Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia Chinnì; Paolo Cubiotti

Annales Polonici Mathematici, Tome 63 (1996), p. 67-80 / Harvested from The Polish Digital Mathematics Library

Résumé

Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_{t}^{k} u + \sum_{j = 0}^{k - 1} \sum_{| α | \leq m} A_{j, α} (D_{t}^{j} D_{x}^{α} u) = f$ in $ℝ^{n + 1}$ , ⎨ ⎩ $D_{t}^{j} u (0, x) = φ_{j} (x)$ in $ℝ^{n}$ , j=0,...,k-1, where each $A_{j, α}$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j, α}$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u \in C^{\infty} (ℝ^{n + 1}, E)$ whose derivatives are equi-bounded on each bounded subset of $ℝ^{n + 1}$ .

Publié le : 1996-01-01

Zbl 0869.35109

EUDML-ID : urn:eudml:doc:269972

@article{bwmeta1.element.bwnjournal-article-apmv65z1p67bwm,
     author = {Antonia Chinn\`\i\ and Paolo Cubiotti},
     title = {Partial differential equations in Banach spaces involving nilpotent linear operators},
     journal = {Annales Polonici Mathematici},
     volume = {63},
     year = {1996},
     pages = {67-80},
     zbl = {0869.35109},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv65z1p67bwm}
}

Antonia Chinnì; Paolo Cubiotti. Partial differential equations in Banach spaces involving nilpotent linear operators. Annales Polonici Mathematici, Tome 63 (1996) pp. 67-80. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv65z1p67bwm/

Bibliographie

[000] [1] L. Cattabriga, On the surjectivity of differential polynomials on Gevrey spaces, Rend. Sem. Mat. Univ. Politec. Torino (1983), special issue on ``Linear partial and pseudodifferential operators'', 81-89. | Zbl 0561.35008

[001] [2] L. Cattabriga and E. De Giorgi, 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.

[002] [3] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 271-355.

[003] [4] B. Ricceri, On the well-posedness of the Cauchy problem for a class of linear partial differential equations of infinite order in Banach spaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991), 623-640. | Zbl 0810.35169

[004] [5] F. Trèves, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966. | Zbl 0164.40602