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

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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
Related Papers

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 and 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.
[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.
[3]	A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods, Mediter. J. Math., 5 (208), 173-185. doi: 10.1007/s00009-008-0142-5.
[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.
[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.
[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.
[7]	C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.
[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.
[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.
[10]	Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-07441-1.
[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.
[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.
[13]	A. Cianchi, On relative isoperimetric inequalities in the plane, Boll. Un. Mat. Ital., 3-B (1989), 289-326.
[14]	A. Cianchi, Elliptic equations on manifolds and isoperimetric inequalities, Proc. Royal Soc. Edinburgh Sect A, 114 (1990), 213-227. doi: 10.1017/S0308210500024392.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	A. Cianchi and L. Pick, Sobolev embeddings into $BMO$, $VMO$ and $L^{\infty}$, Arkiv Mat., 36 (1998), 317-340. doi: 10.1007/BF02384772.
[27]	R. Courant and D. Hilbert, Methoden der mathematischen Physik, Springer-Verlag, Berlin, 1937.
[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.
[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.
[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).
[31]	T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal., 4 (1995), 185-203. doi: 10.1007/BF01275590.
[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.
[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.
[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.
[35]	S. Gallot, Inégalités isopérimétriques et analitiques sur les variétés riemanniennes, Asterisque, 163 (1988), 31-91 (French).
[36]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1983.
[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.
[38]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003. doi: 10.1142/9789812795557.
[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).
[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.
[41]	B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer-Verlag, Berlin, 1985.
[42]	S. Kesavan, Symmetrization & Applications, Series in Analysis 3, World Scientific, Hackensack, 2006. doi: 10.1142/9789812773937.
[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.
[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.
[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.
[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.
[47]	T. Kuusi and G. Mingione, A nonlinear Stein theorem, Calc. Var. Part. Diff. Equat., to appear.
[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.
[49]	O. A. Ladyzenskaya and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
[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.
[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.
[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.
[53]	G. M. Lieberman, The conormal derivative problem for equations of variational type in nonsmooth domains, Trans. Amer. Math. Soc., 330 (1992), 41-67. doi: 10.2307/2154153.
[54]	P.-L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, manuscript.
[55]	P.-L. Lions and F. Pacella, Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc., 109 (1990), 477-485. doi: 10.2307/2048011.
[56]	C. Maderna and S. Salsa, A priori bounds in non-linear Neumann problems, Boll. Un. Mat. Ital., 16 (1979), 1144-1153.
[57]	J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, American Mathematical Society, Providence, 1997. doi: 10.1090/surv/051.
[58]	V. G. Maz'ya, Classes of regions and imbedding theorems for function spaces, Dokl. Akad. Nauk. SSSR, 133 (1960), 527-530 (Russian); English translation: Soviet Math. Dokl., 1 (1960), 882-885.
[59]	V. G. Maz'ya, p-conductivity and theorems on embedding certain functional spaces into a C-space, Dokl. Akad. Nauk. SSSR, 140 (1961), 299-302 (Russian).
[60]	V. G. Maz'ya, Some estimates of solutions of second-order elliptic equations, Dokl. Akad. Nauk. SSSR, 137 (1961), 1057-1059 (Russian); English translation: Soviet Math. Dokl., 2 (1961), 413-415.
[61]	V. G. Maz'ya, On weak solutions of the Dirichlet and Neumann problems, Trusdy Moskov. Mat. Obšč., 20 (1969), 137-172 (Russian); English translation: Trans. Moscow Math. Soc., 20 (1969), 135-172.
[62]	V. G. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.
[63]	V. G. Maz'ya and S. V. Poborchi, Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997.
[64]	N. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206.
[65]	G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-426. doi: 10.1007/s10778-006-0110-3.
[66]	G. Mingione, Gradient estimates below the duality exponent, Math. Ann., 346 (2010), 571-627. doi: 10.1007/s00208-009-0411-z.
[67]	C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966.
[68]	F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023 (Spanish), Laboratoire d'Analyse Numérique de l'Université Paris VI (1993).
[69]	F. Murat, Équations elliptiques non linéaires avec second membre $L^1$ ou mesure, in Actes du 26ème Congrés National d'Analyse Numérique, Les Karellis, France, (1994), A12-A24 (French).
[70]	J. Nečas, Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity, in Theor. Nonlin. Oper., Constr. Aspects. Proc. 4th Int. Summer School., Akademie-Verlag, Berlin, (1975), 197-206.
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[72]	B. Opic and L. Pick, On generalized Lorentz-Zygmund spaces, Math. Ineq. Appl., 2 (1999), 391-467. doi: 10.7153/mia-02-35.
[73]	J. Serrin, Pathological solutions of elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387.
[74]	V. Sverák and X. Yan, Non-Lipschitz minimizers of smooth uniformly convex variational integrals, Proc. Natl. Acad. Sci. USA, 99 (2002), 15269-15276. doi: 10.1073/pnas.222494699.
[75]	G. Talenti, Elliptic equations and rearrangements, Ann. Sc. Norm. Sup. Pisa, 3 (1976), 697-718.
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[77]	G. Trombetti, Symmetrization methods for partial differential equations, Boll. Un. Mat. Ital., 3-B (2000), 601-634.

