VIII Jornada de EDP. Caderno de Resumos

São Carlos, 26-28 de janeiro de 2015 São Carlos, 26-28 de janeiro de 2015 VIII Jornada de EDP Caderno de Resumos São Carlos / 2015 São SãoCarlos / 2015 / 2015 Realização: Realização: Apoio: Apoio: Apoio:

Comitê Científico: Alexandre Nolasco de Carvalho (ICMC-USP) Arnaldo Simal do Nascimento (DM-UFSCar) Jorge Guilhermo Hounie (DM-UFSCar) Olímpio H. Miyagaki (UFJF) Comitê Organizador: Francisco Braun (DM-UFSCar) Gustavo Ferron Madeira (DM-UFSCar) Marcelo José Dias Nascimento (DM-UFSCar) Renato José de Moura (DM-UFSCar) Endereço: VIII Jornada de EDP Departamento de Matemática Universidade Federal de São Carlos Rod. Washington Luís, Km 235 CEP 13565-905 São Carlos, SP (Mapa na última página) Acesso à Internet: Rede: DM-WNET Login: jornadaedp Senha: jornada2015

Contents 1 Minicurso 5 Ezequiel Rodrigues Barbosa (UFMG) Operadores do Tipo Schrödinger e Aplicações Geométricas............ 5 2 Palestras 6 Alexandre Nolasco de Carvalho (ICMC-USP) Skew-product semiflows and their attractors under perturbations......... 6 David Jornet (Universidad Politécnica de Valencia) Wave front sets with respect to the iterates of an operator with constant coefficients 6 Djairo Guedes de Figueiredo (UNICAMP) Nonhomogeneous Critical problems........................ 7 Éder Ritis Aragão Costa (ICMC-USP) Global Hypoellipticity for a class of abstract differential complexes........ 8 Eduardo Teixeira (UFC) Nonlinear elliptic PDEs with high order singularities............... 9 Flank David Morais Bezerra (UFPB) Parabolic approximation of damped wave equations via fractional powers.... 10 Gabriela del Valle Planas (UNICAMP) Limite das equações de Stokes e de Navier-Stokes num domínio perfurado.... 11 Giovany M. Figueiredo (UFPA) Nodal solutions of a NLS equation concentrating on lower dimensional spheres. 12 Grey Ercole (UFMG) On the p-torsion functions of an annulus...................... 12 Jacson Simsen (UNIFEI) Parabolic problems in R n with spatially variable exponents............ 13 Luiz Gustavo Farah (UFMG) On the existence of maximizers for Airy-Strichartz inequalities.......... 14 Marcello D Abbicco (FFCLRP-USP) Energy estimates for the plate equation with time-dependent structural damping. 15 Matheus Cheque Bortolan (UFSC) Recent results on impulsive dynamical systems.................. 16 3

Ma To Fu (ICMC-USP) Time-dependent attractors for wave equations in moving boundary domains... 16 Olivâine Santana de Queiróz (UNICAMP) Free boundary problems with supercharacteristic growth............. 17 Pablo Braz e Silva (UFPE) Some asymptotic supnorm estimates for convection-diffusion equations and systems......................................... 17 Raphael Falcão da Hora (UFSC) Inverse scattering with partial data on asymptotically hyperbolic manifolds.... 18 Sergey Shmarev (UFSC) Parabolic equations with nonstandard growth: solvability and localization properties......................................... 19 Severino Toscano do Rego Melo (IME-USP) Operadores Pseudodiferenciais em Grupoides de Lie............... 20 Tiago Picon (FFCLRP-USP) L 1 estimate for elliptic differential operators.................... 21 Uberlândio Batista Severo (UFPB) An improvement for the Trudinger-Moser inequality and applications...... 21 3 Programação 22 4 Como chegar ao Departamento de Matemática 23

1 Minicurso Operadores do Tipo Schrödinger e Aplicações Geométricas Ezequiel Rodrigues Barbosa Universidade Federal de Minas Gerais ezikielmat@yahoo.com.br Nesse minicurso, estudamos os vários aspectos da geometria das superfícies completas com curvatura média constante e imersas em variedades 3-dimensionais. Obtemos importantes informações geométricas sobre uma superfície com curvatura média constante através da análise dos autovalores, ou do índice do operador de Jacobi, que é um operador do tipo Schrödinger, associado a essa superfície. Por exemplo, no contexto clássico do espaço Euclidiano 3-dimensional, uma superfície completa com curvatura média constante e com operador de Jacobi não-negativo é, necessariamente, um plano; as superfícies mínimas orientáveis cujo operador de Jacobi tem índice 1 são também conhecidas (catenóide e superfície de Enneper); e não existe superfície mínima orientável com índice 2. Além de resultados como os citados acima, vamos abordar resultados envolvendo estimativas para diâmetro, área e característica de Euler, a partir do estudo do operador de Jacobi. Apresentaremos também alguns problemas em aberto. 5

