Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

Let B1 be a ball in ℝN centred at the origin and let B0 be a smaller ball compactly contained in B1. For p ∈ (1, ∞), using the shape derivative method, we show that the first eigenvalue of the p-Laplacian in annulus B1 \B0 strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 and p →∞ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p-Laplacian on bounded radial domains. ©2018 American Mathematical Society.

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Department of Mathematics

Transactions of the American Mathematical Society

On the strict monotonicity of the first eigenvalue of the p-Laplacian on annuli

In this paper, we prove that the second eigenfunctions of the p- Laplacian, p &gt; 1, are not radial on the unit ball in RN, for any N ≥ 2. Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs {τn,Ψn} such that Ψn is nonradial and has exactly 2n nodal domains. A few related open problems are also stated. © 2015 American Mathematical Society.

Fulltext

Proceedings of the American Mathematical Society

On the structure of the second eigenfunctions of the p-laplacian on a ball

We consider the antimaximum principle for the p-Laplacian in the exterior domain: {-Δpu=λK(x)|u|p-2u+h(x)in B1c,u=0on ∂B1, where Δp is the p-Laplace operator with p&gt;1,λ is the spectral parameter and B1c is the exterior of the closed unit ball in RN with N ≥ 1. The function h is assumed to be nonnegative and nonzero, however the weight function K is allowed to change its sign. For K in a certain weighted Lebesgue space, we prove that the antimaximum principle holds locally. A global antimaximum principle is obtained for h with compact support. For a compactly supported K, with N=1 and p=2, we provide a necessary and sufficient condition on h for the global antimaximum principle. In the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems. © 2015 Elsevier Ltd.}, author_keywords={Exterior domains; Local and global antimaximum; p-Laplacian; Positive eigenfunctions; principle; Regularity results}, keywords={Boundary value problems; Eigenvalues and eigenfunctions; Laplace transforms, Exterior domain; Local and global antimaximum; P-Laplacian; principle; Regularity results, Nonlinear equations}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 13-00863S}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={The second author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S.}, funding_text_2={The first, third and fourth authors work is funded by the project EXLIZ-CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.}, references={Allegretto, W., Huang, Y.X., A Picone's identity for the p-Laplacian and applications (1998) Nonlinear Anal., 32 (7), pp. 819-830; Anoop, T.V., Drábek, P., Sasi, S., Weighted quasilinear eigenvalue problems in exterior domains (2015) Calc. Var. Partial Differential Equations, 53 (3-4), pp. 961-975; Birindelli, I., Hopf's lemma and anti-maximum principle in general domains (1995) J. Differential Equations, 119 (2), pp. 450-472; Clément, P., Peletier, L.A., An anti-maximum principle for second-order elliptic operators (1979) J. Differential Equations, 34 (2), pp. 218-229; Drábek, P., Huang, Y.X., Bifurcation problems for the p-Laplacian in RN (1997) Trans. Amer. Math. Soc., 349 (1), pp. 171-188; Fleckinger-Pellé, J., Gossez, J.-P., De Thélin, F., Antimaximum principle in RN: Local versus global (2004) J. Differential Equations, 196 (1), pp. 119-133; Fučík, S., Nečas, J., Souček, J., Souček, V., (1973) Spectral Analysis of Nonlinear Operators, 346. , Lecture Notes in Mathematics Springer-Verlag Berlin, New York; Godoy, T., Gossez, J.-P., Paczka, S., On the antimaximum principle for the p-Laplacian with indefinite weight (2002) Nonlinear Anal., 51 (3), pp. 449-467; Hess, P., An anti-maximum principle for linear elliptic equations with an indefinite weight function (1981) J. Differential Equations, 41 (3), pp. 369-374; Hörmander, L., Lions, J.L., Sur la complétion par rapport à une intégrale de Dirichlet (1956) Math. Scand., 4, pp. 259-270; Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal., 12 (11), pp. 1203-1219; Serrin, J., Local behavior of solutions of quasi-linear equations (1964) Acta Math., 111, pp. 247-302; Shi, J., A new proof of anti-maximum principle via a bifurcation approach (2005) Results Math., 48 (1-2), pp. 162-167; Stavrakakis, N.M., De Thélin, F., Principal eigenvalues and anti-maximum principle for some quasilinear elliptic equations on RN (2000) Math. Nachr., 212, pp. 155-171; Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations (1984) J. Differential Equations, 51 (1), pp. 126-150; Vázquez, J.L., A strong maximum principle for some quasilinear elliptic equations (1984) Appl. Math. Optim., 12 (3), pp. 191-202; Ziemer, W.P., Sobolev spaces and functions of bounded variation (1989) Weakly Differentiable Functions, 120. , Graduate Texts in Mathematics Springer-Verlag New York}, correspondence_address1={Drábek, P.; Department of Mathematics and NTIS, Univerzitní 22, Czech Republic}, publisher={Elsevier Ltd}, issn={0362546X}, coden={NOAND}, language={English}, abbrev_source_title={Nonlinear Anal Theory Methods Appl}, document_type={Article}, source={Scopus},

