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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

Ugur G. Abdulla 1,

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.
Keywords: Dirichlet problem, fine topology, measurable coefficients, Uniformly elliptic equations, regularity (or irregularity) of $\infty$, differential generator., ${\mathcal A}$-thinness, characteristic Markov process, ${\mathcal A}$-harmonic measure, ${\mathcal A}$-super- or subharmonicity, Brownian motion, Newtonian capacity, PWB solution, Wiener test, metric compactification of $R^{N+1}$.
Mathematics Subject Classification: Primary: 35J25, 31C05, 31C15, 31C40; Secondary: 60J45, 60J6.
Citation: Ugur G. Abdulla. Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences. Discrete & 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.  Google Scholar

[2]

U. G. Abdulla, Wiener's criterion at $\infty$ for the heat equation, Advances in Differential Equations, 13 (2008), 457-488.  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), 44-51.  Google Scholar

[4]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.  Google Scholar

[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.  Google Scholar

[6]

E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, 3 (1957), 25-43.  Google Scholar

[7]

J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart," Grundlehren der Mathematischen Wissenschaften, 262, Springer-Verlag, New York, 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, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 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), 151-182. 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), 25-39. doi: 10.1007/BF00280825.  Google Scholar

[12]

K. Itô and H. P. McKean, Jr., Potential and random walk, Illinois J. Math., 4 (1960), 119-132.  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), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[15]

N. S. Landkof, "Foundations of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[16]

P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Mathematica, 155 (1985), 153-171. 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), 43-77.  Google Scholar

[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.  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), 225-242. Google Scholar

[20]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar

[21]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. 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), 258-272. doi: 10.2307/2373060.  Google Scholar

[23]

N. Wiener, Certain notions in potential theory, J. Math. Phys., 3 (1924), 24-51. Google Scholar

[24]

N. Wiener, The dirichlet problem, J. Math. Phys., 3 (1924), 127-146. 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), 563-578. 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), 457-488.  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), 44-51.  Google Scholar

[4]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.  Google Scholar

[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.  Google Scholar

[6]

E. De Giorgi, Sulla differentiabilitá e l'analiticitá delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, 3 (1957), 25-43.  Google Scholar

[7]

J. L. Doob, "Classical Potential Theory and its Probabilistic Counterpart," Grundlehren der Mathematischen Wissenschaften, 262, Springer-Verlag, New York, 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, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 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), 151-182. 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), 25-39. doi: 10.1007/BF00280825.  Google Scholar

[12]

K. Itô and H. P. McKean, Jr., Potential and random walk, Illinois J. Math., 4 (1960), 119-132.  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), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[15]

N. S. Landkof, "Foundations of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[16]

P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Mathematica, 155 (1985), 153-171. 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), 43-77.  Google Scholar

[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.  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), 225-242. Google Scholar

[20]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar

[21]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. 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), 258-272. doi: 10.2307/2373060.  Google Scholar

[23]

N. Wiener, Certain notions in potential theory, J. Math. Phys., 3 (1924), 24-51. Google Scholar

[24]

N. Wiener, The dirichlet problem, J. Math. Phys., 3 (1924), 127-146. Google Scholar

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