Philip L Korman , PHD
Professor
Professor
French Hall
5504
A&S Mathematical Sciences - 0025
Professional Summary
Visit my homepage: https://homepages.uc.edu/~kormanp/
Education
Ph.D.: New York University 1981
Abbreviated Publications
Peer Reviewed Publications
A global solution curve for a class of free boundary value problems arising in plasma physics. Appl. Math. Optim. 71 (2015), no. 1, 25-38.
A global solution curve for a class of periodic problems, including the relativistic pendulum. Appl. Anal. 93 (2014), no. 1, 124–136.
(with Li, Yi) Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities. J. Differential Equations 257 (2014), no. 10,
Global solution curves for self-similar equations. J. Differential Equations 257 (2014), no. 7, 2543–2564.
Curves of equiharmonic solutions, and problems at resonance. Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2847–2860.
(with Li, Yi) Harmonic oscillators at resonance, perturbed by a non-linear friction force. Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no. 4,
Exact multiplicity and numerical computation of solutions for two classes of non-autonomous problems with concave-convex nonlinearities.Nonlinear Analysis 93 (2013), 226-235.
(With Yi Li) "Harmonic oscillators at resonance, perturbed by a non-linear friction force", Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no. 4, 1025–1028.
Nonlinear perturbations of linear elliptic systems at resonance.Proc. Amer. Math. Soc. 140 (2012), no. 7, 2447–2451.
(with Y. Li) A computer assisted study of uniqueness of ground state solutions. J. Comput. Appl. Math. 236 (2012), no. 11, 2838–2843.
Direction of bifurcation for some non-autonomous problems, Electronic J. of Differential Equations (EJDE) Vol. 2012 (2012), No. 214, pp. 1-13.
Adamowicz, Tomasz; Korman, Philip Remarks on time map for quasilinear equations. J. Math. Anal. Appl. 376 (2011), no. 2, 686–695.
Existence and Uniqueness of Solutions for a Class of
p-Laplace Equations on a Ball. Advanced Nonlinear Studies 11 (2011), 875–888.
Curves of equiharmonic solutions and solvability of elliptic systems. Differential and Integral Equations. Volume 24, Numbers 11-12, November/December 2011.
Curves of equiharmonic solutions, and ranges of nonlinear equations. Adv. Differential Equations 14 (2009), no. 9-10, 963–984.
Global solution curves for boundary value problems, with linear part at resonance. Nonlinear Anal. 71 (2009), no. 7-8, 2456–2467.
Korman, Philip Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems. J. Differential Equations 244 (2008), no. 10, 2602–2613.
An oscillatory bifurcation from infinity, and from zero. NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 3, 335–345.
A global approach to ground state solutions. Electron. J. Differential Equations 2008, No. 122, 13 pp.
Korman, Philip; Li, Yi; Ouyang, Tiancheng Verification of bifurcation diagrams for polynomial-like equations. J. Comput. Appl. Math. 212 (2008), no. 2, 187–193.
An exact multiplicity result for a class of symmetric problems. Port. Math. 65 (2008), no. 2, 257–260
Korman, Philip Stability and instability of solutions of semilinear problems. Appl. Anal. 86 (2007), no. 2, 135–147.
Korman, Philip Computation of radial solutions of semilinear equations. Electron. J. Qual. Theory Differ. Equ. 2007, No. 13, 14 pp. (electronic).
Korman, Philip A global solution curve for a class of periodic problems, including the pendulum equation. Z. Angew. Math. Phys. 58 (2007), no. 5, 749–766.
Hou, Xiaojie; Korman, Philip; Li, Yi On the direction of pitchfork bifurcation. Proceedings of the Sixth Mississippi State–UBA Conference on Differential Equations and Computational Simulations, 153–158, Electron. J. Differ. Equ. Conf., 15, Southwest Texas State Univ., San Marcos, TX, 2007.
Galstyan, A., & Li, Y. (2006). On the oscillations of the solution curve for a class of semilinear equations. J. Math. Anal. Applic., 321, 576-588.
(2006). Uniqueness and exact multiplicity of solutions of non-autonomous Dirichlet problems. Advanced Nonlinear Studies, 6, 461-481.
