Regularity of \infty for elliptic equations with measurable coefficients and its consequences

Previous Article
Computation of rotation numbers for a class of PL-circle homeomorphisms
DCDS Home
This Issue
Next Article
First steps in symplectic and spectral theory of integrable systems

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

Abstract
Related Papers

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 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.

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]	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.

[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 and 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]	David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[4]	Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[5]	Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - B, 2016, 21 (10) : 3315-3330. doi: 10.3934/dcdsb.2016099
[8]	Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[9]	Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[10]	Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068
[11]	Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368
[12]	Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321
[13]	Emanuela Caliceti, Sandro Graffi. An existence criterion for the $\mathcal{PT}$-symmetric phase transition. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1955-1967. doi: 10.3934/dcdsb.2014.19.1955
[14]	Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589
[15]	Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121
[16]	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 and Continuous Dynamical Systems, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266
[17]	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 and Management Optimization, 2015, 11 (1) : 13-26. doi: 10.3934/jimo.2015.11.13
[18]	Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, 2021, 29 (3) : 2445-2456. doi: 10.3934/era.2020123
[19]	Salvador Villegas. Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1817-1821. doi: 10.3934/cpaa.2011.10.1817
[20]	Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Ugur G. Abdulla
on Google Scholar
Ugur G. Abdulla

[Back to Top]