## Abstract and Applied Analysis

### Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains

#### Abstract

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 178534, 19 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969165

Digital Object Identifier
doi:10.1155/2008/178534

Mathematical Reviews number (MathSciNet)
MR2407278

Zentralblatt MATH identifier
1187.35065

#### Citation

Amendola, M. E.; Rossi, L.; Vitolo, A. Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains. Abstr. Appl. Anal. 2008 (2008), Article ID 178534, 19 pages. doi:10.1155/2008/178534. https://projecteuclid.org/euclid.aaa/1220969165

#### References

• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 1983.
• M. G. Crandall, H. Ishii, and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,'' Bulletin of the American Mathematical Society, vol. 27, no. 1, pp. 1--67, 1992.
• L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, vol. 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1995.
• M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, NY, USA, 2nd edition, 1984.
• V. A. Kondrate'v and E. M. Landis, Qualitative theory of second order linear partial differential equations,'' in Partial Differential Equations III, Yu. V. Egorov and M. A. Shubin, Eds., Encyclopedia of Mathematical Sciences 32, pp. 87--192, Springer, New York, NY, USA, 1991.
• E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, vol. 171 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1998.
• A. Vitolo, On the Phragmén-Lindelöf principle for second-order elliptic equations,'' Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 244--259, 2004.
• I. Capuzzo Dolcetta and A. Vitolo, A qualitative Phragmen-Lindelof theorem for fully nonlinear elliptic equations,'' Journal of Differential Equations, vol. 243, no. 2, pp. 578--592, 2007.
• I. Capuzzo Dolcetta and A. Vitolo, Local and global estimates for viscosity solutions of fully nonlinear elliptic equations,'' Discrete and Impulsive Systems, Series A, vol. 14, no. S2, pp. 11--16, 2007.
• X. Cabré, On the Alexandroff-Bakel'man-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations,'' Communications on Pure and Applied Mathematics, vol. 48, no. 5, pp. 539--570, 1995.
• H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains,'' Communications on Pure and Applied Mathematics, vol. 47, no. 1, pp. 47--92, 1994.
• A. Vitolo, A note on the maximum principle for second-order elliptic equations in general domains,'' Acta Mathematica Sinica, vol. 23, no. 11, pp. 1955--1966, 2007.
• V. Cafagna and A. Vitolo, On the maximum principle for second-order elliptic operators in unbounded domains,'' Comptes Rendus de L'Académie des Sciences. Series I, vol. 334, no. 5, pp. 359--363, 2002.
• A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains,'' Journal of Differential Equations, vol. 194, no. 1, pp. 166--184, 2003.
• I. Capuzzo-Dolcetta, F. Leoni, and A. Vitolo, The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains,'' Communications in Partial Differential Equations, vol. 30, no. 10--12, pp. 1863--1881, 2005.
• I. Capuzzo Dolcetta and A. Vitolo, On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domains,'' Le Matematiche, vol. 62, no. 2, pp. 69--91, 2007.
• S. Koike and T. Takahashi, Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients,'' Advances in Differential Equations, vol. 7, no. 4, pp. 493--512, 2002.
• T. Shirai, On fully nonlinear uniformly elliptic PDEs with superlinear growth nonlinearlity,'' M.S. thesis, Saitama University, Saitama, Japan, 2002.
• S. Koike and A. Święch, Maximum principle for fully nonlinear equations via the iterated comparison function method,'' Mathematische Annalen, vol. 339, no. 2, pp. 461--484, 2007.
• N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations,'' Inventiones Mathematicae, vol. 61, no. 1, pp. 67--79, 1980.
• I. Capuzzo Dolcetta and A. Cutr\\i, Hadamard and Liouville type results for fully nonlinear partial differential inequalities,'' Communications in Contemporary Mathematics, vol. 5, no. 3, pp. 435--448, 2003.
• A. Cutr\\i and F. Leoni, On the Liouville property for fully nonlinear equations,'' Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 17, no. 2, pp. 219--245, 2000.
• L. Rossi, Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains,'' Communications on Pure and Applied Analysis, vol. 7, no. 1, pp. 125--141, 2008.
• N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations,'' Revista Matemática Iberoamericana, vol. 4, no. 3-4, pp. 453--468, 1988.
• S. Koike, A beginner's guide to the theory of viscosity solutions, vol. 13 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, Japan, 2004.
• S. Koike, Perron's method for $L^p$-viscosity solutions,'' Saitama Mathematical Journal, vol. 23, pp. 9--28, 2005.
• S. Koike and A. Święch, Maximum principle and existence of $L^p$-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms,'' Nonlinear Differential Equations and Applications, vol. 11, no. 4, pp. 491--509, 2004.
• L. A. Caffarelli, Elliptic second order equations,'' Rendiconti del Seminario Matematico e Fisico di Milano, vol. 58, pp. 253--284, 1988.
• L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,'' Annals of Mathematics, vol. 130, no. 1, pp. 189--213, 1989.
• L. A. Caffarelli, M. G. Crandall, M. Kocan, and A. Święch, On viscosity solutions of fully nonlinear equations with measurable ingredients,'' Communications on Pure and Applied Mathematics, vol. 49, no. 4, pp. 365--397, 1996.
• B. Sirakov, Solvability of fully nonlinear elliptic equations with natural growth and unbounded coefficients,'' oai:hal.archives-ouvertes.fr:hal-00115525, version 2, 2007.
• M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations,'' Archiv der Mathematik, vol. 73, no. 4, pp. 276--285, 1999.
• A. Vitolo, Remarks on the maximum principle and uniqueness estimates,'' International Journal of Pure and Applied Mathematics, vol. 9, no. 1, pp. 77--87, 2003.