Grant CNCS-UEFISCDI PN II-ID-PCE-2012-4-0021

Funding Sources: Consiliul National al Cercetarii Stiintifice & Uniunea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii Dezvoltarii si Inovarii

Grant no. 5 from 30/08/2013

Title: "Variable Exponent Analysis: Partial Differential Equations and Calculus of Variations"

Abstract: Variable exponent analysis has its roots in the study of functional spaces involving variable exponents primarily due to Orlicz’s works from the 1930’s. The topic has benefited from constant interest over the years, and in the last decades substantial scholarly attention has been increasingly given to this research area in connection with the study of some PDE’s which can serve as models for different phenomena arising in elasticity theory, fluid dynamics, image processing or mathematical biology. The differential operator usually involved in such equations is the p(x)-Laplace operator and it represents an adequate candidate for modeling nonhomogeneous processes which can occur in different real world applications. At the same time, the competition of growth rates involved in equations with variable exponents enables more situations of study than in the case when constant coefficients are involved. The papers where such kinds of problems are treated usually do not cover all the situations that can occur, generally being just isolated results which are mostly applied on some special cases. There are many open challenging problems and unanswered mathematical questions in this field making the study of equations with variable exponents extremely attractive.

Objectives (pdf)

BUDGET BREAKDOWN: estimated budget for the entire period of the grant

realized budget for 2013

realized budget for 2014

realized budget for 2015

realized budget for 2016

TEAM: Mihai Mihailescu (project leader); Liviu Ignat; Nicusor Costea; Cristian Cazacu; Denisa Stancu-Dumitru; Aurora Marica; Cristian Enache; Tatiana Ignat; Maria Farcaseanu (starting from 1st of September 2015)

HOST INSTITUTION: Institutul de Matematica "Simion Stoilow" al Academiei Romane

FINAL EVALUATION SHEET OF THE PROJECT (IN ROMANIAN): Grade of the evaluation A.

PUBLICATIONS 2013:

  • M. Mihailescu and G. Morosanu, An existence result for a nonhomogeneous problem in R2 related to nonlinear Hencky-type materials, Nonlinear Analysis: Real World Applications 14 (2013), 1466-1476. (pdf)

  • M. Bocea and M. Mihailescu, $\Gamma$-convergence of inhomogeneous functionals in Orlicz-Sobolev spaces, Proceedings of the Edinburgh Mathematical Society 58 (2015), 287-303. (pdf)

CONFERENCES, WORKSHOPS, TALKS 2013:

SCIENTIFIC REPORT FOR 2013

PUBLICATIONS 2014:

  • M. Bocea and M. Mihailescu, Eigenvalue Problems in Orlicz-Sobolev Spaces for Rapidly Growing Operators in Divergence Form, Journal of Differential Equations 256 (2014), 640-657. (pdf)

  • M. Bocea and M. Mihailescu, On the continuity of the Luxemburg norm of the gradient in $L^p(·)$ with respect to $p(·)$, Proceedings of the American Mathematical Society 142 (2014), 507-517. (pdf)

  • C. Enache, Maximum and minimum principles for a class of Monge-Ampere equations in the plane, with applications to surfaces of constant Gauss curvature, Communications on Pure and Applied Analysis, 13(3) (2014), 1347-1359. (pdf)

  • C. Enache, Maximum principles and isoperimetric inequalities for some Monge-Ampere type problems, Comptes Rendus de l’Acadmie des Sciences Paris Series I - Mathematics 352 (2014), 37-42. (pdf)

  • T. I. Ignat, Asymtotics for nonlocal evolution problems by scaling arguments, Differential Equations and Applications, 5(4) (2013), 613-626.(pdf)

  • A. Marica, E. Zuazua, Symmetric Discontinuous Galerkin Methods for 1-D Waves. Fourier Analysis, Propagation, Observability and Applications , Springer Briefs in Mathematics, 2014, ISBN 978-1-4614-5810-4.

CONFERENCES, WORKSHOPS, TALKS 2014:

CONFERENCES ORGANIZED IN 2014:

  • Le 12eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analysis, August 25-30, 2014, University of Lyon, Lyon, France (organizers: Daniel Beltita - Institute of Mathematics of the Romanian Academy; Mihai Mihailescu - Institute of Mathematics of the Romanian Academy & University of Craiova; Emmanuel Russ - Joseph Fourier University, Grenoble).

  • Workshop for Young Researchers in Mathematics, May 22-23, 2014, "Ovidius" University of Constanta, Constanta, Romania (organizers: Adrian Chlinger- University of Missouri; Constantin Costara- "Ovidius" University of Constanta; Cristian Enache- American University of Middle East; Denis Ibadula- "Ovidius" University of Constanta; Liviu Ignat- Institute of Mathematics of the Romanian Academy; Anca-Daniela Macinic- Institute of Mathematics of the Romanian Academy; Sergiu Moroianu- Institute of Mathematics of the Romanian Academy)

SCIENTIFIC REPORT FOR 2014

PUBLICATIONS 2015:

  • M. Bocea and M. Mihailescu, The principal frequency of $\Delta_\infty$ as a limit of Rayleigh quotients involving Luxemburg norms, Bulletin des Sciences Mathematiques, 138 (2014), 236-252. (pdf)

