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
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:
University of Sydney PDE Seminar, September 26, 2013, School of Mathematics and Statistics from the University of Sydney, Sydney, Australia (speaker: Mihai Mihailescu; title of the talk: The asymptotic behavior of some power-law functionals in Sobolev spaces with variable exponents).
The 57th Annual Meeting of the Australian Mathematical Society, September 30 - October 03, 2013, University of Sydney, Sydney, Australia (speaker: Mihai Mihailescu; title of the talk: Eigenvalue problems in Orlicz-Sobolev spaces for rapidly growing operators in divergence form).
Kyushu-Euskadi 2013 Workshop on Applied Mathematics, November 12, 2013, Fukuoka University, Fukuoka, Japan (speaker: Liviu Ignat; title of the talk: Long time behaviour of some nonlocal evolution problems ).
CINTI 2013, 14th IEEE International Symposium on Computational Intelligence and Informatics; Workshop: Analytical and Geometrical Methods for Solving Engineering Problem, November 19-21, 2013, Obuda University, Budapest, Hungary (speaker: Nicusor Costea; title of the talk: Nonlinear Hemivariational Inequalities and Applications).
Seminar on partial differential equations of the Institute of Mathematics ofthe Academy of Sciences of the Czech Republic, October 1, 2013, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Prague, Czech Republic (speaker: Cristian Cazacu; title of the talk: The influence of the Hardy-type inequalities to some singular PDEs ).
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:
Seminar on Nonlinear Operatorsand Differential Equations, March 13, 2014, "Babes-Bolyai" University, Cluj-Napoca, Romania (speaker: Mihai Mihailescu; title of the talk: An eigenvalue problem involving a nonhomogeneous operator in divergence form).
Seminar on Nonlinear Operatorsand Differential Equations, March 13, 2014, "Babes-Bolyai" University, Cluj-Napoca, Romania (speaker: Nicusor Costea; title of the talk: Nonlinear hemivariational inequalities: formulation, existence of solutions and applications).
Workshop for Young Researchers in Mathematics, May 22-23, 2014, “Ovidius” University of Constanta, Constanta, Romania (speaker: Mihai Mihailescu; title of the talk: Eigenvalue problems in Orlicz-Sobolev spaces for rapidly growing operators in divergence form).
Workshop for Young Researchers in Mathematics, May 22-23, 2014, “Ovidius” University of Constanta, Constanta, Romania (speaker: Liviu Ignat; title of the talk:Long-time behavior for nonlocal problems).
Workshop for Young Researchers in Mathematics, May 22-23, 2014, “Ovidius” University of Constanta, Constanta, Romania (speaker: Cristian Cazacu; title of the talk: New estimates of the best constant of a Hardy inequality with multi-singular potential).
Workshop for Young Researchers in Mathematics, May 22-23, 2014, “Ovidius” University of Constanta, Constanta, Romania (speaker: Costea Nicusor; title of the talk: On the existence of solutions for inequality problems of hemivariational type).
Workshop for Young Researchers in Mathematics, May 22-23, 2014, “Ovidius” University of Constanta, Constanta, Romania (speaker: Denisa Stancu-Dumitru; title of the talk: The asymptotic behavior of solutions to non-homogeneous eigenvalue problems in Orlicz-Sobolev spaces).
Le 12eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analysis, August 25-30, 2014, University of Lyon, Lyon, France (speaker: Denisa Stancu-Dumitru; title of the talk: The limiting behavior of solutions to inhomogeneous eigenvalues problems in Orlicz-Sobolev spaces).
Le 12eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analysis, August 25-30, 2014, University of Lyon, Lyon, France (speaker: Liviu Ignat; title of the talk: Long-time behavior for nonlocal problems).
Le 12eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analysis, August 25-30, 2014, University of Lyon, Lyon, France (speaker: Cristian Enache; title of the talk: Maximum principles and isoperimetric estimates for a class of Monge-Ampere
equations, with applications to surfaces of constant Gauss curvature).
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)
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:
Workshop for Young Researchers in Mathematics, May 21-24, 2015, “Ovidius” University of Constanta, Constanta, Romania (speaker: Denisa Stancu-Dumitru; title of the talk: On the spectrum of Baouendi-Grushin type operator in the setting of Orlicz-Sobolev spaces).
Workshop for Young Researchers in Mathematics, May 21-24, 2015, “Ovidius” University of Constanta, Constanta, Romania (speaker: Nicusor Costea; title of the talk: On the weak solvability for some contact models with nonmonotone boundary conditions).
Workshop for Young Researchers in Mathematics, May 21-24, 2015, “Ovidius” University of Constanta, Constanta, Romania (speaker: Aurora Marica; title of the talk: Wave propagation and control on two-dimensional meshes).
