Abstract
In this article we review some of the main contributions of Indian mathematicians in the theoretical analysis of partial differential equations in the last decade.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adimurthi, Nirmalendu Chaudhuri, and Mythily Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130(2) (2002), 489–505.
Adimurthi and G. D. Veerappa Gowda, Conservation law with discontinuous flux, J. Math. Kyoto Univ., 43(1) (2003), 27–70.
Adimurthi and O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality, Comm. Partial Differential Equations, 29(1-2) (2004), 295–322.
Adimurthi and Massimo Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132(4) (2004), 1013–1019.
Adimurthi, Siddhartha Mishra, and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2(4) (2005), 783–837.
Adimurthi, Frdric Robert, and Michael Struwe, Concentration phenomena for Liouville's equation in dimension four, J. Eur. Math. Soc., 8(2) (2006), 171–180.
Adimurthi, Massimo Grossi, and Sanjiban Santra, Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem, J. Funct. Anal., 240(1) (2006), 36–83.
Adimurthi and Anusha Sekar, Role of the fundamental solution in Hardy-Sobolev-type inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 136(6) (2006), 1111–1130.
Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13(5-6) (2007), 585–603.
Adimurthi, Siddhartha Mishra, and G. D. Veerappa Gowda, Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes, Netw. Heterog. Media, 2(1) (2007), 127–157.
Adimurthi, Siddhartha Mishra, and G. D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients, J. Differential Equations, 241(1) (2007), 1–31.
Adimurthi and Kyril Tintarev, Hardy inequalities for weighted Dirac operator, Ann. Mat. Pura Appl. (4), 189(2) (2010), 241–251.
Adimurthi and Kyril Tintarev, On a version of Trudinger-Moser inequality with Mabius shift invariance, Calc. Var. Partial Differential Equations, 39(1–2) (2010), 203–212.
Adimurthi, Rajib Dutta, Shyam Sundar Ghoshal, and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math., 64(1) (2011), 84–115.
Adimurthi, Shyam Sundar Ghoshal, and G. D. Veerappa Gowda, Structure of entropy solutions to scalar conservation laws with strictly convex flux, J. Hyperbolic Differ. Equ., 9(4) (2012), 571–611.
Adimurthi, Shyam Sundar Ghoshal, and G. D. Veerappa Gowda, Exact controllability of scalar conservation laws with strict convex flux, Math. Control Relat. Fields, 4(4) (2014), 401–449.
Adimurthi, Shyam Sundar Ghoshal, and G. D. Veerappa Gowda, Optimal controllability for scalar conservation laws with convex flux, J. Hyperbolic Differ. Equ., 11(3) (2014), 477–491.
Adimurthi, Shyam Sundar Ghoshal, and G. D. Veerappa Gowda, L p stability for entropy solutions of scalar conservation laws with strict convex flux, J. Differential Equations, 256(10) (2014), 3395–3416.
Adimurthi and Cyril Tintarev, On compactness in the Trudinger-Moser inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13(2) (2014), 399–416.
Adimurthi and Cyril Tintarev, Defect of compactness in spaces of bounded variation, J. Funct. Anal., 271(1) (2016), 37–48.
Adimurthi, A. Karthik, and Jacques Giacomoni, Uniqueness of positive solutions of a n-Laplace equation in a ball in n with exponential nonlinearity, J. Differential Equations, 260(11) (2016), 7739–7799.
S. Albeverio and V. M. Shelkovich, On the delta-shock front problem, Analytical approaches to multidimensional balance laws, Nova Science Publ., 2 (2005), 45–88.
Grgoire Allaire and M. Vanninathan, Homogenization of the Schrdinger equation with a time oscillating potential, Discrete Contin. Dyn. Syst. Ser. B, 6(1) (2006), 1–16.
G. Allaire, M. Briane, and M. Vanninathan, A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures, SeMA J., 73(3) (2016), 237–259.
Boris Andreianov, Kenneth Hvistendahl Karlsen, and Nils Henrik Risebro, A theory of L 1-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201(1) (2011), 27–86.
