Global gradient estimates in elliptic problems under minimal data and domain regularity
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A. Alvino, 2010, Well-posed elliptic Neumann problems involving irregular data and domains,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27, 1017, 10.1016/j.anihpc.2010.01.010
A. Alvino, 2000, Estimates for the gradient of solutions of nonlinear elliptic equations with $L^1$ data,, \emph{Ann. Mat. Pura Appl.}, 178, 129, 10.1007/BF02505892
A. Alvino, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods,, \emph{Mediter. J. Math.}, 5, 173, 10.1007/s00009-008-0142-5
A. Ancona, 2009, Elliptic operators, conormal derivatives, and positive parts of functions (with an appendix by Haim Brezis),, \emph{J. Funct. Anal.}, 257, 2124, 10.1016/j.jfa.2008.12.019
A. Banerjee, Gradient bounds for $p$-harmonic systems with vanishing Neumann data in a convex domain,, preprint., 10.1016/j.na.2014.01.009
P. B\'enilan, 1995, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations,, \emph{Ann. Sc. Norm. Sup. Pisa}, 22, 241
C. Bennett, 1988, <em>Interpolation of Operators</em>,, Academic Press
A. Bensoussan, 2002, <em>Regularity Results for Nonlinear Elliptic Systems and Applications</em>,, Springer-Verlag, 10.1007/978-3-662-12905-0
L. Boccardo, 1989, Nonlinear elliptic and parabolic equations involving measure data,, \emph{J. Funct. Anal.}, 87, 149, 10.1016/0022-1236(89)90005-0
M. Carro, 2001, On embeddings between classical Lorentz spaces,, \emph{Math. Inequal. Appl.}, 4, 397, 10.7153/mia-04-37
J. Cheeger, 1970, A lower bound for the smallest eigenvalue of the Laplacian,, in \emph{Problems in Analysis} (Papers dedicated to Salomon Bochner, 195
A. Cianchi, 1989, On relative isoperimetric inequalities in the plane,, \emph{Boll. Un. Mat. Ital.}, 3-B, 289
A. Cianchi, 1990, Elliptic equations on manifolds and isoperimetric inequalities,, \emph{Proc. Royal Soc. Edinburgh Sect A}, 114, 213, 10.1017/S0308210500024392
A. Cianchi, 1992, Maximizing the $L^\infty$ norm of the gradient of solutions to the Poisson equation,, \emph{J. Geom. Anal.}, 2, 499, 10.1007/BF02921575
A. Cianchi, 1996, A sharp embedding theorem for Orlicz-Sobolev spaces,, \emph{Indiana Univ. Math. J.}, 45, 39, 10.1512/iumj.1996.45.1958
A. Cianchi, 1997, Boundedness of solutions to variational problems under general growth conditions,, \emph{Comm. Part. Diff. Eq.}, 22, 1629, 10.1080/03605309708821313
A. Cianchi, 2005, Moser-Trudinger inequalities without boundary conditions and isoperimetric problems,, \emph{Indiana Univ. Math. J.}, 54, 669, 10.1512/iumj.2005.54.2589
A. Cianchi, 2008, Neumann problems and isocapacitary inequalites,, \emph{J. Math. Pures Appl.}, 89, 71, 10.1016/j.matpur.2007.10.001
A. Cianchi, 2011, Global Lipschitz regularity for a class of quasilinear elliptic equations,, \emph{Comm. Part. Diff. Equat.}, 36, 100, 10.1080/03605301003657843
A. Cianchi, 2011, On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds,, \emph{J. Differential Geom.}, 87, 469, 10.4310/jdg/1312998232
A. Cianchi, 2012, Boundedness of solutions to the Schrödinger equation under Neumann boundary conditions,, \emph{J. Math. Pures Appl.}, 98, 654, 10.1016/j.matpur.2012.05.007
A. Cianchi, 2013, Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds,, \emph{Amer. J. Math.}, 135, 579, 10.1353/ajm.2013.0028
A. Cianchi, 2014, Global boundedness of the gradient for a class of nonlinear elliptic systems,, \emph{Arch. Ration. Mech. Anal.}, 212, 129, 10.1007/s00205-013-0705-x
A. Cianchi, 2014, Gradient regularity via rearrangements for $p$-Laplacian type elliptic problem,, \emph{J. Europ. Math. Soc.}, 16, 571, 10.4171/JEMS/440