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.
[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.
[3]	A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods, Mediter. J. Math., 5 (208), 173-185. doi: 10.1007/s00009-008-0142-5.
[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.
[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.
[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.
[7]	C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.
[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.
[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.
[10]	Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-07441-1.
[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.
[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.
[13]	A. Cianchi, On relative isoperimetric inequalities in the plane, Boll. Un. Mat. Ital., 3-B (1989), 289-326.
[14]	A. Cianchi, Elliptic equations on manifolds and isoperimetric inequalities, Proc. Royal Soc. Edinburgh Sect A, 114 (1990), 213-227. doi: 10.1017/S0308210500024392.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	A. Cianchi and L. Pick, Sobolev embeddings into $BMO$, $VMO$ and $L^{\infty}$, Arkiv Mat., 36 (1998), 317-340. doi: 10.1007/BF02384772.
[27]	R. Courant and D. Hilbert, Methoden der mathematischen Physik, Springer-Verlag, Berlin, 1937.
[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.
[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.
[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).
[31]	T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal., 4 (1995), 185-203. doi: 10.1007/BF01275590.
[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.
[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.
[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.
[35]	S. Gallot, Inégalités isopérimétriques et analitiques sur les variétés riemanniennes, Asterisque, 163 (1988), 31-91 (French).
[36]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1983.
[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.
[38]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003. doi: 10.1142/9789812795557.
[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).
[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.
[41]	B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer-Verlag, Berlin, 1985.
[42]	S. Kesavan, Symmetrization & Applications, Series in Analysis 3, World Scientific, Hackensack, 2006. doi: 10.1142/9789812773937.
[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.
[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.
[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.
[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.
[47]	T. Kuusi and G. Mingione, A nonlinear Stein theorem, Calc. Var. Part. Diff. Equat., to appear.
[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.
[49]	O. A. Ladyzenskaya and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
[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.
[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.
[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.
[53]	G. M. Lieberman, The conormal derivative problem for equations of variational type in nonsmooth domains, Trans. Amer. Math. Soc., 330 (1992), 41-67. doi: 10.2307/2154153.
[54]	P.-L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, manuscript.
[55]	P.-L. Lions and F. Pacella, Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc., 109 (1990), 477-485. doi: 10.2307/2048011.
[56]	C. Maderna and S. Salsa, A priori bounds in non-linear Neumann problems, Boll. Un. Mat. Ital., 16 (1979), 1144-1153.
[57]	J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, American Mathematical Society, Providence, 1997. doi: 10.1090/surv/051.
[58]	V. G. Maz'ya, Classes of regions and imbedding theorems for function spaces, Dokl. Akad. Nauk. SSSR, 133 (1960), 527-530 (Russian); English translation: Soviet Math. Dokl., 1 (1960), 882-885.
[59]	V. G. Maz'ya, p-conductivity and theorems on embedding certain functional spaces into a C-space, Dokl. Akad. Nauk. SSSR, 140 (1961), 299-302 (Russian).
[60]	V. G. Maz'ya, Some estimates of solutions of second-order elliptic equations, Dokl. Akad. Nauk. SSSR, 137 (1961), 1057-1059 (Russian); English translation: Soviet Math. Dokl., 2 (1961), 413-415.
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