M. H. C. Anisa and A. R. Aithal, On two functionals connected to the Laplacian in a class of doubly connected domains in space-forms, Proc. Indian Acad. Sci. Math. Sci., 115(1) (2005), 93–102.
T. V. Anoop, P. Drbek, and Sarath Sasi, On the structure of the second eigenfunctions of the p-Laplacian on a ball, Proc. Amer. Math. Soc., 144(6) (2016), 2503–2512.
T. V. Anoop, Vladimir Bobkov, and Sarath Sasi, On the strict monotonicity of the first eigenvalue of the p-Laplacian on annuli, Preprint, arXiv:1611.03532
K. R. Arun and P. Prasad, 3-D kinematical conservation laws (KCL): Evolution of a surface in R3-in particular propagation of a nonlinear wavefront, Wave Motion, 46 (2009), 293–311.
K. R. Arun, M. Luk’avcov’a-Medviov’a, P. Prasad, and S. V. Raghurama Rao, An application of 3-D kinematical conservation laws: Propagation of a three dimensional wavefront, SIAM J. Appl. Math., 70 (2010), 2604–2626.
M. F. Atiyah, R. Bott, and L. Garding, Lacunas for hyperbolic operators with constant coefficients, I. Acta Math., 124 (1970), 109–189.
M. F. Atiyah, R. Bott, and L. Garding, Lacunas for hyperbolic operators with constant coefficients, II. Acta Math., 131 (1973), 145–206.
Saugata Bandyopadhyay, Ana Cristina Barroso, Bernard Dacorogna, and Jos Matias, Differential inclusions for differential forms, Calc. Var. Partial Differential Equations, 28(4) (2007), 449–469.
S. Bandyopadhyay and B. Dacorogna, On the pullback equation φ*(g) = f, Ann. Inst. H. Poincar Anal. Non Linaire, 26(5) (2009), 1717–1741.
Saugata Bandyopadhyay, Bernard Dacorogna, and Olivier Kneuss, Some new results on differential inclusions for differential forms, Trans. Amer. Math. Soc., 367(5) (2015), 3119–3138.
Saugata Bandyopadhyay, Bernard Dacorogna, and Swarnendu Sil, Calculus of variations with differential forms, J. Eur. Math. Soc., 17(4) (2015), 1009–1039.
C. W. Bardos, A. -Y. Leroux, and J. -C. d Nedelec, First order quasilinear equations with boundary conditions, Comm. Part. Diff. Equa., 4 (1979), 1018–1034.
M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz’ya type equations in bounded domains, J. Differential Equations, 247(1) (2009), 119–139.
Mousomi Bhakta and Moshe Marcus, Semilinear elliptic equations admitting similarity transformations, J. Funct. Anal., 267(10) (2014), 3894–3930.
Divyang G. Bhimani and Ratnakumar, Functions operating on modulation spaces and nonlinear dispersive equations, J. Funct. Anal., 270(2) (2016), 621–648.
Imran H. Biswas and Ananta K. Majee, Stochastic conservation laws: Weak-in-time formulation and strong entropy condition, J. Funct. Anal., 267(7) (2014), 2199–2252.
Imran H. Biswas, Ujjwal Koley, and Ananta K. Majee, Continuous dependence estimate for conservation laws with Lv́y noise, J. Differential Equations, 259(9) (2015), 4683–4706.
B. Bougherara, J. Giacomoni, and S. Prashanth, Analytic global bifurcation and infinite turning points for very singular problems, Calc. Var. Partial Differential Equations, 52(3–4) (2015), 829–856.
Haim Brezis and Juan Luis Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Comput. Madrid, 10(2) (1997), 443–469.
D. Castorina, I. Fabbri, G. Mancini, and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246(3) (2009), 1187–1206.
Anisa M. H. Chorwadwala and M. K. Vemuri, Two functionals connected to the Laplacian in a class of doubly connected domains on rank one symmetric spaces of non-compact type, Geom. Dedicata, 167 (2013), 11–21.
