 (2004), 657.  doi: 10.4171/ZAA/1215.  Google Scholar

[16]

A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$,, Studia Math., 127 (1998), 223.   Google Scholar

[17]

N. Fusco, P.-L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases,, Proc. Amer. Math. Soc., 124 (1996), 561.  doi: 10.1090/S0002-9939-96-03136-X.  Google Scholar

[18]

L. Greco, A remark on the equality det $Df$= Det $Df$,, Differential Integral Equations, 6 (1993), 1089.   Google Scholar

[19]

S. Hencl, Sharp generalized Trudinger inequalities via truncation,, J. Math. Anal. Appl., 322 (2006), 336.  doi: 10.1016/j.jmaa.2005.07.041.  Google Scholar

[20]

T. Iwaniec and J. Onninen, Continuity estimates for $n$-harmonic equations,, Indiana Univ. Math. J., 56 (2007), 805.  doi: 10.1512/iumj.2007.56.2987.  Google Scholar

[21]

T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses,, Arch. Rational Mech. Anal., 119 (1992), 129.  doi: 10.1007/BF00375119.  Google Scholar

[22]

T. Iwaniec and C. Sbordone, Quasiharmonic fields,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 519.   Google Scholar

[23]

T. Iwaniec and A. Verde, On the operator $\mathcalL(f)=f$ $\log \|f\|$,, J. Funct. Anal., 169 (1999), 391.  doi: 10.1006/jfan.1999.3443.  Google Scholar

[24]

A. Passarelli di Napoli and C. Sbordone, Elliptic equations with right-hand side in $L(\log L)^{\alpha}$,, Rend. Accad. Sci. Fis. Mat. Napoli (4), 62 (1995), 301.   Google Scholar

[25]

G. Stampacchia, Some limit cases of $L^p$-estimates for solutions of second order elliptic equations,, Comm. Pure Appl. Math., 16 (1963), 505.  doi: 10.1002/cpa.3160160409.  Google Scholar

[26]

E. M. Stein, Editor's note: The differentiability of functions in $\mathbbR^n$,, Ann. of Math. (2), 113 (1981), 383.   Google Scholar

[27]

N. S. Trudinger, On imbeddings into Orlicz spaces and applications,, J. Math. Mech., 17 (1967), 473.   Google Scholar

show all references

References:
[1]

A. Alberico and V. Ferone, Regularity properties of solutions of elliptic equations in $\mathbbR^2$ in limit cases,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 6 (1995), 237.   Google Scholar

[2]

C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces,, Dissertationes Math. (Rozprawy Mat.), 175 (1980), 1.   Google Scholar

[3]

C. Bennett and R. Sharpley, "Interpolation of Operators,", Pure and Applied Mathematics, 129 (1988).   Google Scholar

[4]

L. Boccardo, Quelques problemes de Dirichlet avec donneées dans de grand espaces de Sobolev, (French) [Some Dirichlet problems with data in large Sobolev spaces],, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 1269.   Google Scholar

[5]

C. Capone and A. Fiorenza, On small Lebesgue spaces,, J. Funct. Spaces Appl., 3 (2005), 73.   Google Scholar

[6]

M. Carozza and C. Sbordone, The distance to $L^\infty$ in same function spaces and applications,, Differential Integral Equations, 10 (1997), 599.   Google Scholar

[7]

A. Cianchi, Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 575.   Google Scholar

[8]

M. Cwikel, The dual of Weak $L^p$,, Ann. Inst. Fourier (Grenoble), 25 (1975), 81.  doi: 10.5802/aif.556.  Google Scholar

[9]

G. di Blasio, F. Feo and M. R. Posteraro, Existence results for nonlinear elliptic equations related to Gauss measure in a limit case,, Commun. Pure Appl. Anal., 7 (2008), 1497.  doi: 10.3934/cpaa.2008.7.1497.  Google Scholar

[10]

G. Di Fratta and A. Fiorenza, A direct approach to the duality of grand and small Lebesgue spaces,, Nonlinear Anal., 70 (2009), 2582.  doi: 10.1016/j.na.2008.03.044.  Google Scholar

[11]

D. E. Edmunds and H. Triebel, "Function Spaces, Entropy Numbers, Differential Operators,", Cambridge Tracts in Mathematics, 120 (1996).   Google Scholar

[12]

D. E. Edmunds and H. Triebel, Logarithmic Sobolev spaces and their applications to spectral theory,, Proc. London Math. Soc. (3), 71 (1995), 333.  doi: 10.1112/plms/s3-71.2.333.  Google Scholar

[13]

D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces,, Indiana Univ. Math. J., 44 (1995), 19.  doi: 10.1512/iumj.1995.44.1977.  Google Scholar

[14]

A. Fiorenza, Duality and reflexivity in grand Lebesgue spaces,, Collect. Math., 51 (2000), 131.   Google Scholar

[15]

A. Fiorenza and G. E. Karadzhov, Grand and small Lebesgue spaces and their analogs,, Z. Anal. Anwendungen, 23 (2004), 657.  doi: 10.4171/ZAA/1215.  Google Scholar

[16]

A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$,, Studia Math., 127 (1998), 223.   Google Scholar

[17]

N. Fusco, P.-L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases,, Proc. Amer. Math. Soc., 124 (1996), 561.  doi: 10.1090/S0002-9939-96-03136-X.  Google Scholar

[18]

L. Greco, A remark on the equality det $Df$= Det $Df$,, Differential Integral Equations, 6 (1993), 1089.   Google Scholar

[19]

S. Hencl, Sharp generalized Trudinger inequalities via truncation,, J. Math. Anal. Appl., 322 (2006), 336.  doi: 10.1016/j.jmaa.2005.07.041.  Google Scholar

[20]

T. Iwaniec and J. Onninen, Continuity estimates for $n$-harmonic equations,, Indiana Univ. Math. J., 56 (2007), 805.  doi: 10.1512/iumj.2007.56.2987.  Google Scholar

[21]

T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses,, Arch. Rational Mech. Anal., 119 (1992), 129.  doi: 10.1007/BF00375119.  Google Scholar

[22]

T. Iwaniec and C. Sbordone, Quasiharmonic fields,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 519.   Google Scholar

[23]

T. Iwaniec and A. Verde, On the operator $\mathcalL(f)=f$ $\log \|f\|$,, J. Funct. Anal., 169 (1999), 391.  doi: 10.1006/jfan.1999.3443.  Google Scholar

[24]

A. Passarelli di Napoli and C. Sbordone, Elliptic equations with right-hand side in $L(\log L)^{\alpha}$,, Rend. Accad. Sci. Fis. Mat. Napoli (4), 62 (1995), 301.   Google Scholar

[25]

G. Stampacchia, Some limit cases of $L^p$-estimates for solutions of second order elliptic equations,, Comm. Pure Appl. Math., 16 (1963), 505.  doi: 10.1002/cpa.3160160409.  Google Scholar

[26]

E. M. Stein, Editor's note: The differentiability of functions in $\mathbbR^n$,, Ann. of Math. (2), 113 (1981), 383.   Google Scholar

[27]

N. S. Trudinger, On imbeddings into Orlicz spaces and applications,, J. Math. Mech., 17 (1967), 473.   Google Scholar

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