Gustaf Gripenberg

Aalto University
School of Science and Technology
Department of Mathematics and Systems Analysis
P.O. Box 11100
FI-00076 Aalto
Finland
+358-9-47023025
Send email to: gustaf.gripenberg@aalto.fi

Research interests

• Main current interest:
  • Viscosity solutions to partial differential equations
  • Stochastic processes
• Less current interests:
  • Evolutionary equations
  • Wavelets

Publications

Monograph:

• Gripenberg, G. ; Londen, S.-O. ; Staffans, O. Volterra integral and functional equations. Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. xxii+701 pp. ISBN: 0-521-37289-5

Articles

1. Gripenberg, G. Extending infinity harmonic functions by rotation. Electron. J. Diff. Equ. 149 (2015), pp. 1-6.
2. Gripenberg, G. On the definition of the cone spectral radius. Proc. Amer. Math. Soc, 143 (2015), pp. 1617-1625.
3. Gripenberg, G. White and colored Gaussian noises as limits of sums of random dilations and translations of a single function. Elect. Comm. in Probab., 16 (2011), pp. 507-516. Abstract
   It is shown that a stochastic process obtained by taking random sums of dilations and translations of a given function converges to Gaussian white noise if a dilation parameter grows to infinity and that it converges to Gaussian colored noise if a scaling parameter for the translations grows to infinity. In particular, the question of when one obtains fractional Brownian motion by integrating this colored noise is studied.
4. Gripenberg, G.. Harnack's inequality for viscosity solutions to degenerate elliptic equations in two variables. Proc. Amer. Math. Soc., 137 (2009), pp. 3703-3708. Gripenberg, G. Abstract
   A Harnack inequality is established for viscosity solutions to the elliptic equation $F(\mathbf x,u,Du,D^2u)=0$ in two variables.
5. Gripenberg, G.. Boundary regularity for viscosity solutions to degenerate elliptic PDEs. J. Math. Anal. Appl. 352 (2009), pp. 175-183. Abstract
   Sufficient conditions are given for the solutions to the (fully nonlinear, degenerate) elliptic equation $F(\mathbf{ x},u,Du, D^2u)=0$ in $\Omega$ to satisfy $|u(\mathbf{x})-u(\mathbf{y})| \leq C|\mathbf{x}-\mathbf{y}|^\alpha$ for some $\alpha\in (0,1)$ when $\mathbf{x} \in \Omega$ and $\mathbf{y}\in \partial \Omega$.
6. Gripenberg, G. On the strong maximum principle for degenerate parabolic equations. J. Differential Equations 242 (2007), no. 1, 72--85. Abstract
   Variants of the strong maximum principle are established for subsolutions to degenerate parabolic equations for which the standard version of the strong maximum principle does not hold. The results are formulated for viscosity solutions.
7. Gripenberg, G. Generalized viscosity solutions of elliptic PDEs and boundary conditions. Electron. J. Differential Equations 2006, No. 43, 10 pp. Abstract
   Sufficient conditions are given for a generalized viscosity solution of an elliptic boundary value problem to satisfy the boundary values in the strong sense.
8. Gripenberg, G. Approximation by neural networks with a bounded number of nodes at each level. J. Approx. Theory 122 (2003), no. 2, 260--266. Abstract
   It is shown that the general approximation property of feed-forward multilayer perceptron networks can be achieved in networks where the number of nodes in each layer is bounded, but the number of layers grows to infinity. This is the case provided the node function is twice continuously differentiable and not linear.
9. Gripenberg, G. Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions. Proc. Amer. Math. Soc. 130 (2002), no. 12, 3651--3660. Abstract
   It is shown how one can get upper bounds for $|u-v|$ when $u$ and $v$ are the (viscosity) solutions of \begin{equation*} u_t - \alpha(D_x u) \Delta_x u = 0\quad\text{and}\quad v_t - \beta(D_x v) \Delta_x v = 0, \end{equation*} respectively, in $(0,\infty)\times \Omega$ with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form.
10. Cockburn, B.; Gripenberg, G.; Londen, S.-O. Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations. J. Differential Equations 170 (2001), no. 1, 180--187. Abstract
    This paper establishes an upper bound for $u-v$ where $u$ is a subsolution of $$ u_t + F(u,D_xu,D_x^2 u) = 0, $$ and $v$ is a supersolution of $$ v_t + G(v,D_x v,D_x^2 v) = 0, $$ in $(0,\infty)\times \Omega$ with Neumann bondary conditions and where $\Omega$ is an open convex set.
11. Gripenberg, G. Nonsmoothing in a single conservation law with memory. Electron. J. Differential Equations 2001, No. 8, 8 pp.
12. Gripenberg, G.; Clément, Ph.; Londen, S.-O. Smoothness in fractional evolution equations and conservation laws. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 1, 231--251.
