Bergamasco, Adalberto P.; Petronilho, Gerson On the hypoellipticity of degenerate elliptic boundary value problems. (English) Zbl 0767.35114 J. Math. Anal. Appl. 171, No. 2, 407-417 (1992). The main result of the paper is theTheorem: Let \(U\subset\mathbb{R}^ n\) be a neighborhood of the origin and let \(T>0\). Let \[ P=(\partial_ t-\mu(t,x)D_ x+b(t,x))(\partial_ t +\widetilde\mu(t,x)D_ x+ \widetilde b(t,x)), \] where \(\mu\), \(\widetilde \mu\), \(b\), \(\widetilde b\) are smooth functions on \([0,T]\times \overline{U}\). Assume \(\mu\widetilde\mu>0\) when \(t>0\) and either \(\mu(t,x)=t^ k a(t,x)\), \(\widetilde\mu(t,x)=t^ l \widetilde a(t,x)\) with \(k,l\geq 4\) or \(\mu(t,x)=\psi(t)a(t,x)\), \(\widetilde\mu(t,x)=\widetilde\psi(t)\widetilde a(t,x)\), where \(a\widetilde a>0\), when \(t=0\) and \(\psi\), \(\widetilde\psi\) are arbitrary smooth functions. If \(u\in C^ \infty([0,T],{\mathcal D}'(U))\) satisfies \[ Pu=f\in C^ \infty([0,T]\times U), \qquad u\mid_{t=0}=u_ 0\in C^ \infty(U), \] then \(u\in C^ \infty([0,T]\times U)\) perhaps after shrinking \(T\), \(U\).The theory of pseudo-differential operators is used for proving this theorem. Reviewer: V.S.Rabinovich (Rostov-na-Donu) MSC: 35S05 Pseudodifferential operators as generalizations of partial differential operators 35J70 Degenerate elliptic equations 65H10 Numerical computation of solutions to systems of equations Keywords:smoothness of solutions up to the boundary PDFBibTeX XMLCite \textit{A. P. Bergamasco} and \textit{G. Petronilho}, J. Math. Anal. Appl. 171, No. 2, 407--417 (1992; Zbl 0767.35114) Full Text: DOI References: [1] Bergamasco, A. P.; Gerszonowicz, J. A.; Petronilho, G., On the regularity up to the boundary in the Dirichlet problem for degenerate elliptic equations, Trans. Amer. Math. Soc., 313, 317-329 (1989) · Zbl 0681.35021 [2] Hormander, L., Pseudo-differential operators and hypoelliptic equations, (Proceedings Sympos. Pure Math, Vol. 10 (1966), Amer. Math. Soc.,: Amer. Math. Soc., Providence, RI), 138-183 · Zbl 0212.46601 [3] Treves, F., (Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1 (1980), Plenum: Plenum New York) · Zbl 0453.47027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.