A. Cianchi, 1998, Sobolev embeddings into $BMO$, $VMO$ and $L^{\infty}$,, \emph{Arkiv Mat.}, 36, 317, 10.1007/BF02384772
R. Courant, 1937, <em>Methoden der mathematischen Physik</em>,, Springer-Verlag
A. Dall'Aglio, 1996, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations,, \emph{Ann. Mat. Pura Appl.}, 170, 207, 10.1007/BF01758989
G. Dal Maso, 1999, Renormalized solutions of elliptic equations with general measure data,, \emph{Ann. Sc. Norm. Sup. Pisa Cl. Sci.}, 28, 741
E. De Giorgi, 1968, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico,, \emph{Boll. Un. Mat. Ital.}, 1, 135
T. Del Vecchio, 1995, Nonlinear elliptic equations with measure data,, \emph{Potential Anal.}, 4, 185, 10.1007/BF01275590
G. Dolzmann, 2000, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right-hand side,, \emph{J. Reine Angew. Math.}, 520, 1, 10.1515/crll.2000.022
F. Duzaar, 2010, Local Lipschitz regularity for degenerate elliptic systems,, \emph{Ann. Inst. Henri Poincar\'e}, 27, 1361, 10.1016/j.anihpc.2010.07.002
F. Duzaar, 2010, Gradient continuity estimates,, \emph{Calc. Var. Part. Diff. Equat.}, 39, 379, 10.1007/s00526-010-0314-6
S. Gallot, 1988, Inégalités isopérimétriques et analitiques sur les variétés riemanniennes,, \emph{Asterisque}, 163, 31
M. Giaquinta, 1983, <em>Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems</em>,, Annals of Mathematical Studies
D. Gilbarg, 1983, <em>Elliptic Partial Differential Equations of Second Order</em>,, 2$^{nd}$ edition, 10.1007/978-3-642-61798-0
E. Giusti, 2003, <em>Direct Methods in the Calculus of Variations</em>,, World Scientific, 10.1142/9789812795557
E. Giusti, 1968, Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni,, \emph{Boll. Un. Mat. Ital.}, 1, 219
P. Hai\l asz, 1998, Isoperimetric inequalites and imbedding theorems in irregular domains,, \emph{J. London Math. Soc.}, 58, 425, 10.1112/S0024610798006346
B. Kawohl, 1150, <em>Rearrangements and Convexity of Level Sets in PDE</em>,, Lecture Notes in Math., 1150
S. Kesavan, 2006, <em>Symmetrization & Applications</em>,, Series in Analysis 3, 10.1142/9789812773937
T. Kilpel\, 2000, Sobolev inequalities on sets with irregular boundaries,, \emph{Z. Anal. Anwendungen}, 19, 369, 10.4171/ZAA/956
V. A. Kozlov, 1997, <em>Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations</em>,, Math. Surveys Monographs 52
I. N. Krol', 1972, On the absence of continuity and Hölder continuity of solutions of quasilinear elliptic equations near a nonregular boundary,, \emph{Trudy Moskov. Mat. Os\vs\vc.}, 26, 73
T. Kuusi, 2013, Linear potentials in nonlinear potential theory,, \emph{Arch. Ration. Mech. Anal.}, 207, 215, 10.1007/s00205-012-0562-z
T. Kuusi, A nonlinear Stein theorem,, \emph{Calc. Var. Part. Diff. Equat.}
D. A. Labutin, 1999, Embedding of Sobolev spaces on Hölder domains,, \emph{Proc. Steklov Inst. Math.}, 227, 163
O. A. Ladyzenskaya, 1968, <em>Linear and Quasilinear Elliptic Equations</em>,, Academic Press
G. M. Lieberman, 1986, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable data,, \emph{Comm. Part. Diff. Eq.}, 11, 167, 10.1080/03605308608820422
G. M. Lieberman, 1987, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions,, \emph{Ann. Mat. Pura Appl.}, 148, 77, 10.1007/BF01774284
G. M. Lieberman, 1991, The natural generalization of the natural conditions of Ladyzenskaya and Ural'ceva for elliptic equations,, \emph{Comm. Part. Diff. Eq.}, 16, 311, 10.1080/03605309108820761
G. M. Lieberman, 1992, The conormal derivative problem for equations of variational type in nonsmooth domains,, \emph{Trans. Amer. Math. Soc.}, 330, 41, 10.2307/2154153
P.-L. Lions, Sur les solutions renormalisées d'équations elliptiques non linéaires,, manuscript.