Ouyang, T. (2006). Positivity for the linearized problem for semilinear equations. Topol. Methods Nonlinear Anal., 28(1), 53-60.
Li, Yi, & Ouyang, T. (2005). Computing the location and the direction of bifurcation. Mathematical Research Letters, 12(5 & 6), 933-944.
(2005). On a principle of predatory exclusion. Applicable Analysis, 84(7), 707-712.
Shi, J. (2005). On Lane-Emden type systems. Discrete Contin. Dyn. Syst. 510-517.
(2004). Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. Proc. Roy. Soc. Edinburgh Sect. A, 134(1), 179-190.
(2003). An accurate computation of the global solution curve for the Gelfand problem through a two point approximation. Appl.Math and Computations, 139(2 & 3), 363-369.
(2003). Curves of positive solutions for supercritical problems. Applicable Analysis, 82, 45-54.
Li, Y., & Ouyang, T. (2003). Perturbation of global solution curves for semilinear problems. Adv. Nonlinear Stud., 3(2), 289-299.
(2002). On uniqueness of positive solutions for a class of semilinear problems. Discrete Contin. Dynam. Systems, 8(4), 865-871.
(2002). Curves of sign-changing solutions for semilinear equations. Nonlinear Analysis TMA, 51(5), 801-820.
(2002). Exact multiplicity of solutions for a class of Neumann problems. Comm. Appl. Nonlinear Anal., 9(4), 1-17.
(2002). Global solution curves for semilinear systems. Math. Methods Appl. Sci, 25(1), 3-20.
(2002). A remark on the non-degeneracy condition. Results in Mathematics, 41(3 & 4), 334–336.
(2002). Stability and Morse indices of solutions for two classes of semilinear problems. Comm. Appl. Nonlinear Anal., 9(4), 1-17.
(2001). Monotone approximations of unstable solutions. J. Comput. Appl. Math., 136, 309-315.
(2001). On multiplicity of solutions of semilinear equations. Mathematische Nachrichten, 229, 119-127.
Shi, J. (2001). New exact multiplicity results with an application to a population model. Proc. Royal Soc. Edinburgh Ser. A., 131(5), 1167-1182.
Shi, J. (2000). Instability and exact multiplicity of solutions of semilinear equations. Proceedings of the Conference on Nonlinear Differential Equations 311-322.
Li, Y. (2000). Infinitely many solutions at a resonance. Electron. J. Differ. Equ. Conf., 5.
(1999). Multiplicity of positive solutions for semilinear equations on circular domains. Comm. Appl. Nonlinear Anal., 6(3), 17-35.
Li, Y. (1999). Generalized averages for solutions of two-point Dirichlet problems. J. Math. Anal. Appl., 239(2), 478-484.
Ouyang, T. (1999). Multiplicity and Morse indices of sign-changing solutions for semilinear equations . Canad. Appl. Math. Quart., 7(3), 239-250.
(1999). Exact multiplicity of positive solutions for a class of semilinear equations on a ball. Electron. J. Qual. Theory Differ. Equ., 8.
(1998). The global solution set for a class of semilinear problems. J. Math. Anal. Appl., 226, 101-120.
(1998). A global solution curve for a class of semilinear equations. Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations 119–127.
(1998). Remarks on Nagumo’s condition. Portugaliae Mathematica, 55(1), 1-9.
(1998). Uniqueness and exact multiplicity results for two classes of semilinear problems. Nonlinear Analysis TMA, 31, 849-865.
Li, Y., & Ouyang, T. (1997). An exact multiplicity result for a class of semilinear equations. Commun. in PDE, 22, 661-684.
(1997). Solution curves for semilinear equations on a ball. Proc. of Amer. Math. Soc., 125, 1997-2005.
(1997). Steady states and long time behavior of some convective reaction-diffusion equations . Funkcialaj Ekvacioj, 40, 165-183.
Lazer, A.C., & Li, Y. (1997). On homoclinic and heteroclinic orbits for Hamiltonian systems. Differential and Integral Equations, 10, 357-368.
Ouyang, T. (1996). Solution curves for two classes of boundary-value problems. Nonlinear Analysis Apllic., 27, 1031-1047.