Anisa M. H. Chorwadwala and Rajesh Mahadevan, An eigenvalue optimization problem for the p-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 145(6) (2015), 1145–1151.
Anupam Pal Choudhury, K. T. Joseph, and Manas R. Sahoo, Spherically symmetric solutions of multidimensional zero-pressure gas dynamics system, J. Hyperbolic Differ. Equ., 11(2) (2014), 269–293.
Anupam Pal Choudhury and P. Krishnan Venkateswaran, Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials, J. Math. Anal. Appl., 431(1) (2015), 300–316.
Anupam Pal Choudhury, K. T. Joseph, and Philippe G. LeFloch, The mathematical theory of self-similar boundary layers for nonlinear hyperbolic systems with viscosity and capillarity, Bull. Inst. Math. Acad. Sin. (N.S.), 10(4) (2015), 639–693.
Shirshendu Chowdhury and Mythily Ramaswamy, Optimal control of linearized compressible Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 19(2) (2013), 587–615.
Shirshendu Chowdhury, Debayan Maity, Mythily Ramaswamy, and Jean-Pierre Raymond, Local stabilization of the compressible Navier-Stokes system, around null velocity, in one dimension, J. Differential Equations, 259(1) (2015), 371–407.
S. Chowdhury, M. Ramaswamy, and J. -P. Raymond, Controllability and stabilizability of the linearized compressible Navier Stokes system in one dimension, SIAM J. Control Optim., 50(5) (2012), 2959–2987.
C. Conca, S. Natesan, and M. Vanninathan, Numerical experiments with the Bloch-Floquet approach in homogenization, Internat. J. Numer. Methods Engrg., 65(9) (2006), 1444–1471.
Carlos Conca, Rafael Orive, and Muthusamy Vanninathan, On Burnett coefficients in periodic media, J. Math. Phys., 47(3) (2006), 11 pp.
Gyula Csató, An isoperimetric problem with density and the Hardy Sobolev inequality in ℝ2, Differential Integral Equations, 28(9–10) (2015), 971–988.
Gyula Csató and Prosenjit Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54(2) (2015), 2341–2366.
Gyula Csató and Prosenjit Roy, Singular Moser-Trudinger inequality on simply connected domains, Comm. Partial Differential Equations, 41(5) (2016), 838–847.
C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren Math. Wissenschaften Series, 325, Springer Verlag, 2016.
Umberto De Maio, Akamabadath K. Nandakumaran, and Carmen Perugia, Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition, Evol. Equ. Control Theory, 4(3) (2015), 325–346.
R. Dutta, U. Koley, and N. H. Risebro, Convergence of a higher-order scheme for Korteweg-de Vries equation, SIAM Journal on Numerical Analysis, 53(4) (2015), 1963–1983.
R. Dutta, H. Holden, U. Koley, and N. H. Risebro, Operator splitting for the Benjamin-Ono equation, Journal of Differential Equations, 259 (2015), 6694–6717.
R. Dutta, H. Holden, U. Koley, and N. H. Risebro, Convergence of finite difference schemes for the Benjamin-Ono equation, Numerische Mathematik., 134(2) (2016), 249–274.
S. Ervedoza and M. Vanninathan, Controllability of a simplified model of fluid-structure interaction, ESAIM Control Optim. Calc. Var., 20(2) (2014), 547–575.
P. Esposito, G. Mancini, Sanjiban Santra, and P. N. Srikanth, Asymptotic behavior of radial solutions for a semilinear elliptic problem on an annulus through Morse index, J. Differential Equations, 239(1) (2007), 1–15.
L. Euler, Principes generaux du moument des Fluides, Mèmoires de l’Academie des Sciences de Berlin, 11 (1757), 274–315.
J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Funct. Anal., 255(2) (2008), 313–373.
Ambartsoumian Gaik, Felea Raluca, and P. Krishnan Venkateswaran, Clifford Nolan, Eric Todd Quinto. A class of singular Fourier integral operators in synthetic aperture radar imaging, J. Funct. Anal., 264(1) (2013), 246–269.
