Global gradient estimates in  elliptic  problems under minimal data and domain regularity

Andrea Cianchi; Vladimir Maz'ya

doi:10.3934/cpaa.2015.14.285

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January 2015, 14(1): 285-311. doi: 10.3934/cpaa.2015.14.285

Global gradient estimates in elliptic problems under minimal data and domain regularity

Andrea Cianchi ^1, and Vladimir Maz'ya ^2,

1.	Dipartimento di Matematica "U.Dini", Università di Firenze, Piazza Ghiberti 27, 50122 Firenze
2.	Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden

Received February 2014 Revised March 2014 Published September 2014

Abstract
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This is a survey of some recent contributions by the authors on global integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Minimal assumptions on the regularity of the ground domain and of the prescribed data are pursued.

Keywords: capacity, gradient estimates, rearrangements., Elliptic boundary value problems, perimeter.

Mathematics Subject Classification: Primary: 35J25; Secondary: 35B4.

Citation: Andrea Cianchi, Vladimir Maz'ya. Global gradient estimates in elliptic problems under minimal data and domain regularity. Communications on Pure & Applied Analysis, 2015, 14 (1) : 285-311. doi: 10.3934/cpaa.2015.14.285

References:

[1]	A. Alvino, A. Cianchi, V. Maz'ya and A. Mercaldo, Well-posed elliptic Neumann problems involving irregular data and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017-1054. doi: 10.1016/j.anihpc.2010.01.010. Google Scholar
[2]	A. Alvino, V. Ferone and G.Trombetti, Estimates for the gradient of solutions of nonlinear elliptic equations with $L^1$ data, Ann. Mat. Pura Appl., 178 (2000), 129-142. doi: 10.1007/BF02505892. Google Scholar
[3]	A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods,, \emph{Mediter. J. Math.}, 5 (): 173. doi: 10.1007/s00009-008-0142-5. Google Scholar
[4]	A. Ancona, Elliptic operators, conormal derivatives, and positive parts of functions (with an appendix by Haim Brezis), J. Funct. Anal., 257 (2009), 2124-2158. doi: 10.1016/j.jfa.2008.12.019. Google Scholar
[5]	A. Banerjee and J. Lewis, Gradient bounds for $p$-harmonic systems with vanishing Neumann data in a convex domain,, preprint., (). doi: 10.1016/j.na.2014.01.009. Google Scholar
[6]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Sup. Pisa, 22 (1995), 241-273. Google Scholar
[7]	C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988. Google Scholar
[8]	A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0. Google Scholar
[9]	L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. Google Scholar
[10]	Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-07441-1. Google Scholar
[11]	M. Carro, L. Pick, J. Soria and V. D. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl., 4 (2001), 397-428. doi: 10.7153/mia-04-37. Google Scholar
[12]	J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton (1970), 195-199. Google Scholar
[13]	A. Cianchi, On relative isoperimetric inequalities in the plane, Boll. Un. Mat. Ital., 3-B (1989), 289-326. Google Scholar
[14]	A. Cianchi, Elliptic equations on manifolds and isoperimetric inequalities, Proc. Royal Soc. Edinburgh Sect A, 114 (1990), 213-227. doi: 10.1017/S0308210500024392. Google Scholar
[15]	A. Cianchi, Maximizing the $L^\infty$ norm of the gradient of solutions to the Poisson equation, J. Geom. Anal., 2 (1992), 499-515. doi: 10.1007/BF02921575. Google Scholar
[16]	A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45 (1996), 39-65. doi: 10.1512/iumj.1996.45.1958. Google Scholar
[17]	A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Part. Diff. Eq., 22 (1997), 1629-1646. doi: 10.1080/03605309708821313. Google Scholar
[18]	A. Cianchi, Moser-Trudinger inequalities without boundary conditions and isoperimetric problems, Indiana Univ. Math. J., 54 (2005), 669-705. doi: 10.1512/iumj.2005.54.2589. Google Scholar
[19]	A. Cianchi and V. Maz'ya, Neumann problems and isocapacitary inequalites, J. Math. Pures Appl., 89 (2008), 71-105. doi: 10.1016/j.matpur.2007.10.001. Google Scholar
[20]	A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Part. Diff. Equat., 36 (2011), 100-133. doi: 10.1080/03605301003657843. Google Scholar
[21]	A. Cianchi and V. Maz'ya, On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds, J. Differential Geom., 87 (2011), 469-491. Google Scholar
[22]	A. Cianchi and V. Maz'ya, Boundedness of solutions to the Schrödinger equation under Neumann boundary conditions, J. Math. Pures Appl., 98 (2012), 654-688. doi: 10.1016/j.matpur.2012.05.007. Google Scholar
[23]	A. Cianchi and V. Maz'ya, Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds, Amer. J. Math., 135 (2013), 579-635. doi: 10.1353/ajm.2013.0028. Google Scholar
[24]	A. Cianchi and V. Maz'ya, Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Ration. Mech. Anal., 212 (2014), 129-177. doi: 10.1007/s00205-013-0705-x. Google Scholar
[25]	A. Cianchi and V. Maz'ya, Gradient regularity via rearrangements for $p$-Laplacian type elliptic problem, J. Europ. Math. Soc., 16 (2014), 571-595. doi: 10.4171/JEMS/440. Google Scholar
[26]	A. Cianchi and L. Pick, Sobolev embeddings into $BMO$, $VMO$ and $L^{\infty}$, Arkiv Mat., 36 (1998), 317-340. doi: 10.1007/BF02384772. Google Scholar
[27]	R. Courant and D. Hilbert, Methoden der mathematischen Physik, Springer-Verlag, Berlin, 1937. Google Scholar
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[29]	G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808. Google Scholar
[30]	E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital., 1 (1968), 135-137 (Italian). Google Scholar
[31]	T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal., 4 (1995), 185-203. doi: 10.1007/BF01275590. Google Scholar
[32]	G. Dolzmann, N. Hungerbühler and S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right-hand side, J. Reine Angew. Math., 520 (2000), 1-35. doi: 10.1515/crll.2000.022. Google Scholar
[33]	F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. Henri Poincaré, 27 (2010), 1361-1396. doi: 10.1016/j.anihpc.2010.07.002. Google Scholar
[34]	F. Duzaar and G. Mingione, Gradient continuity estimates, Calc. Var. Part. Diff. Equat., 39 (2010), 379-418. doi: 10.1007/s00526-010-0314-6. Google Scholar
[35]	S. Gallot, Inégalités isopérimétriques et analitiques sur les variétés riemanniennes, Asterisque, 163 (1988), 31-91 (French). Google Scholar
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show all references