Nonlinear Analysis, Theory, Methods and Applications

Antimaximum principle in exterior domains

We consider the following weighted eigenvalue problem in the exterior domain: (Formula presented.),where Δp is the p-Laplace operator with p&gt;1, and (Formula presented.) is the exterior of the closed unit ball in RN with N≥1. There is no restriction on the dimension N in terms of $$p, i.e., we allow both 1&lt;p&lt;N and p≥N. The weight function K is locally integrable on (Formula presented.) and is allowed to change its sign. For some appropriate choice of $$w$$w, a positive weight function on the interval (1,∞), we prove that the Beppo-Levi space (Formula presented.) is compactly embedded into the weighted Lebesgue space Lp(Formula presented.). The existence of the positive eigenvalue for the above problem is proved for K such that (Formula presented.) is of non-zero measure and |K|≤w. Further, we discuss the positivity, the regularity and the asymptotic behaviour at infinity of the first eigenfunctions. © 2014, Springer-Verlag Berlin Heidelberg.

Calculus of Variations and Partial Differential Equations

Weighted quasilinear eigenvalue problems in exterior domains

We study boundary value problems of the form {-δu=λK({pipe}x{pipe})f(u),x∈Ωu=0if{pipe}x{pipe}=r0u→0as{pipe}x{pipe}→∞, where λ is a positive parameter, δu=div(∇;u) is the Laplacian of u, Ω={x∈Rn;n&gt;2,{pipe}x{pipe}&gt;r0}, K belongs to a class of C1 functions such that limr→∞K(r)=0, and f belongs to a class of C1 functions which are negative at the origin and have falling zeros. We discuss the existence and uniqueness of nonnegative radial solutions when λ is large. © 2012 Elsevier Ltd.}, author_keywords={Existence; Exterior domains; Falling zeros; Semipositone problems; Uniqueness}, references={Ali, I., Castro, A., Shivaji, R., Uniqueness and stability of positive solutions for semipositone problems in a ball (1993) Proc. Amer. Math. Soc., 117, pp. 775-782; Ambrosetti, A., Arcoya, D., Biffoni, B., Positive solutions for some semipositone problems via bifurcation theory (1994) Differential Integral Equations, 7 (3), pp. 655-663; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone boundary value problems (1996) Proc. Amer. Math. Soc., 124 (3), pp. 757-763; Castro, A., Gadam, S., Shivaji, R., Positive solution curves of semipositone problems with concave nonlinearities (1997) Proc. Roy. Soc. Edinburgh Sect. A, 127 (5), pp. 921-934; Castro, A., Garner, J.B., Shivaji, R., Existence results for classes of sublinear semipositone problems (1993) Results Math., 23, pp. 214-220; Castro, A., Hassanpour, M., Shivaji, R., Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity (1995) Comm. Partial Differential Equations, 20 (11-12), pp. 1927-1936; Castro, A., Sankar, L., Shivaji, R., Uniqueness of nonnegative solutions for semipositone problems on exterior domains (2012) J. Math. Anal. Appl., 394 (1), pp. 432-437; Castro, A., Shivaji, R., Nonnegative solutions for a class of nonpositone problems (1998) Proc. Roy. Soc. Edinburgh Sect. A, 108, pp. 291-302; Castro, A., Shivaji, R., Nonnegative solutions for a class of radially symmetric nonpositone problems (1989) Proc. Amer. Math. Soc., 106 (3), pp. 735-740; Castro, A., Shivaji, R., Positive solutions for a concave semipositone Dirichlet problem (1998) Nonlinear Anal., 31 (1-2), pp. 91-98; Dancer, E.N., Shi, J., Uniqueness and nonexistence of positive solutions to semipositone problems (2006) Bull. London Math. Soc., 38 (6), pp. 1033-1044; Habets, P., Zanolin, F., Upper an lower solutions for a generalized Emden-Fowler equation (1994) J. Math. Anal. Appl., 181, pp. 684-700; Lee, E., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differential Integral Equations, 24, pp. 861-875; Philippe Clement, I., Sweers, G., Existence and multiplicity results for a semilinear elliptic eigenvalue problem (1987) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1), pp. 97-121; Serrin, J., (1971), Nonlinear equations of second order, in: A.M.S. Sympos. Partial Differential Equations, Berkeley, August}, correspondence_address1={Shivaji, R.; Department of Mathematics and Statistics, , NC 27412, United States; email: shivaji@uncg.edu}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},