Ambartsoumian Gaik and P. Krishnan Venkateswaran, Inversion of a class of circular and elliptical radon transforms, In Complex Analysis and Dynamical Systems VI: Part 1: PDE, Differential Geometry, Radon Transform, Contemp. Math., 653, 1–12, Amer. Math. Soc., Providence, RI, 2015.
Debdip Ganguly and Kunnath Sandeep, Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space, Commun. Contemp. Math., 17(1) (2015), 1450019, 13 pp.
L. Gawarecki, V. Mandrekar, and B. Rajeev, Linear stochastic differential equations in the dual of a multi-Hilbertian space, Theory Stoch. Process, 14(2) (2008), 28–34.
L. Gawarecki, V. Mandrekar, and B. Rajeev, The monotonicity inequality for linear stochastic partial differential equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12(4) (2009), 575–591.
J. Giacomoni, S. Prashanth, and K. A. Sreenadh, Global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type, J. Differential Equations, 232(2) (2007), 544–572.
M. Gisclon and D. Serre, Etude des conditions aux limites pour un systeme strictement hyperbolique vis l'approximation parabolique, C. R. Acad Sc. Paris, Serie 1, 319 (1994), 377–382.
M. Gisclon, Etude des conditions aux limites pour un systeme strictement hyperbolique vis l’approximation parabolique, J.Math.Pures Appl., 75 (1996), 485–508.
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697–715.
Sarika Goyal and K. Sreenadh, On the Fucik spectrum of non-local elliptic operators, NoDEA Nonlinear Differential Equations Appl., 21(4) (2014), 567–588.
Massimo Grossi and S. Prashanth, Local solutions for elliptic problems with exponential nonlinearities via finite dimensional reduction, Indiana Univ. Math. J., 54(2) (2005), 383–415.
S. N. Gurbatov, A. I. Saichev, and S. F. Shandarin, Large-scale structure of the universe. The Zeldovich approximation and adhesion model, Physics-Uspekhi, 55 (2012), 223–249.
H. Holden, U. Koley, and N. H. Risebro, Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation, IMA Journal of Numerical Analysis., 35(3) (2015), 1047–1077.
E. Hopf, The partial differential equation u t + uu x = μu xx, Comm. Pure Appl. Math., 3 (1950), 201–230.
K. T. Joseph, A Riemann problem whose viscosity solutions contain δ-measures, Asymptotic Anal., 7 (1993), 105–120.
K. T. Joseph and G. D. Veerappa Gowda, Explicit formula for the solution of convex conservation laws with boundary condition, Duke Math. J., 62 (1991), 401–416.
K. T. Joseph and P. G. LeFloch, Boundary layers in weak solutions to hyperbolic conservation laws, Preprint #1402, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, May 1996.
K. T. Joseph and P. G. LeFloch, Boundary layers in weak solutions to hyperbolic conservation laws, Arch. Rational Mech Anal., 147 (1999), 47–88.
K. T. Joseph and P. G. LeFloch, Boundary layers in weak solutions of hyperbolic conservation laws. II. Self-similar vanishing diffusion limits, Comm. Pure Appl. Anal., 1 (2002), 51–76.
K. T. Joseph and P. G. LeFloch, Singular limits for the Riemann problem. General diffusion, relaxation, and boundary conditions, In New analytical approach to multidimensional balance laws, O. Rozanova ed., Nova Press, (2006), 143–172.
K. T. Joseph and P. G. LeFloch, Singular limits in phase dynamics with physical viscosity and capillarity, Proc. Royal Soc. Edinburgh, 137A (2007), 1287–1312.
K. T. Joseph and Manas R. Sahoo, Vanishing viscosity approach to a system of conservation laws admitting δ waves, Commun. Pure Appl. Anal., 12(5) (2013), 2091–2118.
Debabrata Karmakar and Kunnath Sandeep, Adams inequality on the hyperbolic space, J. Funct. Anal., 270(5) (2016), 1792–1817.
