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Workgroup Nonlinear Partial Differential Equations

Kollegiengebäude Mathematik (20.30)
Room 3.029

Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe

Office hours:
Mon-Fri 10:00-12:00, Tue+Thu 14:00-16:00

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530



with A. Contreras and D. E. Pelinovsky: Orbital Stability of Domain Walls in Coupled Gross-Pitaevskii Systems. Submitted. http://arxiv.org/abs/1702.00701

with W. Reichel: A breather construction for a semilinear curl-curl wave equation with radially symmetric coefficients. W. J. Elliptic Parabol. Equ. (2016) 2: 371. doi:10.1007/BF03377410

with K. Tanaka, K. Sekine, M. Kashiwagi, and S. Oishi: Verified numerical computation for semilinear elliptic problems with lack of Lipschitz continuity of the first derivative. Submitted. http://arxiv.org/abs/1607.04619

with A. Khrabustovskyi: Spectral properties of elliptic operators with double-contrast coefficients near a hyperplane. Asymptotic Analysis 98 (2016), pp. 91-130, doi: 10.3233/ASY-161363

with M. Brown, V. Hoang, M. Radosz, and I. Wood: Gap localization of TE-Modes by arbitrarily weak defects. To appear in Journal of the London Mathematical Society.

with Y. Watanabe, K. Nagatou, and M. Nakao: Norm bound computation for inverses of linear operators in Hilbert spaces. Journal of Differential Equations 260 (2016), pp. 6363-6374. doi: 10.1016/j.jde.2015.12.041

with F. Pacella and D. Rütters: A computer-assisted existence proof for Emden's equation on an unbounded L-shaped domain. Communications in Contemporary Mathematics, Vol. 19, No. 2 (2017). http://dx.doi.org/10.1142/S0219199717500055

with Th. Bartsch, T. Dohnal, and W. Reichel: Ground states of a nonlinear curl-curl problem in cylindrically symmetric media. Nonlinear Differ. Equ. Appl. (2016) 23: 52. doi:10.1007/s00030-016-0403-0

with B.M. Brown, V. Hoang, and I. Wood: Spectrum created by line defects in
periodic structures. Mathematische Nachrichten 287, No. 17-18, 1972-1985. doi: 10.1002/mana.201300165

with Y. Watanabe, K. Nagatou, and M. T. Nakao: Verified Computations of Eigenvalue Exclosures for Eigenvalue Problems in Hilbert Spaces. SIAM J. Numer. Anal., Vol. 52, No. 2, 975-992, 2014. doi: 10.1137/120894683

with P. J. McKenna, F. Pacella, and D. Roth: A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem. Inequalities and Applications 2010, International Series of Numerical Mathematics, Vol. 161, Part 1, 31-52, 2012. article

with C. Bandle, A. Gilanyi, and L. Losonczi: Inequalities and Applications 2010 (Dedicated to the Memory of Wolfgang Walter), International Series of Numerical Mathematics, Vol. 161, Springer Basel, 2012. book

with M. Brown, V. Hoang, and I. Wood: On the spectrum of waveguides in planar photonic bandgap structures. Proceedings of the Royal Society of London A, 471: 20140673, http://dx.doi.org/10.1098/rspa.2014.0673

with T. Dohnal, K. Nagatou, and W. Reichel: Interfaces supporting surface gap soliton ground states in the 1D nonlinear Schroedinger equation. Journal of Mathematical Analysis and Applications 407 (2013), pp. 425-435. doi: 10.1016/j.jmaa.2013.05.028

with K. Nagatou, M. T. Nakao, and Y. Watanabe: A computer-assisted stability proof for the Orr-Sommerfeld problem with Poiseuille flow. Nonlinear Theory and Its Applications, IEICE 2, 123-127 (2011).