Journal of Mathematical Analysis and Applications

Semipositone problems with falling zeros on exterior domains

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: {-u″=λ[u-u2/K-cu2/1+ u2-∈], x ∈ (0,1) u(0)=0=u(1). Here 1/λ is the diffusion coefficient and K, c and are positive constants. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. In this paper, we discuss the occurrence of a Σ-shaped bifurcation diagram for positive solutions. In particular, for certain parameter values of c,K, ∈ and the diffusion coefficient, we prove the existence of at least four positive solutions. We prove our results by the quadrature method. © 2011 Elsevier Ltd. All rights reserved.}, author_keywords={Σ-shaped bifurcation curves; Ecological systems; Grazing and constant yield harvesting}, keywords={Bifurcation curves; Bifurcation diagram; Dirichlet boundary condition; Ecological models; Ecological systems; Fish populations; Grazing and constant yield harvesting; Logistic growth model; Multiple positive solutions; Parameter values; Positive constant; Positive solution; Quadrature methods; Steady state; Steady state reactions, Boundary conditions; Diffusion; Ecology; Equations of state; Fisheries; Harvesting; Partial differential equations; Potassium, Bifurcation (mathematics)}, references={Causey, R., Sasi, S., Shivaji, R., An ecological model with grazing and constant yield harvesting (2010) Bull. Belg. Math. Soc., 17 (5), pp. 833-839; Lee, E., Sasi, S., Shivaji, R., S-shaped bifurcation curves in ecosystems (2011) J. Math. Anal. Appl., , 10.1016/j.jmaa.2011.03.048; May, R.M., Thresholds and breakpoints in ecosystems with a multiplicity of stable states (1977) Nature, 269, pp. 471-477; Noy-Meir, I., Stability of grazing systems an application of predatorprey graphs (1975) J. Ecol., 63, pp. 459-482; Oruganti, S., Shi, J., Shivaji, R., Diffusive equations with constant yield harvesting, I: Steady states (2002) Trans. Amer. Math. Soc., 354, pp. 3601-3619; Steele, J.H., Henderson, E.W., Modeling long-term fluctuations in fish stocks (1984) Science, 224 (4652), pp. 985-987; Brauer, F., Castillo-Chavez, C., (2001) Mathematical Models in Population Biology and Epidemiology, 40. , Texts in Applied Mathematics Springer-Verlag New York; Clark, C.W., (1990) Mathematical Bioeconomics, the Optimal Management of Renewable Resources, , John Wiley & Sons, Inc. New York; Brown, K.J., Ibrahim, M.M.A., Shivaji, R., S-shaped bifurcation curves (1981) Nonlinear Anal. TMA, 5 (5), pp. 475-486; Laetsch, T., The number of solutions of a nonlinear two point boundary value problem (1970) Indiana Univ. Math. J., 20 (1), pp. 1-13; Jiang, J., Shi, J., Bistability dynamics in some structured ecological models (2009) Spatial Ecology, pp. 33-62. , R.S. Cantrell, C. Cosner, S. Ruan, Chapman & Hall, CRC Mathematical and Computational Biology}, correspondence_address1={Shivaji, R.; Department of Mathematics and Statistics, , Greensboro, NC 27412, United States; email: shivaji@uncg.edu}, issn={14681218}, language={English}, abbrev_source_title={Nonlinear Anal. Real World Appl.}, document_type={Article}, source={Scopus},