2 Palestras Skew-product semiflows and their attractors under perturbations Alexandre Nolasco de Carvalho Universidade de São Paulo andcarva@icmc.usp.br In this lecture we survey our recent results non-autonomous dynamical systems and their attractors. We review the main results on continuity of attractors and their structure including the phase diagram commutativity from the point of view of skew product semi-flows. Wave front sets with respect to the iterates of an operator with constant coefficients David Jornet Universidad Politécnica de Valencia djornet@mat.upv.es We introduce the wave front set W F P (u) with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distribution u D (Ω) in an open set Ω in the setting of ultradifferentiable classes of Braun, Meise and Taylor. We state a version of the microlocal regularity theorem of Hörmander [3, Theorem 5.4] for this new type of wave front set and give some examples and applications of the former result. This talk is based on the recently published paper [1] and the paper in preparation [2]. On joint work with Chiara Boiti and Jordi Juan-Huguet. 6

References [1] C. Boiti, D. Jornet, J. Juan-Huguet, Wave front sets with respect to the iterates of an operator with constant coefficients, Abstr. Appl. Anal. Volume 2014 (2014), Article ID 438716, 17 pages, http://dx.doi.org/10.1155/2014/438716 [2] C. Boiti, D. Jornet, Characterization of wave front sets defined by the iterates of an operator, Preprint. [3] L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear partial differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971), 671 704. Nonhomogeneous Critical problems Djairo Guedes de Figueiredo Universidade Estadual de Campinas djairo@ime.unicamp.br We discuss how assuming some sort of symmetry on the functions we can improve the best imbedding of Sobolev spaces into Lebesgue spaces in the case of dimension larger than 2, and in the case of dimension 2 we discuss imbeddings of the type Trudinger-Moser. 7

Global Hypoellipticity for a class of abstract differential complexes Éder Ritis Aragão Costa Universidade de São Paulo ritis@icmc.usp.br We study sufficient conditions for the global hypoellipticity, in the first degree, of the abstract differential complex given by the following operators L j = + φ (t, A)A, j = 1,..., n. t j t j Where A : D(A) H H is a self-adjoint linear operator, positive with 0 ρ(a), in a Hilbert space H and φ = φ(t, A) is a power series of nonnegative power of A 1 with coefficients in C (Ω), Ω being an open set of R n. We wish to introduce sufficient condition, using dynamics properties of the first coefficient of φ(t, A), to obtain the red hypoellipticity in the elliptic region, which we define bellow, and the techniques we have learned from [1], to overcome the problem out of the elliptic region, in the special case where we suppose A = 1 : H 2 (R n ) L 2 (R n ) L 2 (R n ). Joint work with Adalberto P. Bergamasco (ICMC/USP). References [1] A.P. Bergamasco, P.D. Cordaro, D. Malagutti, P.A.: Globally hypoeliptic systms of vector fields. J. Funct. Anal. 114, n. 2, 267-285 (1993). [2] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, (1981). [3] L. Hörmander. Linear partial differential operators. Springer-Verlag, New York 1963 [4] F. Treves: Topological vector spaces, distributions and kernels. Academic Press, New York 1967. [5] E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24, 2099-2117 (2011). 8

[6] F. Treves, Concatenations of second-order evolution equations applied to local solvability and hypoellipticity, Communications on Pure and Applied Mathematics 26 (1973), 201-250. [7] F. Treves, Study of a Model in the Theory of Complexes of Pseudodifferential Operators, Annals of Mathematics, (2) 104, 269-324 (1976). [8] J. Hounie, Global Hypoelliptic and Global Solvable First Order Evolution Equation, American Mathematical Society, 252, 233-248 (1979). [9] L. C. Yamaoka, Resolubilidade local de uma classe de sistemas subdeterminados abstratos, IME-USP Tese de Doutorado, (2011). [10] Z. Han, Local solvability of analytic pseudo-differential complexes in top degree, Duke Mathematical Journal 87 (1997). [11] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1963. [12] E.R. Aragão-Costa, A.N. Carvalho, P. Marin-Rubio and Gabriela Planas, Gradient-like Nonlinear Semi- groups with Infinitely many Equilibria and Applications to Cascade Systems, Topological Methods in Nonlinear Analysis. Nonlinear elliptic PDEs with high order singularities Eduardo Teixeira Universidade Federal do Ceará teixeira@mat.ufc.br I will discuss about a class of nonlinear elliptic and parabolic partial differential equations presenting high order singular structures. No boundary data are imposed and singularities occur along an a priori unknown interior region (the free boundary of the solution). The key issue we are interested in concerns geometric regularity properties of solutions along their singular sets. 9

