Publications of the Research group for Analysis of Partial Differential Equations

  1. Publications

    1. Preprints

      • Simon Eberle, Alession Figalli, and Georg S. Weiss. Complete classification of global solutions to the obstacle problem. Accepted for publication in the Annals of Mathematics, 2024. [ Ann. Math. ]
      • Anthony Salib and Georg S. Weiss. A singular perturbation approach to the Dirichlet-area minimisation problem. Submitted 2024. [ arXiv ]
      • Anthony Salib and Georg S. Weiss. Classification of global solutions to the obstacle problem in the plane. Submitted 2024. [ arXiv ]
      • Simon Eberle, Arnulf Jentzen, Adrian Riekert, and Georg S. Weiss. Normalized gradient flow optimization in the training of ReLU artificial neural networks. Submitted 2022. [ arXiv ]

      Published

      56 Simon Eberle, Henrik Shahgholian and Georg S. Weiss. The structure of the regular part of the free boundary close to singularities in the obstacle problem. Journal of Differential Equations, 377, 2023. [ Journal of Differential Equations ]
      55 Simon Eberle, Henrik Shahgholian and Georg S. Weiss. On global solutions of the obstacle problem. Duke Mathematical Journal, 172, 2023. [ Duke Mathematical Journal ]
      54 Simon Eberle, Arnulf Jentzen, Adrian Riekert, and Georg S. Weiss. Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation. Electronic Research Archive. Special Issue on Dynamics, Control and Applications of Discontinuous Differential Systems, 31, 2023 [ AIMS Press ]
      53 Simon Eberle. A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities. Journal of Differential Equations, 265, 2018. [ DOI ]
      52 Simon Eberle. A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities in a channel with decreasing cross section. Nonlinear Analysis. Theory, Methods & Application, 172, 2018. [ DOI ]
      51 Simon Eberle. Front blocking versus propagation in the presence of drift disturbance in the direction of propagation. Nonlinear Analysis. Theory, Methods & Application, 176, 2018. [  DOI ]
      50 Simon Eberle, Barbara Niethammer and André Schlichting. Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation. Nonlinear Analysis. Theory, Methods & Application, 158, 2017. [ DOI ]
      49 Gohar Aleksanyan. Optimal regularity in the optimal switching problem. Annales de l'Institut Henri Poincaré Analyse non linéaire, 33(6), 2016. [ DOI ]
      48 Simon Eberle. Explicit formulas for homogenization limits in certain non-periodic problems including ramified domains. Zeitschrift für Angewandte Mathematik und Mechanik, 96(10), 2016. [ DOI ]
      47 Mariana Smit Vega Garcia, Eugen Vărvărucă and Georg S. Weiss. Singularities in axisymmetric free boundaries for electrohydrodynamic equations. Archive for Rational Mechanics and Analysis, 222(2), 2016. [ DOI ]
      46 John Andersson et al. Equilibrium points of a singular cooperative system with free boundary. Advances in Mathematics, 280, 2015. [ DOI ]
      45
      Eugen Vărvărucă and Georg S. Weiss. Singularities of steady axisymmetric free surface flows with gravity. Communications on Pure and Applied Mathematics, 67(8), 2014. [ DOI ]
      44 John Andersson, Henrik Shahgholian and Georg S. Weiss. The singular set of higher dimensional unstable obstacle type problems. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 24(1), 2013. [ DOI ]
      43 Peter V. Gordon and Georg S. Weiss. Convective combustion in porous media: singular limit of high activation energy. Nonlinearity, 26(1), 2013. [ DOI ]
      42 John Andersson, Henrik Shahgholian and Georg S. Weiss. Double obstacle problems with obstacles given by non-C2 Hamilton-Jacobi equations. Archive for Rational Mechanics and Analysis, 206(3), 2012. [ DOI ]
      41 John Andersson, Henrik Shahgholian and Georg S. Weiss. On the singularities of a free boundary through Fourier expansion. Inventiones mathematicae, 187(3), 2012. [ DOI ]
      40 Sagun Chanillo and Georg S. Weiss. A remark on the geometry of uniformly rotating stars. Journal of Differential Equations, 253(2), 2012. [ DOI ]
      39 Eugen Vărvărucă and Georg S. Weiss. The Stokes conjecture for waves with vorticity. Annales de l'Institut Henri Poincaré Analyse non linéaire, 29(6), 2012. [ DOI ]
      38 Georg S. Weiss and Guanghui Zhang. A free boundary approach to two-dimensional steady capillary gravity water waves. Archive for Rational Mechanics and Analysis, 203(3), 2012. [ DOI ]
      37 Georg S. Weiss and Guanghui Zhang. The second variation of the stream function energy of water waves with vorticity. Journal of Differential Equations, 253(9), 2012. [ DOI ]
      36 Eugen Varvaruca and Georg S. Weiss. A geometric approach to generalized Stokes conjectures. Acta Mathetmatica, 206(2), 2011. [ DOI ]
      35