Li, Y., & Ouyang, T. (1996). Exact multiplicity results for boundary value problems with nonlinearities generalizing cubic. Proc. Royal Soc. Edinburgh Ser. A., 126A, 599-616.
Ouyang, T. (1995). Exact multiplicity results for a class of boundary value problems with cubic nonlinearities . J. Math. Anal. Applic., 194, 328-341.
Ouyang, T. (1995). On computation of solution curves for semilinear elliptic problems . Numer. Funct. Anal. and Optimiz., 16(1 & 2), 219-231.
(1995). Symmetry of positive solutions for elliptic problems in one dimension. Applicable Analysis, 58, 351-365.
Ouyang, T. (1995). Exact multiplicity results for two classes of periodic. J. Math. Anal. Applic., 194, 763-779.
Ouyang, T. (1995). Multiplicity results for two classes of boundary-value problems. SIAM J. of Math. Analysis, 26(1), 180-189.
(1994). On the dynamics of two classes of periodic ecological models. J. of Computational and Appl. Math., 52, 267-275.
Lazer, A.C. (1994). Homoclinic orbits for a class of symmetric Hamiltonian systems. EJDE Volumes.
Ouyang, T. (1993). Exact multiplicity results for two classes of boundary value problems. Diff. and Integral Equations, 6(6), 1507-1517.
(1993). An Algorithmfor Computing Unstable Solutions of Semilinear Boundary Value Problems. Computing, 51(3 & 4), 327-334.
(1992). A Computer Assisted Study of Periodic Parabolic Problems. Appl. Math. & Computation, 50(2 & 3), 203-221.
(1992). Dynamics of the Lotka-Volterra Systems with Diffusion. Applicable Analysis, 44(3 & 4), 191-208.
Choudury, G. (1992). On Computation of Solutions of Fully Nonlinear Elliptic Problems. J. of Computational and Appl. Math, 41, 301-311.
(1992). On Periodic Solutions of Singular Perturbation Problems. Proceedings of the Royal Society of Edinburgh, 120A, 143-152.
(1992). Some New Results on the Periodic Competition Model. J. Math. Anal. Applic., 171(1), 131-138.
Barbu, V. (1991). Approximating Optimal Controls of Elliptic Obstacle Problem by Monotone Iteration Schemes. Numer. Funct. Anal. and Optimiz., 12(5 & 6), 429-442.
Choudury, G. (1991). On Computation of Solutions of Nonlinear Boundary Value Problems. Comput. Math. Appl., 22(8), 49-55.
Leung, A., & Stojanovic, S. (1990). Monotone Iterations for Nonlinear Obstacle Problems . J. Austr. Math. Soc. Ser., B 30, 259-276.
(1990). On Existence of Periodic Solutions for a Class of Quasilinear Non- Coercive Problems. Funkcialaj Ekvacioj, 33, 127-138.
(1990). On Periodic Solutions for Singular Perturbation Problems. Nonlinear Analysis TMA, 15(5), 467-478.
Li, Y. (1999). The exactness of S-shaped bifurcation curve. Proc. Amer. Math. Soc., 127, 1011-1020.
(1989). On Computation of Solutions of Elliptic Systems. Numer. Funct. Anal. and Optimiz., 10(9 & 10), 977-990.
(1989). On Existence of Solutions for a Class of Fully Nonlinear Noncoercive Problems. J. Math. Anal. Applic., 137, 477-484.
(1989). On Existence of Solutions for a Class of Non-Coercive Problems. Commun. in PDE, 14(4), 519-539.
(1989). A MaximumPrinciple for Fourth Order Ordinary Differential Equations. Applicable Analysis, 33, 267-273.
(1988). On Blow-Up of Solutions of Nonlinear Evolution Equations. Proc. Amer. Math. Soc., 103, 189-197.
(1988). Computation of Displacements for Nonlinear Elastic Beam Models Using Monotone Iterations. Intern. J. Math. and Math. Sci., 11, 121-128.
(1988). On Existence of Solutions for Several Classes of Free Boundary Problems. SIAM J. Math. Anal., 19, 814-823.
(1987). On the Existence and Uniqueness of Positive Steady States in the Volterra-Lotka Ecological Models with Diffusion . Applicable Analysis, 26, 145-160.