Parabolic approximation of damped wave equations via fractional powers Flank David Morais Bezerra Universidade Federal da Paraíba flank@mat.ufpb.br In this work we consider the problem t 2 u + a t u u = f(u), t > 0, x Ω, u(0, x) = u 0 (x), t u(0, x) = v 0 (x), x Ω, u(t, x) = 0, t 0, x Ω, (1) where Ω R N be a bounded smooth domain, N 3, a > 0, and f C 1 (R) (polynomially growing nonlinearity of power ρ < N+2 N 2 ) is such that lim sup s f(s) s < µ 1, with µ 1 being the first eigenvalue of (the Laplacian with Dirichlet boundary condition in Ω). We study parabolic approximations governed by the fractional powers Λ α, α < 1, of the wave operator Λ, associated to the problem (1). We give explicitly expressions for the fractional powers of the wave operator, compute their resolvent operators, their eigenvalues and exhibit a Lyapunov functional for the approximating equations (with the fractional powers). Finally, we obtain solutions of the semilinear damped wave equation as limit of solutions of the approximating equations. This is a joint work with Alexandre N. Carvalho (ICMC-USP) and Marcelo J. D. Nascimento (DM-UFSCar). References [1] H. Amann, Linear and Quasilinear Parabolic Problems. Volume I: Linear Theory, Birkhäuser Verlag, Basel, 1995. [2] A. N. Carvalho, F. D. M. Bezerra and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of the dynamics, 2013, preprint. 10

[3] S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems. Pacific J. Math. 136 (1989), 15-55. [4] T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad. 36 (1960), 94-96. Limite das equações de Stokes e de Navier-Stokes num domínio perfurado Gabriela del Valle Planas Universidade Estadual de Campinas gplanas@ime.unicamp.br Consideramos as equações de Stokes e de Navier-Stokes em um domínio periódico perfurado bidimensional: Ω r = ( L, L) 2 \ D r, onde D r = B(0, r) é o disco de raio r centrado na origem. Na fronteira de Ω = ( L, L) 2 impomos condições periódicas e na circunferência do disco condições de Dirichlet. As equações são consideradas com uma força externa dada e investigamos o comportamento de soluções quando o raio r tende a 0. Em todos os casos mostramos a convergência das soluções para o problema limite esperado, isto é, o problema colocado em todo Ω com condições de contorno periódicas. Trabalho em colaboração com J.C. Robinson (University of Warwick), M. Chipot e Wei Xue (University of Zurich). 11

Nodal solutions of a NLS equation concentrating on lower dimensional spheres Giovany M. Figueiredo Universidade Federal do Pará giovany@ufpa.br In this work we deal with the following nonlinear Schrödinger equation ɛ 2 u + V (x)u = f(u) in R N u H 1 (R N ), where N 3, f is a subcritical power-type nonlinearity and V is a positive potential satisfying a local condition. We prove the existence and concentration of nodal solutions which concentrate around a k dimensional sphere of R N, where 1 k N 1, as ɛ 0. The radius of such sphere is related with the local minimum of a function which takes into account the potential V. Variational methods are used together with the penalization technique in order to overcome the lack of compactness. Joint work with Marcos Pimenta (UNESP). On the p-torsion functions of an annulus Grey Ercole Universidade Federal de Minas Gerais grey.ercole@gmail.com Let p > 1 and denote, respectively, by u p and h(ω a,b ), the p-torsion function and the Cheeger constant of the annulus Ω a,b = { x R N : a < x < b }, N > 1. Thus, u p is the solution of the p-torsional creep problem ( ) div u p 2 u = 1 in Ω a,b u = 0 on Ω a,b, 12

