American Institute of Mathematical Sciences

Advanced
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 - 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

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. 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

[1]

Roberto Triggiani. A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications. Evolution Equations & Control Theory, 2016, 5 (1) : 185-199. doi: 10.3934/eect.2016.5.185
[2]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281
[3]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[4]

David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[5]

Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5133-5152. doi: 10.3934/dcds.2015.35.5133
[6]

Zhilan Feng, Qing Han, Zhipeng Qiu, Andrew N. Hill, John W. Glasser. Computation of $\mathcal R $ in age-structured epidemiological models with maternal and temporary immunity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 399-415. doi: 10.3934/dcdsb.2016.21.399
[7]

Cameron J. Browne, Sergei S. Pilyugin. Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3315-3330. doi: 10.3934/dcdsb.2016099
[8]

Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589
[9]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068
[10]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[11]

Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321
[12]

Emanuela Caliceti, Sandro Graffi. An existence criterion for the $\mathcal{PT}$-symmetric phase transition. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1955-1967. doi: 10.3934/dcdsb.2014.19.1955
[13]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[14]

Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266
[15]

Zutong Wang, Jiansheng Guo, Mingfa Zheng, Youshe Yang. A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle. Journal of Industrial & Management Optimization, 2015, 11 (1) : 13-26. doi: 10.3934/jimo.2015.11.13
[16]

Salvador Villegas. Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1817-1821. doi: 10.3934/cpaa.2011.10.1817
[17]

Tzong-Yow Lee. Asymptotic results for super-Brownian motions and semilinear differential equations. Electronic Research Announcements, 1998, 4: 56-62.

[18]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281
[19]

Gilberto M. Kremer, Filipe Oliveira, Ana Jacinta Soares. $\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases. Kinetic & Related Models, 2009, 2 (2) : 333-343. doi: 10.3934/krm.2009.2.333
[20]

Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences