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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), 563-578.
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), 457-488. |
[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), 44-51. |
[4] |
R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. |
[5] |
M. Brelot, "On Topologies and Boundaries in Potential Theory," Enlarged edition of a course of lectures delivered in 1966, Lecture Notes in Mathematics, 175, Springer-Verlag, Berlin-New York, 1971. |
[6] |
E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, 3 (1957), 25-43. |
[7] |
J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart," Grundlehren der Mathematischen Wissenschaften, 262, Springer-Verlag, New York, 1984. |
[8] | |
[9] |
J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. |
[10] |
E. Fabes, D. Jerison and C. Kenig, The Wiener test for degenerate elliptic equations, Annales de l'Institut Fourier (Grenoble), 32 (1982), 151-182.
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), 25-39.
doi: 10.1007/BF00280825. |
[12] |
K. Itô and H. P. McKean, Jr., Potential and random walk, Illinois J. Math., 4 (1960), 119-132. |
[13] |
K. Ito and H. P. McKean, Jr., "Diffusion Processes and Their Sample Paths," Springer, 1996. |
[14] |
T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Mathematica, 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[15] |
N. S. Landkof, "Foundations of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. |
[16] |
P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Mathematica, 155 (1985), 153-171.
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), 43-77. |
[18] |
J. Malý and W. P. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations," Mathematical Surveys and Monographs, 51, American Mathematical Society, Providence, RI, 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), 225-242. |
[20] |
J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.
doi: 10.1002/cpa.3160140329. |
[21] |
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.
doi: 10.2307/2372841. |
[22] |
J. Serrin and H. F. Weinberger, Isolated singularities of solutions of linear elliptic equations, Amer. J. Math., 88 (1966), 258-272.
doi: 10.2307/2373060. |
[23] |
N. Wiener, Certain notions in potential theory, J. Math. Phys., 3 (1924), 24-51. |
[24] |
N. Wiener, The dirichlet problem, J. Math. Phys., 3 (1924), 127-146. |
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), 563-578.
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), 457-488. |
[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), 44-51. |
[4] |
R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. |
[5] |
M. Brelot, "On Topologies and Boundaries in Potential Theory," Enlarged edition of a course of lectures delivered in 1966, Lecture Notes in Mathematics, 175, Springer-Verlag, Berlin-New York, 1971. |
[6] |
E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, 3 (1957), 25-43. |
[7] |
J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart," Grundlehren der Mathematischen Wissenschaften, 262, Springer-Verlag, New York, 1984. |
[8] | |
[9] |
J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. |
[10] |
E. Fabes, D. Jerison and C. Kenig, The Wiener test for degenerate elliptic equations, Annales de l'Institut Fourier (Grenoble), 32 (1982), 151-182.
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), 25-39.
doi: 10.1007/BF00280825. |
[12] |
K. Itô and H. P. McKean, Jr., Potential and random walk, Illinois J. Math., 4 (1960), 119-132. |
[13] |
K. Ito and H. P. McKean, Jr., "Diffusion Processes and Their Sample Paths," Springer, 1996. |
[14] |
T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Mathematica, 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[15] |
N. S. Landkof, "Foundations of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. |
[16] |
P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Mathematica, 155 (1985), 153-171.
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), 43-77. |
[18] |
J. Malý and W. P. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations," Mathematical Surveys and Monographs, 51, American Mathematical Society, Providence, RI, 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), 225-242. |
[20] |
J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.
doi: 10.1002/cpa.3160140329. |
[21] |
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.
doi: 10.2307/2372841. |
[22] |
J. Serrin and H. F. Weinberger, Isolated singularities of solutions of linear elliptic equations, Amer. J. Math., 88 (1966), 258-272.
doi: 10.2307/2373060. |
[23] |
N. Wiener, Certain notions in potential theory, J. Math. Phys., 3 (1924), 24-51. |
[24] |
N. Wiener, The dirichlet problem, J. Math. Phys., 3 (1924), 127-146. |
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