and { } E h(ω a,b ) := min : E Ω a,b E where E and E denote, respectively, the (N 1)-dimensional Lebesgue perimeter of E in R N and the N-dimensional Lebesgue volume of the smooth subset E Ω a,b. We prove that lim u p 1 p p 1 + = lim u p 1 p p 1 + = N bn 1 + a N 1 b N a N = Ω a,b Ω a,b and combine this fact with a characterization of the Cheeger constant that we proved in a previous paper, to give a new proof of the calibrability of Ω a,b, that is, h(ω a,b ) = Ω a,b Ω a,b. Moreover, we prove that u p is concave and satisfies lim p 1 +(u p(x)/ u p ) = 1, uniformly in the set a + ɛ x b ɛ, for all ɛ > 0 sufficiently small. Our results rely on estimates for m p, the radius of the sphere on which u p assumes its maximum value. We derive these estimates by combining Pohozaev s identity for the p- torsional creep problem with a kind of L Hospital rule for monotonicity. Acknowledgments: the authors thank the support of CNPq and FAPEMIG, Brazil. Parabolic problems in R n with spatially variable exponents Jacson Simsen Universidade Federal de Itajubá jacson@unifei.edu.br We study the asymptotic behavior of parabolic p(x)-laplacian problems of the form u λ t div(dλ u λ p(x) 2 u λ ) + a u λ p(x) 2 u λ = B(u λ ) in L 2 (R n ), where n 1, p L (R n ) such that 2 < p := ess inf p(x) p(x) p + := ess sup p(x), D λ L (R n ), > M D λ (x) σ > 0 a.e. in R n, λ [0, ), B : L 2 (R n ) L 2 (R n ) is a globally Lipschitz map and a : R n R is a non-negative 13

continuous function such that there exists R 1 > 0 with {x R n ; a(x) = 0} B R1 (0), inf x R n \B R1 (0) a(x) > 0, and R n \B R1 (0) 1 dx < +. a(x) 2/(p(x) 2) We also study the sensitivity of the problem according to the variation of the diffusion coefficients. This is a joint work with Claudianor O. Alves and Mariza S. Simsen. This work was partially supported by the Brazilian research agencies FAPEMIG grant CEX-APQ-04098-10 and CAPES - PVE - Process 88881.0303888/2013-01. C.O. Alves was partially supported by CNPq - Grant 304036/2013-7. On the existence of maximizers for Airy-Strichartz inequalities Luiz Gustavo Farah Universidade Federal de Minas Gerais lgfarah@gmail.com Recently, in a joint work with Ademir Pastor [1], we give a simple proof of the classical Kenig, Ponce and Vega well-posedness result for the generalized KdV equation [2] t u + xu 3 + x (u k+1 ) = 0, x R, t > 0, k 4, u(x, 0) = u 0 (x). The key ingredient in the proof is the following Airy-Strichartz estimate U(t)u 0 5k/4 L x L 5k/2 C k u 0 Ḣs, k t x where k > 4, s k = (k 4)/2k and U(t) denotes the linear propagator for the KdV equation. Our goal here is to prove the existence of maximizers for the above inequality. The main tool we use is a linear profile decomposition for the Airy equation with initial data in Ḣs k x (R). As a consequence, we also establish the existence of maximizers for a more general class of Strichartz type inequalities associated to the Airy equation. This is a joint work with Henrique Versieux (UFRJ). The author was partially supported by CNPq/Brazil and FAPEMIG/Brazil. 14

References [1] L.G. Farah and A. Pastor, On well-posedness and wave operator for the gkdv equation, Bulletin des Sciences Mathématiques 137 (2013), 229-241. [2] C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Communications on Pure and Applied Mathematics 46 (1993), 527 620. Energy estimates for the plate equation with time-dependent structural damping Marcello D Abbicco Universidade de São Paulo m.dabbicco@gmail.com In this talk we derive long-time energy estimates for the Cauchy problem u tt κ u tt + 2 u + b(t)( ) θ u t = 0, t 0, x R n, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), where b(t) > 0 is an increasing C 1 function with 1/b L 1, θ (0, 1], and κ {0, 1}. The term b(t)( ) θ u t represents a structural damping. For κ = 1, the term u tt corresponds to the rotational inertia, for which a regularity loss decay appears, exception given for the special case θ = 1. We assume initial data in the energy space H 2 L 2 and we consider extra L 1 regularity. The obtained decay estimates are optimal in the special case b(t) = µ(1+t) α, with µ > 0 and α (0, 1]. These results are contained in a joint paper with M.R. Ebert [2] and in a joint paper with R.C. Charão and C.R. Da Luz [1]. The speaker is supported by São Paulo Research Foundation (Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP), grants 2013/15140-2 and 2014/02713-7, JP - Programa Jovens Pesquisadores em Centros Emergentes, research project Decay estimates for semilinear hyperbolic equations 15

