# American Institute of Mathematical Sciences

October  2012, 32(10): 3379-3397. doi: 10.3934/dcds.2012.32.3379

## 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

Received  January 2011 Revised  March 2012 Published  May 2012

This paper introduces a notion of regularity (or irregularity) of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset {\mathbb R}^{N}$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the ${\mathcal A}$- harmonic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of ${\mathbb R}^{N}, N\ge 3$ is established in terms of the Wiener test for the regularity of $\infty$. It coincides with the Wiener test for the regularity of $\infty$ in the case of Laplace equation. From the topological point of view, the Wiener test at $\infty$ presents thinness criteria of sets near $\infty$ in fine topology. Precisely, the open set is a deleted neigborhood of $\infty$ in fine topology if and only if $\infty$ is irregular.
Citation: Ugur G. Abdulla. Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3379-3397. doi: 10.3934/dcds.2012.32.3379
##### 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.  Google Scholar [2] U. G. Abdulla, Wiener's criterion at $\infty$ for the heat equation,, Advances in Differential Equations, 13 (2008), 457.   Google Scholar [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.   Google Scholar [4] R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar [5] M. Brelot, "On Topologies and Boundaries in Potential Theory,", Enlarged edition of a course of lectures delivered in 1966, 175 (1966).   Google Scholar [6] E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino, 3 (1957), 25.   Google Scholar [7] J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984).   Google Scholar [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).   Google Scholar [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.  Google Scholar [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.  Google Scholar [12] K. Itô and H. P. McKean, Jr., Potential and random walk,, Illinois J. Math., 4 (1960), 119.   Google Scholar [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.  Google Scholar [15] N. S. Landkof, "Foundations of Modern Potential Theory,", Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar [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.  Google Scholar [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.   Google Scholar [18] J. Malý and W. P. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 (1997).   Google Scholar [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.  Google Scholar [21] J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931.  doi: 10.2307/2372841.  Google Scholar [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.  Google Scholar [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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##### 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.  Google Scholar [2] U. G. Abdulla, Wiener's criterion at $\infty$ for the heat equation,, Advances in Differential Equations, 13 (2008), 457.   Google Scholar [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.   Google Scholar [4] R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar [5] M. Brelot, "On Topologies and Boundaries in Potential Theory,", Enlarged edition of a course of lectures delivered in 1966, 175 (1966).   Google Scholar [6] E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino, 3 (1957), 25.   Google Scholar [7] J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984).   Google Scholar [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).   Google Scholar [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.  Google Scholar [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.  Google Scholar [12] K. Itô and H. P. McKean, Jr., Potential and random walk,, Illinois J. Math., 4 (1960), 119.   Google Scholar [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.  Google Scholar [15] N. S. Landkof, "Foundations of Modern Potential Theory,", Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar [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.  Google Scholar [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.   Google Scholar [18] J. Malý and W. P. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 (1997).   Google Scholar [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.  Google Scholar [21] J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931.  doi: 10.2307/2372841.  Google Scholar [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.  Google Scholar [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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