13. Clément, Ph.; Gripenberg, G.; Londen, S.-O. Schauder estimates for equations with fractional derivatives. Trans. Amer. Math. Soc. 352 (2000), no. 5, 2239--2260.
14. Cockburn, B.; Gripenberg, G. Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations 151 (1999), no. 2, 231--251.
15. Gripenberg, G.; Londen, S.-O.; Prüss, J. On a fractional partial differential equation with dominating linear part. Math. Methods Appl. Sci. 20 (1997), no. 16, 1427--1448.
16. Gripenberg, G. A lower bound for the order parameter in the one-dimensional contact process. Stochastic Process. Appl. 63 (1996), no. 2, 211--219.
17. Cockburn, B.; Gripenberg, G.; Londen, S.-O. On convergence to entropy solutions of a single conservation law. J. Differential Equations 128 (1996), no. 1, 206--251.
18. Gripenberg, G.; Norros, Ilkka On the prediction of fractional Brownian motion. J. Appl. Probab. 33 (1996), no. 2, 400--410.
19. Gripenberg, G. On subdivision interpolation schemes. SIAM J. Math. Anal. 27 (1996), no. 2, 578--593.
20. Gripenberg, G. Computing the joint spectral radius. Linear Algebra Appl. 234 (1996), 43--60.
21. Gripenberg, G. Compensated compactness and one-dimensional elastodynamics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 2, 227--240.
22. Gripenberg, G.; Londen, S.-O. Fractional derivatives and smoothing in nonlinear conservation laws. Differential Integral Equations 8 (1995), no. 8, 1961--1976.
23. Gripenberg, G. Approximations of wavelet projections. Appl. Comput. Harmon. Anal. 2 (1995), no. 3, 257--264.
24. Gripenberg, G. A necessary and sufficient condition for the existence of a father wavelet. Studia Math. 114 (1995), no. 3, 207--226.
26. Gripenberg, G. Nonlinear Volterra equations of parabolic type due to singular kernels. J. Differential Equations 112 (1994), no. 1, 154--169.
27. Gripenberg, G. On the uniqueness and nonuniqueness of weak solutions of hyperbolic-parabolic Volterra equations. Differential Integral Equations 7 (1994), no. 2, 509--522.
28. Gripenberg, G. Weak solutions of hyperbolic-parabolic Volterra equations. Trans. Amer. Math. Soc. 343 (1994), no. 2, 675--694.
29. Gripenberg, G. Wavelet bases in $L^ p(\mathbb {R})$. Studia Math. 106 (1993), no. 2, 175--187.
30. Gripenberg, G. Stability of Volterra equations with measure kernels in Banach spaces. J. Math. Anal. Appl. 178 (1993), no. 1, 156--164.
31. Gripenberg, G. Unconditional bases of wavelets for Sobolev spaces. SIAM J. Math. Anal. 24 (1993), no. 4, 1030--1042.
32. Gripenberg, G. Global existence of solutions of Volterra integrodifferential equations of parabolic type. J. Differential Equations 102 (1993), no. 2, 382--390.
33. Gripenberg, G. Unimodality and viscoelastic pulse propagation. Quart. Appl. Math. 51 (1993), no. 1, 183--189.
34. Gripenberg, G. Confidence intervals for partial rank correlations. J. Amer. Statist. Assoc. 87 (1992), no. 418, 546--551. 62F25 (62G05)
35. Gripenberg, G. Thermostat systems having stable periodic solutions with short periods. SIAM J. Math. Anal. 22 (1991), no. 5, 1332--1348.
36. Gripenberg, G. On the uniqueness of solutions of Volterra equations. J. Integral Equations Appl. 2 (1990), no. 3, 421--430.
37. Gripenberg, G. Stability of periodic solutions in thermostat control. SIAM J. Math. Anal. 20 (1989), no. 6, 1459--1471.
38. Gripenberg, G. On periodic solutions of a thermostat equation. SIAM J. Math. Anal. 18 (1987), no. 3, 694--702.
39. Gripenberg, G. Asymptotic behaviour of resolvents of abstract Volterra equations. J. Math. Anal. Appl. 122 (1987), no. 2, 427--438.
40. Gripenberg, G. Volterra integro-differential equations with accretive nonlinearity. J. Differential Equations 60 (1985), no. 1, 57--79.
41. Gripenberg, G. Extinction in a model for the growth of a population subject to catastrophes. Stochastics 14 (1985), no. 2, 149--163.
42. Gripenberg, G. An eigenvalue problem for a Volterra integral operator with infinite delay. SIAM J. Math. Anal. 16 (1985), no. 3, 541--547.
43. Gripenberg, G. On a nonlinear integral equation modelling an epidemic in an age-structured population. J. Reine Angew. Math. 341 (1983), 54--67.
44. Gripenberg, G. An estimate for the solution of a Volterra equation describing an epidemic. Nonlinear Anal. 7 (1983), no. 2, 161--165.
45. Gripenberg, G. A stationary distribution for the growth of a population subject to random catastrophes. J. Math. Biol. 17 (1983), no. 3, 371--379.
46. Gripenberg, G. Stability of the expected value of the solution of a random Volterra integral equation. J. Integral Equations 5 (1983), no. 2, 159--173.
47. Gripenberg, G. Two Tauberian theorems for nonconvolution Volterra integral operators. Proc. Amer. Math. Soc. 89 (1983), no. 2, 219--225.