References:

[1]	A. Alvino, A. Cianchi, V. Maz'ya and A. Mercaldo, Well-posed elliptic Neumann problems involving irregular data and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017-1054. doi: 10.1016/j.anihpc.2010.01.010. Google Scholar
[2]	A. Alvino, V. Ferone and G.Trombetti, Estimates for the gradient of solutions of nonlinear elliptic equations with $L^1$ data, Ann. Mat. Pura Appl., 178 (2000), 129-142. doi: 10.1007/BF02505892. Google Scholar
[3]	A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods,, \emph{Mediter. J. Math.}, 5 (): 173. doi: 10.1007/s00009-008-0142-5. Google Scholar
[4]	A. Ancona, Elliptic operators, conormal derivatives, and positive parts of functions (with an appendix by Haim Brezis), J. Funct. Anal., 257 (2009), 2124-2158. doi: 10.1016/j.jfa.2008.12.019. Google Scholar
[5]	A. Banerjee and J. Lewis, Gradient bounds for $p$-harmonic systems with vanishing Neumann data in a convex domain,, preprint., (). doi: 10.1016/j.na.2014.01.009. Google Scholar
[6]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Sup. Pisa, 22 (1995), 241-273. Google Scholar
[7]	C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988. Google Scholar
[8]	A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0. Google Scholar
[9]	L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. Google Scholar
[10]	Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-07441-1. Google Scholar
[11]	M. Carro, L. Pick, J. Soria and V. D. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl., 4 (2001), 397-428. doi: 10.7153/mia-04-37. Google Scholar
[12]	J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton (1970), 195-199. Google Scholar
[13]	A. Cianchi, On relative isoperimetric inequalities in the plane, Boll. Un. Mat. Ital., 3-B (1989), 289-326. Google Scholar
[14]	A. Cianchi, Elliptic equations on manifolds and isoperimetric inequalities, Proc. Royal Soc. Edinburgh Sect A, 114 (1990), 213-227. doi: 10.1017/S0308210500024392. Google Scholar
[15]	A. Cianchi, Maximizing the $L^\infty$ norm of the gradient of solutions to the Poisson equation, J. Geom. Anal., 2 (1992), 499-515. doi: 10.1007/BF02921575. Google Scholar
[16]	A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45 (1996), 39-65. doi: 10.1512/iumj.1996.45.1958. Google Scholar
[17]	A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Part. Diff. Eq., 22 (1997), 1629-1646. doi: 10.1080/03605309708821313. Google Scholar
[18]	A. Cianchi, Moser-Trudinger inequalities without boundary conditions and isoperimetric problems, Indiana Univ. Math. J., 54 (2005), 669-705. doi: 10.1512/iumj.2005.54.2589. Google Scholar
[19]	A. Cianchi and V. Maz'ya, Neumann problems and isocapacitary inequalites, J. Math. Pures Appl., 89 (2008), 71-105. doi: 10.1016/j.matpur.2007.10.001. Google Scholar
[20]	A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Part. Diff. Equat., 36 (2011), 100-133. doi: 10.1080/03605301003657843. Google Scholar
[21]	A. Cianchi and V. Maz'ya, On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds, J. Differential Geom., 87 (2011), 469-491. Google Scholar
[22]	A. Cianchi and V. Maz'ya, Boundedness of solutions to the Schrödinger equation under Neumann boundary conditions, J. Math. Pures Appl., 98 (2012), 654-688. doi: 10.1016/j.matpur.2012.05.007. Google Scholar
[23]	A. Cianchi and V. Maz'ya, Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds, Amer. J. Math., 135 (2013), 579-635. doi: 10.1353/ajm.2013.0028. Google Scholar
[24]	A. Cianchi and V. Maz'ya, Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Ration. Mech. Anal., 212 (2014), 129-177. doi: 10.1007/s00205-013-0705-x. Google Scholar
[25]	A. Cianchi and V. Maz'ya, Gradient regularity via rearrangements for $p$-Laplacian type elliptic problem, J. Europ. Math. Soc., 16 (2014), 571-595. doi: 10.4171/JEMS/440. Google Scholar