  • M. Bocea, M. Mihailescu and D. Stancu-Dumitru, The limiting behavior of solutions to inhomogeneous eigenvalue problems in Orlicz-Sobolev spaces, Advanced Nonlinear Studies 14 (2014), 977-990. (pdf)

  • C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary , SIAM J. Control Optim., 52 (2014), no. 4, 2055-2089. (pdf)

  • M. Farcaseanu, M. Mihailescu and D. Stancu-Dumitru, On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition, Nonlinear Analysis, 116 (2015), 19-25. (pdf)

  • L. I. Ignat, A. Pozo and E. Zuazua, Large-time asymtotics, vanishing viscosity and numerics for 1-D scalar conservation laws , Mathematics of Computation, 84 (2015), 1633-1662. (pdf)

  • D. Stancu-Dumitru, Multiplicity of solutions for an anisotropic problem in Orlicz-Sobolev spaces , Journal of Nonlinear and Convex Analysis, 16 (2015), 815-834. (pdf)

  • M. Mihailescu, Classification of isolated singularities for nonhomogeneous operators in divergence form, Journal of Functional Analysis, 268 (2015), 2336-2355. (pdf)

  • C. Enache, A Liouville type theorem for p-Laplace equations, Electronic Journal of Differential Equations, 128 (2015), 1-7. (pdf)

  • C. Enache and G. A. Philippin, On some isoperimetric inequalities involving eigenvalues of fixed membranes, Applied Mathematics Letters, 50 (2015), 10-15. (pdf)

  • M. Mihailescu, D. Stancu-Dumitru, and C. Varga, On the spectrum of a Baouendi-Grushin type operator: an Orlicz-Sobolev space setting approach, Nonlinear Differential Equations and Applications (NoDEA), 22 (2015), 1067-1087. (pdf)

  • L. I. Ignat, T. I. Ignat and D. Stancu-Dumitru, A compactness tool for the analysis of nonlocal evolution problems, SIAM Journal on Mathematical Analysis, 47 (2015), 1330-1354. (pdf)

  • C. Nicusor, M. Csirik and C. Varga: Weak solvability via bipotential method for contact models with nonmonotone boundary conditions, Zeitschrift fur angewandte Mathematik

  • und Physik ZAMP, 66 (2015), 2787–2806. (pdf)

CONFERENCES, WORKSHOPS, TALKS 2015:

SCIENTIFIC REPORT FOR 2015

PUBLICATIONS 2016:

  • C. Cazacu, New estimates for the Hardy constants of multipolar Schrodinger operators, Communications in Contemporary Mathematics, 18 (2016), No. 5, 1550093 (28 pages). (pdf)

  • M. Farcaseanu and D. Stancu-Dumitru, Existence of solutions for quasilinear elliptic equations involving a nonlocal term, Electronic Journal of Differential Equations, 293 (2015), 1-8. (pdf)

  • L. Barbu and C. Enache, Maximum principle, Liouvilles type theorems and symmetry results for a general class of quasilinear anisotropic equations, Advances in Nonlinear Analysis, in press. (pdf)

  • M. Bocea and M. Mihailescu, Existence of nonnegative viscosity solutions for a class of problems involving the $\infty$-Laplacian, Nonlinear Differential Equations and Applications (NoDEA), 23 (2016), 1-21. (pdf)

  • M. Mihailescu and G. Morosanu, Eigenvalues of $-\Delta_p -\Delta_q$ under Neumann boundary condition, Canadian Mathematical Bulletin 59 (2016), 606-616. (http://dx.doi.org/10.4153/CMB-2016-025-2) (pdf)

  • M. Mihailescu and D. Stancu-Dumitru, A perturbed eigenvalue problem on general domains, Annals of Functional Analysis 7 (2016), 529-542. (http://dx.doi.org/10.1215/20088752-3660738) (pdf)

  • M. Farcaseanu, An eigenvalue problem involving an anisotropic differential operator, Complex Variables and Elliptic Equations, in press. (pdf)

  • S. Ervedoza, A. Marica, and E. Zuazua, Numerical meshes ensuring uniform observability of 1d waves: construction and analysis, IMA J. Numer. Anal. 36 (2016), 503-542. (pdf)

  • N. Costea, M. Csirik, and C. Varga, Linking-type results in nonsmooth critical point theory and applications, Set-Valued and Variational Analysis, in press. (pdf)

  • M. Farcaseanu, M. Mihailescu, and D. Stancu-Dumitru, A maximum principle for a class of first order differential operators, Proceedings of the 8th Congress of Romanian Mathematicians, 2016. (DOI: 10.1142/9789813142862_0007) (pdf)

  • L. Barbu and C. Enache, A Liouville type theorem for a class of anisotropic equations, Analele Stiintifice ale Universitatii Ovidius - Seria Matematica, in press. (pdf)

  • M. Farcaseanu and M. Mihailescu, Continuity of the first eigenvalue for a family of degenerate eigenvalue problems, Archiv der Mathematik, in press. (DOI: 10.1007/s00013-016-0976-1) (pdf)

CONFERENCES, WORKSHOPS, TALKS 2016:

SCIENTIFIC REPORT FOR 2016