with K. Nagatou und M. T. Nakao: Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 468, Issue 2138 (2012), pp. 545-562; published online before print 10.2011: doi: 10.1098/rspa.2011.0159; pdf

with T. Dohnal and W. Reichel: Surface gap soliton ground states for the nonlinear Schrödinger equation. Comm. Math. Phys., Volume 308, Issue 2 (2011), 511-542; http://www.springerlink.com/content/j01k11547j2p8083/ - pdf

with W. Dörfler, A. Lechleiter, G. Schneider, and Ch. Wieners: Photonic Crystals: Mathematical Analysis and Numerical Approximation. Birkh"auser, 2010; Oberwolfach Seminars 42.

with T. Dohnal and W. Reichel: Localized modes of the linear periodic Schrödinger operator with a nonlocal perturbation. SIAM J. Math. Anal., Volume 41, Issue 5 (2009), 1967-1993. pdf

mit with B. M. Brown, V. Hoang, I. G. Wood: Floquet-Bloch Theory for Elliptic Problems with Discontinuous Coefficients. In: Spectral theory and analysis, 1–20, Oper. Theory Adv. Appl., 214, Birkhäuser/Springer, Basel, 2011.

with P. J. McKenna, F. Pacella, D. Roth: A Uniqueness Result for a Semilinear Elliptic Problem: A Computer-assisted Proof. J. Differential Equations 247 (2009) 2140-2162; doi:10.1016/j.jde.2009.06.023.

Computer-assisted Proofs for Semilinear Elliptic Boundary Value Problems. Japan Journal for Industrial and Applied Mathematics, Vol. 26, No. 2-3 (2009); doi: 10.1007/BF03186542.

with V. Hoang, Ch. Wieners: A computer-assisted proof for photonic band gaps. Z. angew. Math. Phys. 60 (2009) 1035-1052; 0044-2275/09/061035-18; DOI 10.1007/s00033-008-8021-2; www.birkhauser.ch/ZAMP.

with B. M. Brown, V. Hoang, I.G. Wood: On Spectral Bounds for Photonic Crystal Waveguides. In: Inequalities and Applications (C. Bandle, A. Gilànyi, L. Losonczi, Z. Pales, M. Plum eds.), International Series of Numerical Mathematics, Vol. 157, 23-30 (2009).

with Y. Watanabe, M. T. Nakao: A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow. Z. Angew. Math. Mech. 89, No. 1, 5-18 (2009)/DOI 10.1002/zamm.200700158.

Existence and Multiplicity Proofs for Semilinear Elliptic Boundary Value Problems by Computer Assistance. DMV Jahresbericht: JB 110, Heft 1 (2008) 19-54.
Download: emp.ps emp.pdf

mit W. N. Everitt, L. Markus, M. Muzzulini: A continuum of unusual selfadjoint linear partial differential operators.
Journal of Computational and Applied Mathematics 208, 164-175 (2007).

with Ch. Wieners: Numerical enclosures for variational inequalities. Computational Methods in Applied Mathematics, Vol. 7, No. 4, 376-388 (2007).

with B. Breuer, J. Horak, P.J. McKenna: A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam.
Download: travelling.pdf

with R. C. Brown: An Opial Type Inequality with an Integral Boundary Condition.
To appear in Proc. R. Soc. Lond. A.
Download: opial_type.pdf

with W.N. Everitt, L. Markus: An Unusual Self-Adjoint Linear Partial Differential Operator.
Trans. Amer. Math. Soc. 367, 1303-1324 (2005).
Download: self-adjoint.pdf

with E. B. Davies: Spectral Pollution.
IMA Journal of Numerical Analysis 24, 417-438 (2004).

with B. Breuer, P.J. McKenna: Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof.
J. Differential Equations 195, 243-269 (2003).