Leung, A. (1986). A General Monotone Scheme for Elliptic Systems with Applications to Ecological Models . Proc. Royal Soc. Edinburgh, 102A, 315-325.
(1986). Existence and Approximation of Solutions of Inverse-Positive Problems. Nonlinear Analysis TMA, 10, 863-872.
(1986). Existence of Solutions for a Class of Semilinear Noncoercive Problems. Nonlinear Analysis TMA, 10, 1471-1476.
(1984). On Application of Monotone Iteration Scheme to Nonlinear Non-Coercive Problems. Nonlinear Analysis TMA, 8, 97-105.
(1983). Existence of Periodic Solutions for a Class of Nonlinear Problems. Non-linear Analysis TMA, 7, 873-879.
(1983). Existence of Solutions for a Class of Nonlinear Non-Coercive Problems. Comm. Partial Diff. Eqs., 8, 819-846.
Non-existence of solutions for non-autonomous elliptic systems of Hamiltonian type, Comm. Appl. Nonlinear Anal. 21 (2014), no. 3, 13–20.
(with Li, Yi) Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities. J. Differential Equations 257 (2014), no. 10,
Regularization of radial solutions of $p$-Laplace equations, and computations using infinite series. Electron. J. Qual. Theory Differ. Equ. 2015,
Non-singular solutions of two-point problems, with multiple changes of sign in the nonlinearity. Proc. Amer. Math. Soc. 144 (2016), no. 6, 2539–2546.
Nonlinear elliptic equations and systems with linear part at resonance Electron. J. Diff. Equ. Vol. 2016 (2016), No. 67, pp. 1-17.
(with D. Schmidt) Infinitely many solutions and asymptotics for resonant oscillatory problems, Electron. J. Diff. Equ., Special Issue 01, 301-313 (2021)..
(with D. Schmidt) Continuation of global solution curves using global parameters, J. Comput. Appl. Math. 440 (2024), Paper No. 115621, 18 pp.
Book
GLOBAL SOLUTION CURVES for SEMILINEAR ELLIPTIC EQUATIONS
World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
LECTURES ON DIFFERENTIAL EQUATIONS
AMS/MAA Textbooks 54, MAA Press, Providence, RI, 2019. xii+399 pp.
LECTURES ON LINEAR ALGEBRA and its APPLICATIONS
In the series De Gruyter Textbook, Walter de Gruyter, Oct 24, 2023 - Mathematics - 258 pages.
Book Chapter
(2006). Handbook of Differential Equations, Ordinary Differential Equations. In A. Canada, P. Drabek and A. Fonda (Eds.), Global solution branches and exact multiplicity of solutions for two point boundary value problems (pp. 547-606). North Holland: Elsevier Science.
Other Publication
(2002). The global solution set for a class of semilinear problems. Pure and Appl. Math., 225. New York: Dekker.
(2001). Similarity of solution branches for two-point semilinear problems. Proceedings of the Fifth Mississippi State Conference on Differential Equations and Computational Simulations. Mississippi State.