      John Andersson, Henrik Shahgholian and Georg S. Weiss. Regularity below the C2 threshold for a torsion problem, based on regularity for Hamilton-Jacobi equations. Nonlinear partial differential equations and related topics. American Mathematical Society Translations, 229, 2010. [ DOI ]
      34 John Andersson, Henrik Shahgholian and Georg S. Weiss. Uniform regularity close to cross singularities in an unstable free boundary problem. Communications in Mathematical Physics, 296(1), 2010). [ DOI ]
      33 Georg S. Weiss and Guanghui Zhang. Existence of a degenerate singularity in the high activation energy limit of a reaction-diffusion equation. Communications in Partial Differential Equations 35(1), 2010. [ DOI ]
      32 John Andersson and Georg S. Weiss. A parabolic free boundary problem with Bernoulli type condition on the free boundary. Journal für die reine und angewandte Mathematik, 627, 2009. [ DOI ]
      31 Régis Monneau and Georg S. Weiss. Pulsating traveling waves in the singular limit of a reaction-diffusion system in solid combustion. Annales de l'Institut Henri Poincaré Analyse non linéaire, 26(4), 2009. [ DOI ]
      30 Henrik Shahgholian, Nina Uraltseva and Georg S. Weiss. A parabolic two-phase obstacle-like equation. Advances in Mathematics, 221(3), 2009. [ DOI ]
      29 Régis Monneau and Georg S. Weiss. An unstable elliptic free boundary problem arising in solid combustion. Duke Mathematical Journal, 136(2), 2007. [ DOI ]
      28 Régis Monneau and Georg S. Weiss. Self-propagating high temperature synthesis (SHS) in the high activation energy regime. Acta Math. Univ. Comenian. (N.S.), 76(1), 2007. [ PDF ]
      27

      Henrik Shahgholian, Nina Uraltseva and Georg S. Weiss. The two-phase membrane problem--regularity of the free boundaries in higher dimensions. International Mathematics Research Notices, 2007, 2007. [ DOI ]
      26 Henrik Shahgholian and Georg S. Weiss. Aleksandrov and Kelvin reflection and the regularity of free boundaries. Free boundary problems. Theory and Applications, 154, 2007. [ DOI ]
      25 John Andersson and Georg S. Weiss. Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem. Journal of Differential Equations, 228(2), 2006. [ DOI ]
      24 Henrik Shahgholian and Georg S. Weiss. The two-phase membrane problem--an intersection-comparison approach to the regularity at branch points. Advanced Mathematics, 205(2), 2006. [ DOI ]
      23 Georg S. Weiss. Regularity in free boundary problems. Selected Papers on Differential Equations and Analysis. American Mathematical Society Translations, 215, 2005. [ DOI ]
      22 Henrik Shahgholian, Nina Uraltseva and Georg S. Weiss. Global solutions of an obstacle-problem-like equation with two phases. Monatshefte Mathematik, 142(1-2), 2004. [ DOI ]
      21 Georg S. Weiss. A parabolic free boundary problem with double pinning. Nonlinear Analysis, 57(2), 2004. [ DOI ]
      20 Georg S. Weiss. Boundary monotonicity formulae and applications to free boundary problems. I. The elliptic case. Electronic Journal of Differential Equations, 44, 2004. [ PDF ]
      19 Hi Jun Choe and Georg S. Weiss. A semilinear parabolic equation with free boundary. Indiana University Mathematical Journal, 52(1), 2003. [ DOI ]
      18 Georg S. Weiss. A singular limit arising in combustion theory: fine properties of the free boundary. Calculus of Variations and Partial Differential Equations, 17(3), 2003. [ DOI ]
      17 Georg S. Weiss. Regularity in free boundary problems. Sugaku, 54(3), 2002. [ PDF ] (Japanese)
      16 Georg S. Weiss. A singular limit arising in combustion theory: fine properties of the free boundary. International Conference on Reaction-Diffusion Systems: Theory and Applications, 1249, pp.126-132, 2001.
      15 Georg S. Weiss. A gradient flow approach to a free boundary problem with volume constraint. Numerical solution of partial differential equations and related topics, 1198, 2001. [ PDF ]
      14 Georg S. Weiss. An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary. Interfaces Free Bound, 3(2), 2001. [ DOI ]
      13 Georg S. Weiss. A singular limit arising in combustion theory: identification of the limit. Nonlinear diffusive systems--dynamics and asymptotic analysis, 1178, 2000. [ PDF ]
      12 Georg S. Weiss. The free boundary of a thermal wave in a strongly absorbing medium. Journal of Differential Equations, 160(2), 2000. [ DOI ]
      11 Georg S. Weiss. A homogeneity improvement approach to the heat equation with strong absorption. Variational problems and related topics, 1076, 1999. [ PDF ]
      10 Georg S. Weiss. On the two-phase obstacle problem. Variational problems and related topics, 1117, 1999. [ PDF ]
      9 Georg S. Weiss. Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems. SIAM Journal on Mathematical Analysis, 30(3), 1999. [ DOI ]
      8 Georg S. Weiss. A homogeneity improvement approach to the obstacle problem. Inventiones mathematicae, 138(1), 1999. [ DOI ]
      7 Georg S. Weiss. Partial regularity for a minimum problem with free boundary. The Journal Geometric Analysis, 9(2), 1999. [ DOI ]
      6 Georg S. Weiss. Partial regularity for weak solutions of an elliptic free boundary problem. Communications in Partial Differential Equations 23(3&4), 1998. [ DOI ]
      5 Georg S. Weiss. Structural properties of a semilinear parabolic equation with free boundary--regularity of singular lines. Proceedings of the International Conference on Asymptotics in Nonlinear Diffusive Systems, Tohoku Mathematical Publications, 8, pp.180-197, 1998.
      4 Georg S. Weiss and Yasumasa Nishiura. The singular limit of the Cahn-Hilliard equation with a nonlocal term. Variational problems and related topics, 951, 1996. [ PDF ]
      3 Georg S. Weiss. Partial regularity for electrochemical machining with threshold current. Nonlinear evolution equations and their applications, 966, 1996. [ PDF ]
      2 Georg S. Weiss. A free boundary problem for non-radial-symmetric quasi-linear elliptic equations. Advances in Applied Mathematics, 5(2), 1995, pp. 497–555.
      1 Georg S. Weiss. Shape optimization with respect to the boundary condition of elliptic-parabolic systems. Advances in Applied Mathematics, 5(2), 1995, pp. 717–741.