with B.M. Brown, W.D. Evans: Titchmarsh-Sims-Weyl theory for Complex Hamiltonian Systems.
Proc. London Math. Soc. (3) 87, 419-450 (2003).
Download: titchmarsh.pdf

with J.-R. Lahmann: A computer-assisted instability proof for the Orr-Sommerfeld equation with Blasius profile.
ZAMM 84, No. 3, 188-204 (2004).
Download: orr-sommerfeld.pdf

with E. Adams: Free Vibrations of Trusses by Fourier Decomposition, and Homotopy Methods for Nonlinear Matrix Eigenvalue Problems, I and II.
J. Math. Anal. Appl. 275, 333-353 (2002) and 276, 64-79 (2002).

with Ch. Wieners: New solutions of the Gelfand problem.
J. Math. Anal. Appl. 269, 588-606 (2002).

with Ch. Wieners: Optimal a priori estimates for interface problems.
Numer. Math. 95, 735-759 (2003).

with P. J. McKenna: Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof.
J. Math. Anal. Appl. 251, No. 2, 710-715 (2000).

with B. Breuer, P.J. McKenna: Inclusions and Existence Proofs for Solutions of a Nonlinear Boundary Value Problem by Spectral Numerical Methods.
Computing Suppl. 15, 61-77 (2001).

with Ch. Hipp: Optimal investment for investors with state dependent income, and for insurers.
Finance and Stochastics, Vol. 7, No. 3, 299-321 (2003).

Safe Numerical Error Bounds for Solutions of Nonlinear Elliptic Boundary Value Problems.
In: Symbolic Algebraic Methods and Verification Methods (Eds.: G. Alefeld, J. Rohn, S. Rump, T. Yamamoto), Springer Wien New York, 195-207 (2001).

with Ch. Hipp: Optimal investment for insurers.
Insurance, Mathematics and Economics 27, 215-228 (2000).

with H. Behnke, U. Mertins, Ch. Wieners: Eigenvalue inclusions via domain decomposition.
Proc. R. Soc. Lond. A. 456, 2717-2730 (2000).

Computer-assisted Enclosure Methods for Elliptic Differential Equations.
Linear Algebra and its Applications 324, 147-187 (2001).

with J. Lahmann: Das Spektrum der Orr-Sommerfeld-Gleichung mit Blasius-Strömungsprofil.
ZAMM 78, Suppl. 3, S987 - S988 (1998).

with B. Breuer: Lösungseinschließungen bei einem nichtlinearen Randwertproblem mittels eines Fourierreihenansatzes.
Internal Report.

with R. M. Redheffer: A Class of Second-order Differential Equations.
J. Differential Equations 154, 454-469 (1999).

with J. Lahmann: On the spectrum of the Orr-Sommerfeld equation on the semi-axis.
Math. Nachr. 216, 145-153 (2000).

with B.M. Brown, D.K.R. McCormack, W.D. Evans: On the spectrum of second order differential operators with complex coeficients.
Proc. Royal Soc. London. A 455, 1235-1257 (1999).

with B.M. Brown, M.S.P. Eastham, D.K.R. McCormack: On a new algorithm for the computation of enclosures for the Titchmarsh-Weyl m-function.
Result. Math. 33, 50-64 (1998).

Eigenvalue Problems for Differential Equations.
In: Wavelets, Multilevel Methods and Elliptic PDEs (Eds.: M. Ainsworth , J. Levesley, M. Marletta, W.A. Light), Clarendon Press, Oxford, 39-83 (1997).

Guaranteed Numerical Bounds for Eigenvalues.
In: Spectral theory and computational methods of Sturm-Liouville problems (Eds.: D. Hinton, P. Schaefer), Marcel Dekker, New York, 313-332 (1997).

Enclosure Methods with Existence Proof for Elliptic Differential Equations. Proceedings zu "General Inequalities 7" (Oberwolfach 1995), International Series of Numerical Mathematics, Vol. 123, 347-368 (1997).

Enclosures for Two-Point Boundary Value Problems near Bifurcation Points.
In: Scientific Computing and Validated Numerics, Proceedings of the International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics SCAN-95, held in Wuppertal, Germany, September 26-29, 1995 (Eds.: G. Alefeld, A. Frommer, B. Lang), Mathematical Research, Volume 90, Akademie Verlag, 265-279 (1996).