[26]	A. Cianchi and L. Pick, Sobolev embeddings into $BMO$, $VMO$ and $L^{\infty}$, Arkiv Mat., 36 (1998), 317-340. doi: 10.1007/BF02384772. Google Scholar
[27]	R. Courant and D. Hilbert, Methoden der mathematischen Physik, Springer-Verlag, Berlin, 1937. Google Scholar
[28]	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. Google Scholar
[29]	G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808. Google Scholar
[30]	E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital., 1 (1968), 135-137 (Italian). Google Scholar
[31]	T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal., 4 (1995), 185-203. doi: 10.1007/BF01275590. Google Scholar
[32]	G. Dolzmann, N. Hungerbühler and S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right-hand side, J. Reine Angew. Math., 520 (2000), 1-35. doi: 10.1515/crll.2000.022. Google Scholar
[33]	F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. Henri Poincaré, 27 (2010), 1361-1396. doi: 10.1016/j.anihpc.2010.07.002. Google Scholar
[34]	F. Duzaar and G. Mingione, Gradient continuity estimates, Calc. Var. Part. Diff. Equat., 39 (2010), 379-418. doi: 10.1007/s00526-010-0314-6. Google Scholar
[35]	S. Gallot, Inégalités isopérimétriques et analitiques sur les variétés riemanniennes, Asterisque, 163 (1988), 31-91 (French). Google Scholar
[36]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1983. Google Scholar
[37]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. Google Scholar
[38]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003. doi: 10.1142/9789812795557. Google Scholar
[39]	E. Giusti and M. Miranda, Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll. Un. Mat. Ital., 1 (1968), 219-226 (Italian). Google Scholar
[40]	P. Haiłasz and P. Koskela, Isoperimetric inequalites and imbedding theorems in irregular domains, J. London Math. Soc., 58 (1998), 425-450. doi: 10.1112/S0024610798006346. Google Scholar
[41]	B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer-Verlag, Berlin, 1985. Google Scholar
[42]	S. Kesavan, Symmetrization & Applications, Series in Analysis 3, World Scientific, Hackensack, 2006. doi: 10.1142/9789812773937. Google Scholar
[43]	T. Kilpeläinen and J. Malý, Sobolev inequalities on sets with irregular boundaries, Z. Anal. Anwendungen, 19 (2000), 369-380. doi: 10.4171/ZAA/956. Google Scholar
[44]	V. A. Kozlov, V. G. Maz'ya and J. Rossman, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Math. Surveys Monographs 52, Amer. Math. Soc., Providence, RI, 1997. Google Scholar
[45]	I. N. Krol' and V. G. Maz'ya, On the absence of continuity and Hölder continuity of solutions of quasilinear elliptic equations near a nonregular boundary, Trudy Moskov. Mat. Osšč., 26 (1972) (Russian); English translation: Trans. Moscow Math. Soc., 26 (1972), 73-93. Google Scholar
[46]	T. Kuusi and G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal., 207 (2013), 215-246. doi: 10.1007/s00205-012-0562-z. Google Scholar
[47]	T. Kuusi and G. Mingione, A nonlinear Stein theorem,, \emph{Calc. Var. Part. Diff. Equat.}, (). Google Scholar
[48]	D. A. Labutin, Embedding of Sobolev spaces on Hölder domains, Proc. Steklov Inst. Math., 227 (1999), 163-172 (Russian); English translation: Trudy Mat. Inst., 227 (1999), 170-179. Google Scholar
[49]	O. A. Ladyzenskaya and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Google Scholar
[50]	G. M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable data, Comm. Part. Diff. Eq., 11 (1986), 167-229. doi: 10.1080/03605308608820422. Google Scholar
[51]	G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl., 148 (1987), 77-99. doi: 10.1007/BF01774284. Google Scholar
[52]	G. M. Lieberman, The natural generalization of the natural conditions of Ladyzenskaya and Ural'ceva for elliptic equations, Comm. Part. Diff. Eq., 16 (1991), 311-361. doi: 10.1080/03605309108820761. Google Scholar
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