References [1] R.C. Charão, M. D Abbicco, C.R. Da Luz, Optimal decay for plate dynamics with time-dependent increasing structural damping, preprint. [2] M. D Abbicco, M.R. Ebert, A classification of structural dissipations for evolution operators, preprint, submitted. Recent results on impulsive dynamical systems Matheus Cheque Bortolan Universidade Federal de Santa Catarina mbortolan@gmail.com Impulsive dynamical systems model various real world phenomena, in which we have abrupt changes of state. In this lecture we will give the basic definitions and discuss about recent developments and results on this theory. This work is in collaboration with Everaldo M. Bonotto, Alexandre N. Carvalho, both from ICMC-USP São Carlos, and Radoslaw Czaja, from University of Silesia - Poland. Time-dependent attractors for wave equations in moving boundary domains Ma To Fu Universidade de São Paulo matofu@icmc.usp.br In this work we study a weakly dissipative wave equation defined on time-varying domains. In a context of evolution process defined on time-varying metric spaces we prove the existence of a pullback attractor. 16

Free boundary problems with supercharacteristic growth Olivâine Santana de Queiróz Universidade Estadual de Campinas olivaine@ime.unicamp.br We study a minimum problem for a nondifferentiable functional whose reaction term does not have scaling properties. Specifically, the model problem is u = χ {u>0} log u, which becomes singular along the free boundary {u > 0}. Notice that the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a supercharacteristic growth like r 2 log r. Some asymptotic supnorm estimates for convection-diffusion equations and systems Pablo Braz e Silva Universidade Federal de Pernambuco pablo@dmat.upfe.br We discuss a direct method to obtain general large time bounds for the supnorms of solutions for various 1D convection-diffusion initial value problems(as, for example, in [1], [2]. We also mention some open interesting problems. 17

References [1] P. Braz e Silva and P. R. Zingano, Some asymptotic properties for solutions of onedimensional advection-diffusion equations with Cauchy data in L p (R), C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 465-467. [2] P. Braz e Silva, P. R. Zingano and W. G. Melo, An asymptotic sup norm estimate for solutions of 1 D systems of convection-diffusion equations, Journal of Differential Equations, to appear. Inverse scattering with partial data on asymptotically hyperbolic manifolds Raphael Falcão da Hora Universidade Federal de Santa Catarina rhora@mtm.ufsc.br We prove a local support theorem for the radiation fields on asymptotically hyperbolic manifolds and use it to show that the scattering operator of an asymptotically hyperbolic manifold, restricted to an open subset of its boundary, determines the manifold modulo isometries that are equal to the identity on the open subset where the scattering operator is known. References [1] R. Hora, A. Sá Barreto, Inverse Scattering with Partial Data on Asymptotically Hyperbolic Manifolds. arxiv:1307.8402. [2] A. Katchalov, Y. Kurylev, M. Lassas, Inverse Boundary Spectral Problems. Monographs and Surveys in Pure and Applied Mathematics 123, Chapman Hall/CRC-press, 2001, xi+290 pp. [3] Y. Kurylev, M. Lassas, Hyperbolic inverse problem with data on a part of the boundary. AMS/IP Stud. Adv. Math. 16 (2000), 259-272, Amer. Math. Soc., Proceedings of Differential equations and mathematical physics (Birmingham, AL, 1999). 18

[4] A. Sá Barreto, Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds. Duke Math. Journal Vol 129, No. 3, 407-480, 2005. [5] D. Tataru, Unique continuation for solutions to PDE s; between Hormander s theorem and Holmgren s theorem. Comm. Partial Differential Equations 20 (1995), no. 5-6, 855-884. Parabolic equations with nonstandard growth: solvability and localization properties Sergey Shmarev University of Oviedo - Spain shmarev@uniovi.es We present recent results on the solvability and localization properties of solutions of parabolic equations of the type t u n i=1 ) D i ( D i u pi(x,t) 2 D i u + f(x, t, u) = 0 where (x, t) denotes the points of the cylinder Q = Ω (0, T ), p i (x, t) > 1 are given functions. The function f(x, t, u) models the presence of absorption or reaction. The following issues are discussed: sufficient conditions for the existence of weak (energy) solutions, energy estimates, sufficient conditions of extinction in a finite time (f < 0), nonexistence of global in time solutions (blow-up, f > 0), the possibility of extinction in a finite time in the limit cases when f 0, p i (x, t) 2 as t, and the equation eventually becomes linear, non-propagation of disturbances from the data caused by the anisotropy. The presentation is partly based on results of the papers [1, 2, 3, 4], obtained in collaboration with S. Antontsev. 19