S. Kesavan, On two functionals connected to the Laplacian in a class of doubly connected domains, Proc. Roy. Soc. Edinburgh Sect. A, 133(3) (2003), 617–624.
Srinivasan Kesavan, On Poincarè’s and J. L. Lions’ lemmas, C. R. Math. Acad. Sci. Paris, 340(1) (2005), 27–30.
S. Kesavan and T. Muthukumar, Homogenization of an optimal control problem with state-constraints, Differ. Equ. Dyn. Syst., 19(4) (2011), 361–374.
P. D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10 (1957), 537–566.
Chang-Shou Lin and Jyotshana V. Prajapat, Harnack type inequality and a priori estimates for solutions of a class of semilinear elliptic equations, J. Differential Equations, 244(3) (2008), 649–695.
Chang-Shou Lin and Jyotshana V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288(1) (2009), 311–347.
T. P. Liu, Admissible solutions of hyperbolic conservation laws, Mem. Amer. Math. Soc., 30 (1981).
Marcello Lucia and S. Prashanth, Simplicity of principal eigenvalue for p-Laplace operator with singular indefinite weight, Arch. Math. (Basel), 86(1) (2006), 79–89.
R. Magnanini, J. Prajapat, and S. Sakaguchi, Stationary isothermic surfaces and uniformly dense domains, Trans. Amer. Math. Soc., 358(11) (2006), 4821–4841.
Rajesh Mahadevan and T. Muthukumar, Homogenization of some cheap control problems, SIAM J. Math. Anal., 43(5) (2011), 2211–2229.
G. Mancini, I. Fabbri, and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations, 224(2) (2006), 258–276.
Gianni Mancini and Kunnath Sandeep, On a semilinear elliptic equation in ℍn, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7(4) (2008), 635–671.
G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12(6) (2010), 1055–1068.
Utpal Manna and Manil T. Mohan, Shell model of turbulence perturbed by Lvy noise, NoDEA Nonlinear Differential Equations Appl., 18(6) (2011), 615–648.
Utpal Manna, Manil T. Mohan, and Sivaguru S. Sritharan, Stochastic Navier-Stokes equations in unbounded channel domains, J. Math. Fluid Mech., 17(1) (2015), 47–86.
B. B. Manna and P. N. Srikanth, On the solutions of a singular elliptic equation concentrating on two orthogonal spheres, NoDEA Nonlinear Differential Equations Appl., 21(6) (2014), 915–927.
Pawan K. Mishra and Konijeti Sreenadh, Existence and multiplicity results for fractional p-Kirchhoff equation with sign changing nonlinearities, Adv. Pure Appl. Math., 7(2) (2016), 97–114.
Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond, Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension, Math. Control Relat. Fields, 5(2) (2015), 259–290.
Roberta Musina and K. Sreenadh, Radially symmetric solutions to the Hnon-Lane-Emden system on the critical hyperbola, Commun. Contemp. Math., 16(3) (2014), 1350030, 16 pp.
T. Muthukumar and A. K. Nandakumaran, Homogenization of low-cost control problems on perforated domains, J. Math. Anal. Appl., 351(1) (2009), 29–42.
A. K. Nandakumaran, Ravi Prakash, and Bidhan Chandra Sardar, Periodic controls in an oscillating domain: controls via unfolding and homogenization, SIAM J. Control Optim., 53(5) (2015), 3245–3269.
A. K. Nandakumaran and M. Rajesh, Homogenization of a nonlinear degenerate parabolic differential equation, Electron. J. Differential Equations, 17 (2001), 19 pp.
Filomena Pacella and P. N. Srikanth, A reduction method for semilinear elliptic equations and solutions concentrating on spheres, J. Funct. Anal., 266(11) (2014), 6456–6472.
Manoj Pandey and V. D. Sharma, Kinematics of a shock wave of arbitrary strength in a non-ideal gas, Quart. Appl. Math., 67(3) (2009), 401–418.
