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First steps in symplectic and spectral theory of integrable systems
Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences
1. | Department of Mathematics, Florida Institute of Technology, Melbourne, Florida 32901, United States |
References:
[1] |
U. G. Abdulla, Wiener's criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries,, Nonlinear Analysis, 67 (2007), 563.
doi: 10.1016/j.na.2006.06.004. |
[2] |
U. G. Abdulla, Wiener's criterion at $\infty$ for the heat equation,, Advances in Differential Equations, 13 (2008), 457.
|
[3] |
U. G. Abdulla, Wiener's criterion at $\infty$ for the heat equation and its measure-theoretical counterpart,, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 44.
|
[4] |
R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).
|
[5] |
M. Brelot, "On Topologies and Boundaries in Potential Theory,", Enlarged edition of a course of lectures delivered in 1966, 175 (1966).
|
[6] |
E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino, 3 (1957), 25.
|
[7] |
J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984).
|
[8] |
E. B. Dynkin, "Markov Processes,", Springer-Verlag, (1965). Google Scholar |
[9] |
J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", Oxford Mathematical Monographs, (1993).
|
[10] |
E. Fabes, D. Jerison and C. Kenig, The Wiener test for degenerate elliptic equations,, Annales de l'Institut Fourier (Grenoble), 32 (1982), 151.
doi: 10.5802/aif.883. |
[11] |
R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations,, Arch. for Rational Mech. Anal., 67 (1977), 25.
doi: 10.1007/BF00280825. |
[12] |
K. Itô and H. P. McKean, Jr., Potential and random walk,, Illinois J. Math., 4 (1960), 119.
|
[13] |
K. Ito and H. P. McKean, Jr., "Diffusion Processes and Their Sample Paths,", Springer, (1996). Google Scholar |
[14] |
T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Mathematica, 172 (1994), 137.
doi: 10.1007/BF02392793. |
[15] |
N. S. Landkof, "Foundations of Modern Potential Theory,", Die Grundlehren der mathematischen Wissenschaften, (1972).
|
[16] |
P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations,, Acta Mathematica, 155 (1985), 153.
doi: 10.1007/BF02392541. |
[17] |
W. Littman, G. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 17 (1963), 43.
|
[18] |
J. Malý and W. P. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 (1997).
|
[19] |
V. G. Maz'ya, On the continuity at a boundary point of solutions of quasi-linear elliptic equations,, Vestnik Leningrad University: Mathematics, 3 (1976), 225. Google Scholar |
[20] |
J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577.
doi: 10.1002/cpa.3160140329. |
[21] |
J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931.
doi: 10.2307/2372841. |
[22] |
J. Serrin and H. F. Weinberger, Isolated singularities of solutions of linear elliptic equations,, Amer. J. Math., 88 (1966), 258.
doi: 10.2307/2373060. |
[23] |
N. Wiener, Certain notions in potential theory,, J. Math. Phys., 3 (1924), 24. Google Scholar |
[24] |
N. Wiener, The dirichlet problem,, J. Math. Phys., 3 (1924), 127. Google Scholar |
show all references
References:
[1] |
U. G. Abdulla, Wiener's criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries,, Nonlinear Analysis, 67 (2007), 563.
doi: 10.1016/j.na.2006.06.004. |
[2] |
U. G. Abdulla, Wiener's criterion at $\infty$ for the heat equation,, Advances in Differential Equations, 13 (2008), 457.
|
[3] |
U. G. Abdulla, Wiener's criterion at $\infty$ for the heat equation and its measure-theoretical counterpart,, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 44.
|
[4] |
R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).
|
[5] |
M. Brelot, "On Topologies and Boundaries in Potential Theory,", Enlarged edition of a course of lectures delivered in 1966, 175 (1966).
|
[6] |
E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino, 3 (1957), 25.
|
[7] |
J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984).
|
[8] |
E. B. Dynkin, "Markov Processes,", Springer-Verlag, (1965). Google Scholar |
[9] |
J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", Oxford Mathematical Monographs, (1993).
|
[10] |
E. Fabes, D. Jerison and C. Kenig, The Wiener test for degenerate elliptic equations,, Annales de l'Institut Fourier (Grenoble), 32 (1982), 151.
doi: 10.5802/aif.883. |
[11] |
R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations,, Arch. for Rational Mech. Anal., 67 (1977), 25.
doi: 10.1007/BF00280825. |
[12] |
K. Itô and H. P. McKean, Jr., Potential and random walk,, Illinois J. Math., 4 (1960), 119.
|
[13] |
K. Ito and H. P. McKean, Jr., "Diffusion Processes and Their Sample Paths,", Springer, (1996). Google Scholar |
[14] |
T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Mathematica, 172 (1994), 137.
doi: 10.1007/BF02392793. |
[15] |
N. S. Landkof, "Foundations of Modern Potential Theory,", Die Grundlehren der mathematischen Wissenschaften, (1972).
|
[16] |
P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations,, Acta Mathematica, 155 (1985), 153.
doi: 10.1007/BF02392541. |
[17] |
W. Littman, G. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 17 (1963), 43.
|
[18] |
J. Malý and W. P. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 (1997).
|
[19] |
V. G. Maz'ya, On the continuity at a boundary point of solutions of quasi-linear elliptic equations,, Vestnik Leningrad University: Mathematics, 3 (1976), 225. Google Scholar |
[20] |
J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577.
doi: 10.1002/cpa.3160140329. |
[21] |
J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931.
doi: 10.2307/2372841. |
[22] |
J. Serrin and H. F. Weinberger, Isolated singularities of solutions of linear elliptic equations,, Amer. J. Math., 88 (1966), 258.
doi: 10.2307/2373060. |
[23] |
N. Wiener, Certain notions in potential theory,, J. Math. Phys., 3 (1924), 24. Google Scholar |
[24] |
N. Wiener, The dirichlet problem,, J. Math. Phys., 3 (1924), 127. Google Scholar |
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