# 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 and Continuous Dynamical Systems, 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), 563-578. 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] E. B. Dynkin, "Markov Processes," Springer-Verlag, 1965. [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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##### 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] E. B. Dynkin, "Markov Processes," Springer-Verlag, 1965. [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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