Nonlinear Analysis: Real World Applications

An ecological model with a Σ-shaped bifurcation curve

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form:. Here δu=div(∇u) is the Laplacian of u, 1λ is the diffusion coefficient, K and c are positive constants and Ω⊂RN is a smooth bounded region with ∂. Ω in C2. This model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. In this paper we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of sub-supersolutions. © 2011 Elsevier Inc.

S-shaped bifurcation curves in ecosystems

We study positive solutions to the steady state reaction diffusion equation with Dirichlet boundary condition of the form: {-Δu=au-bu2-c up/1+up - K, x ∈ ω u=0, x ∈ ∂ω Here Δu = div(▽u) is the Laplacian of u, a,b,c,p,K are positive constants with p ≥ 2 and ω is a smooth bounded region with ∂ω in C2. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. We study the existence of positive solutions to this model. We prove our results by the method of sub-super solutions.}, references={Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces (1976) SIAM Rev., 18, pp. 620-709; Berestycki, H., Caffarelli, L.A., Nirenberg, L., Inequalities for second-order elliptic equations with applications to unbounded domains (1996) Duke Mathematical Journal, 81 (2), pp. 467-494. , DOI 10.1215/S0012-7094-96-08117-X; Clement, Ph., Peletier, L.A., An anti-maximum principle for second order elliptic operators (1979) J. Diff. Eqns., 34 (2), pp. 218-229; Lions, P.L., On the existence of positive solutions of semilinear elliptic equations (1982) SIAM Rev., 24, pp. 441-467; May, R.M., Thresholds and breakpoints in ecosystems with a multiplicity of stable states (1977) Nature, 269, pp. 471-477; Noy-Meir, I., Stability of grazing systems an application of predator-prey graphs (1975) Journal of Ecology, 63, pp. 459-482; Oruganti, S., Shi, J., Shivaji, R., Diffusive equations with constant yield harvesting, I: Steady states (2002) Trans. Amer. Math. Soc., 354, pp. 3601-3619; Steele, J.H., Henderson, E.W., Modelling long term fluctuations in fish stocks (1984) Science, 224, pp. 985-987}, correspondence_address1={Causey, R.; Department of Mathematics and Statistics, , Mississippi State 39759, United States; email: rmc107@msstate.edu}, publisher={Belgian Mathematical Society}, issn={13701444}, language={English}, abbrev_source_title={Bull. Belg. Math. Soc. Simon Stevin}, document_type={Article}, source={Scopus},

Bulletin of the Belgian Mathematical Society - Simon Stevin

An ecological model with grazing and constant yield harvesting

Electronic Journal of Qualitative Theory of Differential Equations

Population dynamics with symmetric and asymmetric harvesting

Communications on Pure and Applied Analysis

Bifurcation, uniqueness and multiplicity results for classes of reaction-diffusion equations arising in ecology with nonlinear boundary conditions

We consider semi-linear elliptic equations of the following form:\begin{equation*}\left\{\begin{aligned}-\Delta u &amp;= \lambda[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:\lambda f_h(u) &amp;&amp;\textcolor{black}{\rm{in}} ~ \Omega,\\frac{\partial u}{\partial \eta}&amp;+qu = 0 &amp;&amp;\textcolor{black}{\rm{on}} ~ \partial\Omega,\end{aligned}\right.\end{equation*}where, $h\in U=\{h\in L^2(\Omega): 0\leq h(x)\leq H\}.$ We prove the existence and uniqueness of the positive solution for large $\lambda.$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional$J(h)=\int_{\Omega}h(x)u_h(x)~{\rm{d}}x-\int_{\Omega}(B_1+B_2 h(x))h(x)~{\rm{d}}x$over $U$, where $u_h$ is the unique positive solution of the above problem associated with $h$, $B_1&gt;0$ is the cost per unit effort when the level of effort is low and $B_2&gt;0$ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system.