A partial differential operator which is surjective on Gevrey classes Γd(ℝ³) with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6
Braun, Rüdiger
Studia Mathematica, Tome 104 (1993), p. 157-169 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

It is shown that the partial differential operator P(D)=∂⁴/∂x⁴-∂²/∂y²+i∂/∂z:Γd(ℝ³)→Γd(ℝ³) is surjective if 1 ≤ d < 2 or d ≥ 6 and not surjective for 2 ≤ d < 6.

Publié le : 1993-01-01
EUDML-ID : urn:eudml:doc:216027 
@article{bwmeta1.element.bwnjournal-article-smv107i2p157bwm,
     author = {R\"udiger Braun},
     title = {A partial differential operator which is surjective on Gevrey classes $$\Gamma$^{d}($\mathbb{R}$$^3$)$ with 1 $\leq$ d < 2 and d $\geq$ 6 but not for 2 $\leq$ d < 6},
     journal = {Studia Mathematica},
     volume = {104},
     year = {1993},
     pages = {157-169},
     zbl = {0810.35008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv107i2p157bwm}
}

Braun, Rüdiger. A partial differential operator which is surjective on Gevrey classes $Γ^{d}(ℝ³)$ with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6. Studia Mathematica, Tome 104 (1993) pp. 157-169. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv107i2p157bwm/

[00000] [1] L. V. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973.

[00001] [2] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 (1990), 206-237. | Zbl 0735.46022

[00002] [3] R. W. Braun, R. Meise and D. Vogt, Applications of the projective limit functor to convolution and partial differential equations, in: Advances in the Theory of Fréchet Spaces, Proc. Istanbul 1987, T. Terzioğlu (ed.), NATO Adv. Sci. Inst. Ser. C 287, Kluwer, 1989, 29-46.

[00003] [4] R. W. Braun, R. Meise and D. Vogt, Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type on ℝN, preprint. | Zbl 0848.35023

[00004] [5] L. Cattabriga, Solutions in Gevrey spaces of partial differential equations with constant coefficients, in: Analytic Solutions of Partial Differential Equations, Proc. Trento 1981, L. Cattabriga (ed.), Astérisque 89/90 (1981), 129-151. | Zbl 0496.35018

[00005] [6] L. Cattabriga, On the surjectivity of differential polynomials on Gevrey spaces, in: Atti del Convegno: 'Linear Partial and Pseudodifferential Operators', Rend. Sem. Mat. Univ. Politec. Torino, fascicolo speziale, 1983, 81-89.

[00006] [7] E. De Giorgi e L. Cattabriga, Una dimostrazione diretta dell'esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti, Boll. Un. Mat. Ital. (4) 4 (1971), 1015-1027.

[00007] [8] L. Hörmander, On the existence of real-analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-183. | Zbl 0282.35015

[00008] [9] R. Meise, B. A. Taylor and D. Vogt, Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier (Grenoble) 40 (1990), 619-655. | Zbl 0703.46025

[00009] [10] L. C. Piccinini, Non surjectivity of the Cauchy-Riemann operator on the space of the analytic functions on Rn. Generalization to the parabolic operators, Boll. Un. Mat. Ital. (4) 7 (1973), 12-28. | Zbl 0264.35003

[00010] [11] G. Zampieri, An application of the Fundamental Principle of Ehrenpreis to the existence of global Gevrey solutions of linear partial differential equations, ibid. (6) 5-B (1986), 361-392. | Zbl 0624.35011