Existence Proofs for Elliptic Boundary Value Problems by Numerical Means.
ZAMM 75 SII, S429-S432 (1995).

Lösungseinschließung mit automatischem Existenzbeweis bei elliptischen Randwertproblemen.
In: Jahrbuch Überblicke Mathematik 1995 (Hrsg.: A. Beutelspacher, S.D. Chatterji, U. Kulisch, R. Liedl), Vieweg, Braunschweig, 103 - 124 (1995).

Enclosures for Weak Solutions of Nonlinear Elliptic Boundary Value Problems.
In: Inequalities and Applications (Ed.: Ravi P. Agarwal), World Scientific Series In Applicable Analysis, Vol. 3, 505-521 (1994).

Inclusion Methods for Elliptic Boundary Value Problems.
In: Topics in Validated Computations (Ed.: J. Herzberger), Studies in Computational Mathematics 5, North Holland (Elsevier), Amsterdam, 323 - 379 (1994).

Existence and Enclosure Results for Continua of Solutions of Parameter-Dependent Nonlinear Boundary Value Problems.
Journal of Computational and Applied Mathematics 60, 187 - 200 (1995).

with B. Kalkofen: Optimal Pricing Rules for Private-Value Auctions with Incomplete Information.
Ifo-Studien 1/1996, 77 - 100.

Enclosures for Solutions of Parameter-dependent Nonlinear Elliptic Boundary Value Problems: Theory and Implementation on a Parallel Computer.
Interval Computations 3 (Special Issue on Parallel Algorithms), 106 - 121 (1994).

with B.M. Brown, W.D. Evans und V.G. Kirby: Safe numerical bounds for the Titchmarsh-Weyl m-function.
Mathematical Proceedings of the Cambridge Philosophical Society 113, 583-599 (1993).

An Existence and Inclusion Method for Two-Point Boundary Value Problems with Turning Points.
ZAMM 74, 615 - 623 (1994).

Numerical Existence Proofs and Explicit Bounds for Solutions of Nonlinear Elliptic Boundary Value Problems.
Computing 49, 25-44 (1992).

Characterization and Computation of Nash-Equilibria for Auctions with Incomplete Information.
International Journal of Game Theory 20, 393-418 (1992).

Continuous Sealed-bid Auctions with Asymmetric Incomplete Information.
Methods of Operations Research 63, 303-312.

Bounds for eigenvalues of second-order elliptic differential operators.
ZAMP 42, 848 - 863 (1991).

Existence proofs in combination with error bounds for approximate solutions of weakly nonlinear second-order elliptic boundary value problems.
ZAMM 71, T 660 - T 662 (1991).

Explicit H2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems.
J. Math. Anal. Appl. 165, 36-61 (1992).

Computer-assisted existence proofs for two-point boundary value problems.
Computing 46, 19-34 (1991).

Verified existence and inclusion results for two-point boundary value problems.
IMACS Annals on Computing and Applied Mathematics, Baltzer, Vol. 7, 341-355 (1990).

with M. Göhlen und J. Schröder: A Programmed Algorithm for Existence Proofs for Two-Point Boundary Value Problems.
Computing 44, 91-132 (1990).

Existence and inclusion for two-point boundary value problems by numerical means.
In: Computational Ordinary Differential Equations (Eds.: J. R. Cash, I. Gladwell), Clarendon Press, Oxford, 415-423 (1992).

Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method.
ZAMP 41, 205-226 (1990).

Shape-Invariant Bounds for Reaction-Diffusion Systems with Unequal Diffusion Coeficients.
J. Differential Equations 73, 82-103 (1988).

Pointwise Bounds for Linear Reaction-Diffusion Systems and an Extension to Nonlinear Problems.
J. Math. Anal. Appl. 135, 88-111 (1988).