By Giovanna Citti, Maria Manfredini, Daniele Morbidelli, Sergio Polidoro, Francesco Uguzzoni

The research of PDEs is a well known self-discipline in arithmetic learn, either when it comes to its theoretical facets and its relevance in functions. lately, the geometric homes of linear and nonlinear moment order PDEs of elliptic and parabolic kind were widely studied by means of many remarkable researchers. This publication collects contributions from a specific team of top specialists who took half within the INdAM assembly "Geometric equipment in PDEs", at the get together of the seventieth birthday of Ermanno Lanconelli. They describe a couple of new achievements and/or the state-of-the-art of their self-discipline of study, offering readers an summary of contemporary development and destiny learn developments in PDEs. specifically, the quantity collects major effects for sub-elliptic equations, power thought and diffusion equations, with an emphasis on evaluating various methodologies and on their implications for thought and purposes.

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Barbatis and A.  / 0ÄÂ Ä 2 : =2. In (40) This is true since Ä C Ä ˇ .  C 2 / 0 for all 0 Ä Â Ä =2 and the RHS is clearly non-negative. (ii) Let P be one of the parabola segments P1 ; : : : ; Pm . OAC [ OA /. sin ; cos / and E has equation ˇ x cos C y sin C c D 0 for some c 2 R. Then 2 Œ 2 ; . We note that the axis of the parabola has an asymptote at angle  D 32 . Indeed we shall prove the required inequality for all  2 Œ 2 ; 32  Œ 2 ; ˇ2 .  C / (41) Hence, noting that proof. 2. 1; see Fig.

More recently, Lu and Tang [50] have established a singular version of sharp Moser-Trudinger inequalities on hyperbolic spaces with exact growth. Thus, the result of [50] extends those of [49] in the same spirit of [27, 54]. To describe the main result in [50], we need to introduce some notions. Let Bn D fx 2 Rn W jxj < 1g denote the unit open ball in the Euclidean space n R . The space Bn endowed with the Riemannian metric gij D . 1 1jxj2 /2 ıij is called the ball model of the hyperbolic space Hn .

J. Funct. Anal. 264, 837–893 (2013) 10. : Lecture Notes, Warwick, 3–8 April 2005 (unpublished). ac. pdf 11. : Hardy inequalities for simply connected planar domains. In: Suslina, T. ) Spectral Theory of Differential Operators. American Mathematical Society Translation Seriers 2, vol. 225, pp. 133–140. American Mathematical Society, Providence, RI (2008) 12. : Improved Lp Hardy inequalities. D. Thesis, Stockholm University (2005) Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth Nguyen Lam and Guozhen Lu Dedicated to Ermanno Lanconelli on the occasion of his 70th birthday, with friendship Abstract The main purpose of this paper is two fold.

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