Presentations
Invited Presentations
(10-19-2012. ) Resonance in nonlinear equations and systems .Wright State University, Dayton, OH. Other Institution. . Level:State
(03-22-2016. ) Some Special Equations .Joint Analysis/PDE seminar, Ohio State University. Other Institution. . Level:National
Lecture
UC. Level:Department
UC. Level:Department
Symposium
(03-09-2011. ) UC. Level:International
Service
Reviewer Type:Editorial Service Level:International 1/1/2024 12:00:00 AM -12/30/2024 12:00:00 AM
Journal of Differential Equations (Refereed two papers ) Peer Review/Referee Type:Editorial Service Level:International 03-10-2015
Journal of Differential Equations Peer Review/Referee Type:Editorial Service Level:International 02-28-2013
Mathematische Nachrichten Peer Review/Referee Type:Editorial Service Level:International 02-04-2013
Peer Review/Referee Type:Editorial Service Level:International 01-28-2013
Peer Review/Referee Type:Editorial Service Level:International 01-12-2013
Journal of Differential Equations Peer Review/Referee Type:Editorial Service Level:International 01-07-2013
MathSciences Reviewer Type:Editorial Service Level:International 02-12-2012
Nonlinear Analysis, TMA Peer Review/Referee Type:Editorial Service Level:International 09-21-2011 -01-13-2012
Mathematica Slovaca (Slovakia, published by Springer Verlag) Editorial Board Member Type:Editorial Service Level:International 01-01-2010
Communications on Applied Nonlinear Analysis Editorial Board Member Type:Editorial Service Level:International
Electronic Journal of Differential Equations, Problem Section Editorial Board Member Type:Editorial Service Level:International
SIAM Review, Problem Section Editorial Board Member Type:Editorial Service Level:International
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Journal of Partial Differential Equations (China) Peer Review/Referee Type:Editorial Service Level:International
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(RPT, Colloquium Coordinator, Math Bowl, Calculus Contest, PDE preliminary exams. I wrote several letters of recommendation. ) Member Type:Departmental Service Level:Department
AMS Reviewer Type:Editorial Service Level:International
Reviewed a paper in Mathematical Modelling and Analysis for MathSciences Reviewer Type:Editorial Service Level:International 05-10-2015 -05-29-2015
Refereed a paper for Jourmal of Math. Analysis and Applications Reviewer Type:Editorial Service Level:International 05-10-2015 -07-29-2015
Refereed a paper for Journal of Applied Analysis and Computation Peer Review/Referee Type:Editorial Service Level:International 05-02-2015 -07-08-2015
Mathematical Reviews Reviewer Type:Editorial Service Level:International 08-04-2015 -Bad Format: 20152104
(Wrote Evaluation Letter for C. Zhao promotiom to full professor at Georgia Southern University ) Reviewer Type:Other Level:National 07-21-2015 -08-24-2015
Mathematische Nachrichten Peer Review/Referee Type:Editorial Service Level:International 06-04-2015 -09-10-2015
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Communications in Pure and Applied Analysis Peer Review/Referee Type:Editorial Service Level:International 07-02-2015 -10-15-2015
(Wrote Evaluation Letter for X. Hou promotiom to full professor at the University of North Carolina, Wilmington ) Type:Other Level:National 10-20-2015 -12-20-2015
Refereed JMAA-15-2160 (Journal of Mathematical Analysis and Applications ) Peer Review/Referee Type:Editorial Service Level:International 10-09-2015 -12-23-2015
Communications in Pure and Applied Analysis Peer Review/Referee Type:Editorial Service Level:International 09-01-2015 -01-06-2016
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(Master's exam for Brian Lunn ) Committee Member Type:Departmental Service Level:Department 04-27-2016 -04-27-2016
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(Served on three master's exam committees ) Committee Member Type:Departmental Service Level:Department 04-19-2017 -04-26-2017
Reviewed a paper for MathSciences Type:Editorial Service Level:International 06-01-2017 -06-24-2017
Refereed for EJDE Peer Review/Referee Type:Editorial Service Level:International 03-13-2017 -08-07-2017
Acta Mathematica Scientia (China) Peer Review/Referee Type:Editorial Service Level:International 08-01-2017 -08-15-2017
Keywords
Differential Equations
Courses Taught
-MATH-1061 CALCULUS I Level:Undergraduate
15-MATH-553 PDE & FOURIER ANAL Level:Both
15-MATH-273 DIFFERENTIAL EQNS
15-MATH-251 CALCULUS I Level:Undergraduate
15-MATH-629 PART DIFFER EQUATNS Level:Graduate
15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate
APP LIN ALGEBRA Level:Both
ORDINARY DIFF EQ Level:Graduate
15-MATH-568 APPL COMPLX ANALYS Level:Both
15-MATH-553 PDE & FOURIER ANAL Level:Both
15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate
15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate
15-MATH-554 PDE & FOURIER ANAL Level:Both
15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate
Applied ODE Level:Both
15-MATH-252 CALCULUS II Level:Undergraduate
15-MATH-352 LINEAR ALGEBRA II Level:Undergraduate
15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate
15-MATH-264 CALC & AN GEOM IV Level:Undergraduate
15-MATH-568 APPL COMPLX ANALYS Level:Both
15-MATH-355 INTRO ORD DIFF EQU Level:Undergraduate
15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate