On local solvability of pseudo-differential equations

Menikoff, A.

doi:10.1090/S0002-9939-1974-0338844-1

On local solvability of pseudo-differential equations

Author: A. Menikoff
Journal: Proc. Amer. Math. Soc. 43 (1974), 149-154
MSC: Primary 47G05; Secondary 35S05
DOI: https://doi.org/10.1090/S0002-9939-1974-0338844-1
MathSciNet review: 0338844
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Abstract: A sufficient condition for the local solvability of the equation ${u_t} - A(x,t,{D_x})u = f(x,t)$ is proved, where is a first order pseudo-differential operator with real symbol. This is a special case of the local solvability conjecture of Nirenberg and Treves.

[1] Richard Beals and Charles Fefferman, On local solvability of linear partial differential equations, Ann. of Math. (2) 97 (1973), 482–498. MR 352746, https://doi.org/10.2307/1970832
[2] Lars Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. (2) 83 (1966), 129–209. MR 233064, https://doi.org/10.2307/1970473
[3] P. D. Lax and L. Nirenberg, On stability for difference schemes: A sharp form of Gȧrding’s inequality, Comm. Pure Appl. Math. 19 (1966), 473–492. MR 206534, https://doi.org/10.1002/cpa.3160190409
[4] Louis Nirenberg and François Trèves, On local solvability of linear partial differential equations. I. Necessary conditions, Comm. Pure Appl. Math. 23 (1970), 1–38. MR 264470, https://doi.org/10.1002/cpa.3160230102
[5] L. Nirenberg and J. F. Treves, Remarks on the solvability of linear equations of evolution, Symposia Mathematica, Vol. VII (Convegno sulle Problemi di Evoluzione, INDAM, Rome, 1970) Academic Press, London, 1971, pp. 325–338. MR 0333846