Representation formulae and inequalities for solutions of a class of second order partial differential equations
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- by Lorenzo D’Ambrosio, Enzo Mitidieri and Stanislav I. Pohozaev PDF
- Trans. Amer. Math. Soc. 358 (2006), 893-910 Request permission
Abstract:
Let $L$ be a possibly degenerate second order differential operator and let $\Gamma _\eta =d^{2-Q}$ be its fundamental solution at $\eta$; here $d$ is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of $-Lu\ge f(\xi ,u)\ge 0$ on ${\mathbb {R}}^N$ to satisfy the representation formula \[ (\mbox R)\qquad \qquad \qquad \qquad \qquad u(\eta )\ge \int _{\mathbb {R}^N} \Gamma _\eta f(\xi ,u) d\xi .\qquad \qquad \qquad \qquad \qquad \qquad \] We prove that (R) holds provided $f(\xi ,\cdot )$ is superlinear, without any assumption on the behavior of $u$ at infinity. On the other hand, if $u$ satisfies the condition \[ \liminf _{R\rightarrow \infty } {-\!\!\!\!\!\!\int }_{R\le d(\xi )\le 2R}|u(\xi )|d\xi =0,\] then (R) holds with no growth assumptions on $f(\xi ,\cdot )$.References
- William Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1233–1246. MR 1709740, DOI 10.1090/S0002-9939-00-05630-6
- A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Uniform Gaussian estimates for the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), no. 10, 1153–1192. MR 1919700
- Haïm Brezis and Shoshana Kamin, Sublinear elliptic equations in $\textbf {R}^n$, Manuscripta Math. 74 (1992), no. 1, 87–106. MR 1141779, DOI 10.1007/BF02567660
- Lorenzo D’Ambrosio and Sandra Lucente, Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Differential Equations 193 (2003), no. 2, 511–541. MR 1998967, DOI 10.1016/S0022-0396(03)00138-4
- G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
- G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
- Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
- Léonard Gallardo, Capacités, mouvement brownien et problème de l’épine de Lebesgue sur les groupes de Lie nilpotents, Probability measures on groups (Oberwolfach, 1981) Lecture Notes in Math., vol. 928, Springer, Berlin-New York, 1982, pp. 96–120 (French, with English summary). MR 669065
- Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
- Juha Heinonen, Calculus on Carnot groups, Fall School in Analysis (Jyväskylä, 1994) Report, vol. 68, Univ. Jyväskylä, Jyväskylä, 1995, pp. 1–31. MR 1351042, DOI 10.1530/jrf.0.0680001
- È. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1–384 (Russian, with English and Russian summaries); English transl., Proc. Steklov Inst. Math. 3(234) (2001), 1–362. MR 1879326
- E. Mitidieri, S.I. Pohozaev, Positivity property of solutions of some elliptic inequalities on $\mathbb {R}^n$, Doklady Math. 68 (2003), 339–344.
Additional Information
- Lorenzo D’Ambrosio
- Affiliation: Dipartimento di Matematica, via E. Orabona 4, Università degli Studi di Bari, I-70125 Bari, Italy
- MR Author ID: 688653
- ORCID: 0000-0003-0677-056X
- Email: dambros@dm.uniba.it
- Enzo Mitidieri
- Affiliation: Dipartimento di Scienze Matematiche, via A. Valerio 12/1, Università degli Studi di Trieste, I-34127 Trieste, Italy
- Email: mitidier@units.it
- Stanislav I. Pohozaev
- Affiliation: Steklov Institute of Mathematics, Gubkina Str. 8, 117966 Moscow, Russia
- Email: pohozaev@mi.ras.ru
- Received by editor(s): April 19, 2004
- Published electronically: April 22, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 893-910
- MSC (2000): Primary 35H10, 35C15, 26D10
- DOI: https://doi.org/10.1090/S0002-9947-05-03717-7
- MathSciNet review: 2177044