48. Gripenberg, G. The construction of the solution of an optimal control problem described by a Volterra integral equation. SIAM J. Control Optim. 21 (1983), no. 4, 582--597.
49. Gripenberg, G. Stability analysis of a distributed parameter model for the growth of micro-organisms. Hyperbolic partial differential equations. Comput. Math. Appl. 9 (1983), no. 3, 431--442.
50. Gripenberg, G. Stability of periodic solutions of some integral equations. J. Reine Angew. Math. 331 (1982), 16--31.
51. Gripenberg, G. Volterra integral operators and logarithmic convexity. Math. Scand. 50 (1982), no. 2, 209--220.
52. Gripenberg, G. Optimization of crystal growth with diffusion. Quart. Appl. Math. 40 (1982/83), no. 3, 297--310.
53. Gripenberg, G. Nonexistence of smooth solutions for shearing flows in a nonlinear viscoelastic fluid. SIAM J. Math. Anal. 13 (1982), no. 6, 954--961.
54. Gripenberg, G. Asymptotic estimates for resolvents of Volterra equations. J. Differential Equations 46 (1982), no. 2, 230--243.
55. Gripenberg, G. Decay estimates for resolvents of Volterra equations. J. Math. Anal. Appl. 85 (1982), no. 2, 473--487.
56. Gripenberg, G. Unique solutions of some Volterra integral equations. Math. Scand. 48 (1981), no. 1, 59--67.
57. Gripenberg, G. Asymptotic solutions of some nonlinear Volterra integral equations. SIAM J. Math. Anal. 12 (1981), no. 4, 595--602.
58. Gripenberg, G. Integrability of resolvents of systems of Volterra equations. SIAM J. Math. Anal. 12 (1981), no. 4, 585--594.
59. Gripenberg, G. On some epidemic models. Quart. Appl. Math. 39 (1981/82), no. 3, 317--327.
60. Gripenberg, G. On the behavior of solutions of Volterra equations with nonconvolution kernels. J. Integral Equations 3 (1981), no. 1, 83--92. 45D05
61. Gripenberg, G. On the convergence of solutions of Volterra equations to almost-periodic functions. Quart. Appl. Math. 39 (1981/82), no. 3, 363--373.
62. Gripenberg, G. A Tauberian problem for a Volterra integral operator. Proc. Amer. Math. Soc. 82 (1981), no. 4, 576--582.
63. Gripenberg, G. On the resolvents of Volterra equations with nonincreasing kernels. J. Math. Anal. Appl. 76 (1980), no. 1, 134--145.
64. Gripenberg, G. On some positive definite forms and Volterra integral operators. Applicable Anal. 11 (1980/81), no. 3, 211--222.
65. Gripenberg, G. Periodic solutions of an epidemic model. J. Math. Biol. 10 (1980), no. 3, 271--280.
66. Gripenberg, G. On Volterra equations of the first kind. Integral Equations Operator Theory 3 (1980), no. 4, 473--488.
67. Gripenberg, G. On the resolvents of nonconvolution Volterra kernels. Funkcial. Ekvac. 23 (1980), no. 1, 83--95.
68. Gripenberg, G. Stability problems for some nonlinear Volterra equations. J. Integral Equations 2 (1980), no. 3, 247--258.
69. Gripenberg, G. On nonlinear Volterra equations with nonintegrable kernels. SIAM J. Math. Anal. 11 (1980), no. 4, 668--682.
70. Gripenberg, G. On the asymptotic behavior of resolvents of Volterra equations. SIAM J. Math. Anal. 11 (1980), no. 4, 654--662.
71. Gripenberg, G. On a frequency domain condition used in the theory of Volterra equations. SIAM J. Math. Anal. 10 (1979), no. 4, 839--843.
72. Gripenberg, G. An abstract nonlinear Volterra equation. Israel J. Math. 34 (1979), no. 3, 198--212 (1980).
73. Gripenberg, G. On the asymptotic behaviour of solutions of nonlinear integral equations in Banach spaces. Math. Nachr. 92 (1979), 185--194.
74. Gripenberg, G. On the asymptotic behaviour of nonlinear contraction semigroups. Math. Scand. 44 (1979), no. 2, 385--397.
75. Gripenberg, G. On the boundedness of solutions of Volterra equations. Indiana Univ. Math. J. 28 (1979), no. 2, 279--290.
76. Gripenberg, G. On an abstract integral equation. SIAM J. Math. Anal. 10 (1979), no. 5, 1017--1021.
77. Gripenberg, G. On rapidly decaying resolvents of Volterra equations. J. Integral Equations 1 (1979), no. 3, 241--247.
78. Gripenberg, G. A Volterra equation with nonintegrable resolvent. Proc. Amer. Math. Soc. 73 (1979), no. 1, 57--60.
79. Gripenberg, G. On positive, nonincreasing resolvents of Volterra equations. J. Differential Equations 30 (1978), no. 3, 380--390.
80. Gripenberg, G. On some integral and integro-differential equations in a Hilbert space. Ann. Mat. Pura Appl. (4) 118 (1978), 181--198.
81. Gripenberg, G. On a nonlinear Volterra integral equation in a Banach space. J. Math. Anal. Appl. 66 (1978), no. 1, 207--219.
82. Gripenberg, G. An existence result for a nonlinear Volterra integral equation in a Hilbert space. SIAM J. Math. Anal. 9 (1978), no. 5, 793--805.
83. Gripenberg, G. Bounded solutions of a Volterra equation. J. Differential Equations 28 (1978), no. 1, 18--22.
84. Gripenberg, G. On the accretivity of the inverse of an accretive relation. Proc. Amer. Math. Soc. 64 (1977), no. 1, 91--92.