Manoj Pandey, R. Radha, and V. D. Sharma, Symmetry analysis and exact solutions of magnetogasdynamic equations, Quart. J. Mech. Appl. Math., 61(3) (2008), 291–310.
S. Prashanth, Exact multiplicity result for the perturbed scalar curvature problem in ℝN(N ≥ 3), Proc. Amer. Math. Soc., 135(1) (2007), 201–209.
S. Prashanth, Sanjiban Santra, and Abhishek Sarkar, On the perturbed Q-curvature problem on S 4, J. Differential Equations, 255(8) (2013), 2363–2391.
R. Radha and V. D. Sharma, Interaction of a weak discontinuity with elementary waves of Riemann problem, J. Math. Phys., 53(1) (2012), 013506.
T. Raja Sekhar and V. D. Sharma, Wave interactions for the pressure gradient equations, Methods Appl. Anal., 17(2) (2010), 165–178.
T. Raja Sekhar and V. D. Sharma, Riemann problem and elementary wave interactions in isentropic magnetogasdynamics, Nonlinear Anal. Real World Appl., 11(2) (2010), 619–636.
B. Rajeev and K. Suresh Kumar, A class of stochastic differential equations with pathwise unique solutions, Indian J. Pure Appl. Math., 47(2) (2016), 343–355.
Mythily Ramaswamy and Sanjiban Santra, Uniqueness and profile of positive solutions of a critical exponent problem with Hardy potential, J. Differential Equations, 254(11) (2013), 4347–4372.
P. K. Ratnakumar and Vijay Kumar Sohani, Nonlinear Schrodinger equation for the twisted Laplacian, J. Funct. Anal., 265(1) (2013), 1–27.
J. -P. Raymond and M. Vanninathan, Null controllability in a fluid-solid structure model, J. Differential Equations, 248(7) (2010), 1826–1865.
B. Riemann, Ueber die Fortpflanzung ebener LUftwellen von endlicher Schwingungweite, Gött. Abh. Math. Cl., 8 (1860), 43–65.
Bernhard Ruf and P. N. Srikanth, Singularly perturbed elliptic equations with solutions concentrating on a 1-dimensional orbit, J. Eur. Math. Soc., 12(2) (2010), 413–427.
Bernhard Ruf and P. N. Srikanth, Concentration on Hopf-fibres for singularly perturbed elliptic equations, J. Funct. Anal., 267(7) (2014), 2353–2370.
V. D. Sharma and R. Radha, Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis, Z. Angew. Math. Phys., 59(6) (2008), 1029–1038.
Sista Sivaji Ganesh and Muthusamy Vanninathan, Bloch wave homogenization of linear elasticity system, ESAIM Control Optim. Calc. Var., 11(4) (2005), 542–573.
K. Sreenadh and Sweta Tiwari, Global multiplicity results for p(x)-Laplacian equation with nonlinear Neumann boundary condition, Differential Integral Equations, 26(7–8) (2013), 815–836.
K. Sreenadh and Sweta Tiwari, On W 1;p(x) versus C 1 local minimizers of functionals related to p(x)-Laplacian, Appl. Anal., 92(6) (2013), 1271–1282.
Tuhin Ghosh and P. Krishnan Venkateswaran, Determination of lower order perturbations of the polyharmonic operator from partial boundary data, Appl. Anal., 95(11) (2016), 2444–2463.
P. Krishnan Venkateswaran, A support theorem for the geodesic ray transform on functions, J. Fourier Anal. Appl., 15(4) (2009), 515–520.
Acknowledgement
The authors are grateful to Prof. A. K. Nandakumaran, Dr. K. R. Arun, Dr. Venkateswaran P. Krishnan and Dr. U. Koley, for their help in writing up topics in homogenization, kinematic theory, inverse problems and dispersive equations respectively. We also thank the referee for the constructive criticism which improved the presentation of this paper.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Joseph, K.T., Sandeep, K. Theoretical developments in the study of partial differential equations. Indian J Pure Appl Math 50, 681–704 (2019). https://doi.org/10.1007/s13226-019-0349-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-019-0349-0