INdAM Intensive Period - Geometric Properties of Nonlinear Local and Nonlocal Problems

REFERENCES FOR COURSES OF THE INTENSIVE PERIOD

REFERENCES FOR THE FIRST WEEK

[1] P. Gyrya and L. Saloff-Coste, Neumann and Dirichlet Heat Kernels in Inner Uniform Domains, 1-114.
[2] L. Saloff-Coste, The heat kernel and its estimates, (1st MJS-SI) Advanced Studies in Pure Mathematics, 1-32, (2009).
[3] L. Saloff-Coste, Sobolev Inequalities in Familiar and Unfamilar settings, Sobolev Spaces in Mathematics I, International Series in Mathematics , Springer, Berlin (2009), Vladimir Maz'ya, Editor, 299-343.
[4] L. Saloff-Coste, Pseudo-Poincaré inequalities and applications to Sobolev inequalities, 1-23.

All these files can be downloaded from the web-page http://www.math.cornell.edu/~lsc/lau.html.

REFERENCES FOR THE SECOND WEEK

[1] E. Acerbi and G. Mingione, Gradient estimates for the p(x)-Laplacean system, J.reine ang. Math. (Crelles J.) 584 (2005), 117-148
[2] E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), 285-320
[3] L.A. Caffarelli and I. Peral, On W^1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1-21
[4] E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993), 1107-1134
[5] F. Duzaar and G. Mingione, Gradient continuity estimates, In preparation
[6] F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, Preprint 2009
[7] T. Iwaniec, Projections onto gradient fields and L^p-estimates for degenerated elliptic operators, Studia Math. 75 (1983), 293-312
[8] G. Mingione, The Calderon-Zygmund theory for elliptic problems with measure data, Ann. Scu. Norm. Sup. Pisa Cl. Sci. (5) 6 (2007), 195-261
[9] G. Mingione, Gradient estimates below the duality exponent, Mathematische Annalen, to appear
[10] G. Mingione, Gradient potential estimates, Preprint 2008

REFERENCES FOR THE THIRD WEEK

[1] G. Allaire, Homogenization and two-scale convergence, S.I.A.M. J. Math. Anal. 23 (1992) 1482–1518
[2] D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math. 2 (2002) 35–86
[3] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, S.I.A.M. J. Math. Anal. 20 (1989) 608–623
[4] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var. 12 (2006) 371–397
[5] A. Visintin, Homogenization of a doubly-nonlinear Stefan-type problem, S.I.A.M. J. Math. Anal. 39 (2007) 987–1017
[6] A. Visintin, Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity, Royal Soc. Edinburgh Proc. A 138 (2008) 1363–1401
[7] A. Visintin, Electromagnetic processes in doubly-nonlinear composites, Communications in P.D.E.s 33 (2008) 1–34

REFERENCES FOR THE SIXTH WEEK

[1] E. Ferretti and M.V. Safonov, Growth theorems and Harnack inequality for second order parabolic equations. Harmonic analysis and boundary value problems (Fayetteville, AR, 2000), 87-112, Contemp. Math., 277, Amer. Math. Soc., Providence, RI, 2001.
[2] E.B. Fabes, M.V. Safonov and Yu Yuan, Behavior near the boundary of positive solutions of second order parabolic equations. II, Trans. Amer. Math. Soc. 351 (1999), no. 12, 4947-4961.
[3] M.V. Safonov, Boundary estimates for positive solutions to second order elliptic equations, preprint, 1-20, (2008).
[4] J. Húska, P. Poláčik, M.V. Safonov, Principal Eigenvalues, Spectral Gaps and Exponential Separation Between Positive and Sign-Changing Solutions of Parabolic Equations, Discrete Contin. Dyn. Syst. 2005, suppl., 427--435.
[5] J. Húska, P. Poláčik and M.V. Safonov, Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. \ 5, 711--739.
[6] M.V. Safonov, Harnack's Inequality for Elliptic Equations and the Hoelder Property of their Solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980), 272--287, 312.
[7] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983.
[8] E.M. Landis, Second order equations of elliptic and parabolic type. Translated from the 1971 Russian original by Tamara Rozhkovskaya. With a preface by Nina Uralʹtseva. Translations of Mathematical Monographs, 171. American Mathematical Society, Providence, RI, 1998.
[9] M.V. Safonov, Non-divergence Elliptic Equations of Second Order with Unbounded Drift, preprint (2009), 1-21.