References [1] S. Antontsev and S. Shmarev Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 53 (2009), no.2, pp. 355 399. [2] S. Antontsev and S. Shmarev Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234 (9), 2010, 263 2645. [3] S. Antontsev and S. Shmarev Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2), 2010, 371-391. [4] S. Antontsev and S. Shmarev Localization of solutions of anisotropic parabolic equations, Nonlinear Anal., 71 (12), 2009, e725 e737. Operadores Pseudodiferenciais em Grupoides de Lie Severino Toscano do Rego Melo Universidade de São Paulo toscano@ime.usp.br Meu objetivo será explicar o enunciado do resultado central do artigo de Claire Debord e Georges Skandalis Adiabatic groupoid, crossed product by R + and pseudodifferential calculus (Advances in Mathematics, 2014) e sua relação com um resultado mais antigo, obtido em colaboração com Aastrup, Monthubert e Schrohe (Journal of Noncommutative Geometry, 2010). A maior parte do tempo será dedicada a preliminares sobre grupoides e operadores pseudodiferenciais em grupoides. 20

L 1 estimate for elliptic differential operators Tiago Picon Universidade de São Paulo picon@ffclrp.usp.br In this lecture we present a characterization of L 1 Sobolev-Gagliardo-Nirenberg estimate for elliptic differential operators with variable coefficients. As application we obtain the previous results in [HP1] and [HP2]. This is joint work with Jorge Hounie (UFSCar). References [HP1] J. Hounie and T. Picon Local Gagliardo-Nirenberg estimates for elliptic of vector fields, Math. Res. Lett. 18 (2011), no. 04, 791 804. [HP2] J. Hounie and T. Picon Local L 1 estimates for elliptic systems of complex vector fields, Proc. Amer. Math. Soc. (to appear) [VS] J. Van Schaftingen, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc. 5 (2013), no. 3, 877 921. An improvement for the Trudinger-Moser inequality and applications Uberlândio Batista Severo Universidade Federal da Paraíba uberlandio@mat.ufpb.br In line with the Concentration Compactness Principle due to P. -L. Lions, we study the lack of compactness of Sobolev imbedding of W 1,n (R n ), n 2, into the Orlicz space L Φα determined by the Young function Φ α (s) behaving like e α s n/(n 1) 1 as s +. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of Trudinger-Moser type in the whole space R n. 21

3 Programac a o PROGRAMAÇÃO DA VIII JORNADA DE EDP 26/01 - Segunda-feira 27/01 - Terça-feira 28/01 - Quarta-feira Chairman Arnaldo Nascimento Jorge Houine Severino Toscano Auditório 8:20-9:00 Alexandre N. de Carvalho Djairo G. Figueiredo Giovany Figueiredo ICMC/USP UNICAMP UFPA Auditório 9:00-9:40 Olivâine S. de Queiróz Eduardo Teixeira Ma To Fu UNICAMP UFC ICMC/USP Auditório 9:40-10:20 Uberlândio B. Severo Sergey Shmarev Luiz Gustavo Farah UFPB Oviedo University UFMG 10:20-10:50 Café Café Café Chairman Uberlândio Batista Eduardo Teixeira Giovany Figueiredo Auditório 10:50-11:30 Marcello D'Abbicco Severino Toscano Gabriela Planas FFCLRP/USP IME/USP UNICAMP Auditório 11:30-12:00 David Jornet Jacson Simsen Pablo Braz e Silva Valencia University UNIFEI UFPE Almoço Almoço Almoço Ezequiel R. Barboza Ezequiel R. Barboza Ezequiel R. Barboza UFMG Minicurso UFMG Minicurso UFMG Minicurso 16:00-16:30 Café Café Café 16:00-16:30 Exposição de Trabalhos Exposição de Trabalhos Exposição de Trabalhos Chairman Luiz G. Farah Gabriela Planas Matheus C. Bortolan Auditório 16:30-17:00 Raphael F. da Hora Matheus C. Bortolan Tiago Picon UFSC UFSC FFCLRP/USP Auditório 17:00-17:30 Flank D. M. Bezerra Grey Ercole Éder R. A. Costa UFPB UFMG ICMC/USP 12:00-14:00 Auditório 14:00-15:30 22

4 Como chegar ao Departamento de Matemática 23