P.-L. Lions, 1990, Isoperimetric inequalities for convex cones,, \emph{Proc. Amer. Math. Soc.}, 109, 477, 10.2307/2048011
C. Maderna, 1979, A priori bounds in non-linear Neumann problems,, \emph{Boll. Un. Mat. Ital.}, 16, 1144
J. Mal\'y, 1997, <em>Fine Regularity of Solutions of Elliptic Partial Differential Equations</em>,, American Mathematical Society, 10.1090/surv/051
V. G. Maz'ya, 1960, Classes of regions and imbedding theorems for function spaces,, \emph{Dokl. Akad. Nauk. SSSR}, 133, 527
V. G. Maz'ya, 1961, p-conductivity and theorems on embedding certain functional spaces into a C-space,, \emph{Dokl. Akad. Nauk. SSSR}, 140, 299
V. G. Maz'ya, 1961, Some estimates of solutions of second-order elliptic equations,, \emph{Dokl. Akad. Nauk. SSSR}, 137, 1057
V. G. Maz'ya, 1969, On weak solutions of the Dirichlet and Neumann problems,, \emph{Trusdy Moskov. Mat. Ob\v s\v c.}, 20, 137
V. G. Maz'ya, 2011, <em>Sobolev Spaces with Applications to Elliptic Partial Differential Equations</em>,, Springer, 10.1007/978-3-642-15564-2
V. G. Maz'ya, 1997, <em>Differentiable Functions on Bad Domains</em>,, World Scientific
N. Meyers, 1963, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations,, \emph{Ann. Scuola Norm. Sup. Pisa}, 17, 189
G. Mingione, 2006, Regularity of minima: An invitation to the dark side of the calculus of variations,, \emph{Appl. Math.}, 51, 355, 10.1007/s10778-006-0110-3
G. Mingione, 2010, Gradient estimates below the duality exponent,, \emph{Math. Ann.}, 346, 571, 10.1007/s00208-009-0411-z
C. B. Morrey, 1966, <em>Multiple Integrals in the Calculus of Variations</em>,, Springer-Verlag
F. Murat, 1993, Soluciones renormalizadas de EDP elipticas no lineales,, Preprint 93023 (Spanish)
F. Murat, 1994, Équations elliptiques non linéaires avec second membre $L^1$ ou mesure,, in \emph{Actes du 26\`eme Congr\'es National d'Analyse Num\'erique
J. Ne\v cas, 1975, Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity,, in \emph{Theor. Nonlin. Oper., 197
R. O'Neil, 1963, Convolution operators in $L(p,q)$ spaces,, \emph{Duke Math. J.}, 30, 129, 10.1215/S0012-7094-63-03015-1
B. Opic, 1999, On generalized Lorentz-Zygmund spaces,, \emph{Math. Ineq. Appl.}, 2, 391, 10.7153/mia-02-35
J. Serrin, 1964, Pathological solutions of elliptic partial differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa}, 18, 385
V. Sver\'ak, 2002, Non-Lipschitz minimizers of smooth uniformly convex variational integrals,, \emph{Proc. Natl. Acad. Sci. USA}, 99, 15269, 10.1073/pnas.222494699
G. Talenti, 1976, Elliptic equations and rearrangements,, \emph{Ann. Sc. Norm. Sup. Pisa}, 3, 697
G. Talenti, 1979, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces,, \emph{Ann. Mat. Pura Appl.}, 120, 159, 10.1007/BF02411942
G. Trombetti, 2000, Symmetrization methods for partial differential equations,, \emph{Boll. Un. Mat. Ital.}, 3-B, 601