Other

1. Clément, Ph.; Gripenberg, G.; Londen, S.-O. Regularity properties of solutions of fractional evolution equations. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 235--246, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001.
2. Clément, Ph.; Gripenberg, G.; Högnäs, V. Some remarks on the method of sums. Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999), 125--134, CMS Conf. Proc., 29, Amer. Math. Soc., Providence, RI, 2000.
3. Clément, Ph.; Gripenberg, G.; Londen, S.-O. Hölder regularity for a linear fractional evolution equation. Topics in nonlinear analysis, 69--82, Progr. Nonlinear Differential Equations Appl., 35, Birkhäuser, Basel, 1999.
4. Gripenberg, G. Stability problems in thermostat control. Models of hysteresis (Trento, 1991), 20--29, Pitman Res. Notes Math. Ser., 286, Longman Sci. Tech., Harlow, 1993. 93C15 (34K35 93D05)
5. Gripenberg, G. Optimal control and integral equations. Differential equations (Xanthi, 1987), 283--290, Lecture Notes in Pure and Appl. Math., 118, Dekker, New York, 1989.
6. Gripenberg, G. Stationary distributions for populations subject to random catastrophes. Mathematics in biology and medicine (Bari, 1983), 60--65, Lecture Notes in Biomath., 57, Springer, Berlin, 1985. 92A15