International Conference on Nonlinear Operators, Differential Equations and Applications (ICNODEA), July 14-17, 2015, "Babes Bolyai" University, Cluj-Napoca, Romania (speaker: Denisa Stancu-Dumitru; title of the talk: The asymptotic behavior of solution to nonhomogeneous eigenvalue problems in Orlicz-Sobolev spaces)
The Eighth Congress of Romanian Mathematicians - Session: Ordinary and Partial Differential Equations, Variational Methods, Optimal Control, June 26 - July 1, 2015, "Alexandru Ioan Cuza" University, Iasi, Romania (speaker: Denisa Stancu-Dumitru; title of the talk: A Baouendi-Grushin type operator in Orlicz-Sobolev spaces and applications to PDEs)
The Eighth Congress of Romanian Mathematicians - Session: Ordinary and Partial Differential Equations, Variational Methods, Optimal Control, June 26 - July 1, 2015, "Alexandru Ioan Cuza" University, Iasi, Romania (speaker: Liviu Ignat; title of the talk: Dispersion property for Schrodinger equations)
The Eighth Congress of Romanian Mathematicians - Session: Ordinary and Partial Differential Equations, Variational Methods, Optimal Control, June 26 - July 1, 2015, "Alexandru Ioan Cuza" University, Iasi, Romania (speaker: Mihai Mihailescu; title of the talk: On the asymptotic behavior of some classes of nonlinear eigenvalue problems involving the $p$-Laplacian)
The Eighth Congress of Romanian Mathematicians - Session: Real and Complex Analysis, Potential Theory, June 26 - July 1, 2015, "Alexandru Ioan Cuza" University, Iasi, Romania (speaker: Cristian Cazacu; title of the talk: Optimal Hardy constants for Schrodinger operators with multi-singular inverse-square potentials)
The Eighth Congress of Romanian Mathematicians - Session: Ordinary and Partial Differential Equations, Variational Methods, Optimal Control, June 26 - July 1, 2015, "Alexandru Ioan Cuza" University, Iasi, Romania (speaker: Aurora Marica; title of the talk: Numerical meshes ensuring uniform observability of 1d waves)
AMS Central Fall Sectional Meeting - Special Session on Nonlinear PDEs and Calculus of Variations, October 2 - 4, 2015, Loyola University Chicago, Chicago, IL, USA (speaker: Mihai Mihailescu; title of the talk: Classification of isolated singularities for inhomogeneous operators in divergence form)
AMS Central Fall Sectional Meeting - Special Session on Nonlinear PDEs and Calculus of Variations, October 2 - 4, 2015, Loyola University Chicago, Chicago, IL, USA (speaker: Cristian Enache; title of the talk: Maximum principles and isoperimetric estimates for a class of Monge-Ampere equations, with applications to surfaces of constant Gauss curvature)
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:
Geometry and Analysis Seminar, March 4, 2016, Tohoku University, Sendai, Japan (speaker: Cristian Enache; title of the talk: On some recent maximum principles for P-functions and their applications)
Analysis Seminar, April 8, 2016, University of Craiova, Craiova, Romania, (speaker: Cristian Enache; title of the talk: Maximum and minimum principles for Monge-Ampere equations, with applications to surfaces of constant Gauss curvature)
Workshop for Young Researchers in Mathematics, May 19-22, 2016, “Ovidius” University of Constanta, Constanta, Romania (speaker: Mihai Mihailescu; title of the talk: Classification of isolated singularities for inhomogeneous operators in divergence form)
Workshop for Young Researchers in Mathematics, May 19-22, 2016, “Ovidius” University of Constanta, Constanta, Romania (speaker: Maria Farcaseanu; title of the talk: On the convergence of the sequence of solutions for a family of eigenvalue problems)
Workshop for Young Researchers in Mathematics, May 19-22, 2016, “Ovidius” University of Constanta, Constanta, Romania (speaker: Tatiana Ignat; title of the talk: Long-time behaviour for nonlocal problems)
Workshop for Young Researchers in Mathematics, May 19-22, 2016, “Ovidius” University of Constanta, Constanta, Romania (speaker: Nicusor Costea; title of the talk: Bounded saddle-point methods for locally Lipschitz functionals)
Workshop for Young Researchers in Mathematics, May 19-22, 2016, “Ovidius” University of Constanta, Constanta, Romania (speaker: Aurora Marica; title of the talk: Wave propagation on irregular grids)
Workshop for Young Researchers in Mathematics, May 19-22, 2016, “Ovidius” University of Constanta, Constanta, Romania (speaker: Cristian Cazacu; title of the talk: Decay rates for the heat evolution of the magnetic Laplacian with Hardy potential)
The 6th Workshop Series on Mathematics, June 3-4, 2016, University of Pitesti, Pitesti, Romania (speaker: Mihai Mihailescu; title of the talk: On the asymptotic behavior of some classes of nonlinear eigenvalue problems involving the $p$-Laplacian)
Le 13eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analyse et Controle des EDP, August 25-29, 2016, Universitatea A. I. Cuza din Iasi, Iasi, Romania (speaker: Denisa Stancu-Dumitru; title of the talk: A perturbed eigenvalue problem on general domains).
Le 13eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analyse et Controle des EDP, August 25-29, 2016, Universitatea A. I. Cuza din Iasi, Iasi, Romania (speaker: Mihai Mihailescu; title of the talk: Classification of isolated singularities for inhomogeneous operators in divergence form).
Le 13eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analyse et Controle des EDP, August 25-29, 2016, Universitatea A. I. Cuza din Iasi, Iasi, Romania (speaker: Maria Farcaseanu; title of the talk: On the convergence of the sequence of solutions for a family of eigenvalue problems).
Le 13eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analyse et Controle des EDP, August 25-29, 2016, Universitatea A. I. Cuza din Iasi, Iasi, Romania (speaker: Cristian Cazacu; title of the talk: Controllability results for a Kuramoto-Sivashinsky model on trees).
Le 13eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analyse et Controle des EDP, August 25-29, 2016, Universitatea A. I. Cuza din Iasi, Iasi, Romania (speaker: Aurora Marica; title of the talk: Wave propagation on irregular grids).
Le 13eme Colloque Franco-Roumain en Mathematiques Appliquees - Special Session on Analyse Non-lisse et Optimisation, August 25-29, 2016, Universitatea A. I. Cuza din Iasi, Iasi, Romania (speaker: Nicusor Costea; title of the talk: Bounded saddle-point methods for locally Lipschitz functionals).