-
Previous Article
Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems
- CPAA Home
- This Issue
-
Next Article
Statistical exponential formulas for homogeneous diffusion
Global gradient estimates in elliptic problems under minimal data and domain regularity
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 |
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. |
[71] |
R. O'Neil, Convolution operators in $L(p,q)$ spaces, Duke Math. J., 30 (1963), 129-142. |
[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. |
[76] |
G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl., 120 (1979), 159-184.
doi: 10.1007/BF02411942. |
[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. |
[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. |
[71] |
R. O'Neil, Convolution operators in $L(p,q)$ spaces, Duke Math. J., 30 (1963), 129-142. |
[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. |
[76] |
G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl., 120 (1979), 159-184.
doi: 10.1007/BF02411942. |
[77] |
G. Trombetti, Symmetrization methods for partial differential equations, Boll. Un. Mat. Ital., 3-B (2000), 601-634. |
[1] |
Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations and Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271 |
[2] |
Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 |
[3] |
Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347 |
[4] |
Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819 |
[5] |
Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059 |
[6] |
Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 |
[7] |
Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure and Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399 |
[8] |
Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107 |
[9] |
Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380 |
[10] |
Ha Tuan Dung, Nguyen Thac Dung, Jiayong Wu. Sharp gradient estimates on weighted manifolds with compact boundary. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4127-4138. doi: 10.3934/cpaa.2021148 |
[11] |
Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure and Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521 |
[12] |
Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747 |
[13] |
Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155 |
[14] |
Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, 2021, 29 (5) : 3361-3382. doi: 10.3934/era.2021043 |
[15] |
Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51 |
[16] |
Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 |
[17] |
Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897 |
[18] |
Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141 |
[19] |
Luigi Ambrosio, Michele Miranda jr., Diego Pallara. Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 591-606. doi: 10.3934/dcds.2010.28.591 |
[20] |
Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]