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**On closed boundary value problems for equations of mixed elliptic-hyperbolic type.**
*(English)*
Zbl 1125.35066

The purpose of this work is to examine the question of well-posedness of boundary value problems for linear partial differential equations of second order of the form
\[
\begin{aligned} Lu & =K(y)u_{xx}+u_{yy} =f \text{ in }\Omega,\tag{1}\\ Bu & =g\quad\text{ on } \partial\Omega, \tag{2}\end{aligned}
\]
where \(K\in C^1(\mathbb R^2)\) satisfies
\[
K(0)=0\text{ and }yK(y)>0\text{ for }y\neq 0.\tag{3}
\]
\(\Omega\) is a bounded, open, and connected subset of \(\mathbb R^2\) with piecewise \(C^1\) boundary, \(f\) and \(g\) are given functions, and \(B\) is some given boundary operator. It is assumed throughout that
\[
\Omega^\pm:=\Omega\cap \mathbb R_\pm^2\neq 0, \tag{4}
\]
so that equation (1) is of mixed elliptic-hyperbolic type. \(\Omega\) is called a mixed domain if (4) holds. Such a boundary value problem is called closed in the sense that the boundary condition (2) is imposed on the entire boundary as opposed to an open problem, in which (2) is imposed on a proper subset \(\Gamma\subset \partial\Omega\). For these equations results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet-conormal types are proved. Such problems are of interest for applications to transonic flow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the \(a-b-c\) integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In the important special case of the Tricomi equation some interior regularity results are given.

Reviewer: Elena Gavrilova (Sofia)

### MSC:

35M10 | PDEs of mixed type |

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\textit{D. Lupo} et al., Commun. Pure Appl. Math. 60, No. 9, 1319--1348 (2007; Zbl 1125.35066)

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### References:

[1] | Agmon, Comm Pure Appl Math 6 pp 455– (1953) |

[2] | Barros-Neto, Duke Math J 98 pp 465– (1999) |

[3] | Expansions in eigenfunctions of selfadjoint operators. Translations of Mathematical Monographs, 17. American Mathematical Society, Providence, R.I., 1968. |

[4] | Mathematical aspects of subsonic and transonic gas dynamics. Surveys in Applied Mathematics, 3. Wiley, New York; Chapman & Hall, London, 1958. · Zbl 0083.20501 |

[5] | Didenko, Ukrain Math J 25 pp 10– (1973) |

[6] | Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, R.I., 1998. |

[7] | Fichera, Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8) 50 pp 6– (1971) |

[8] | Friedrichs, Trans Amer Math Soc 55 pp 132– (1944) |

[9] | Friedrichs, Comm Pure Appl Math 11 pp 333– (1958) |

[10] | Germain, ONERA Publ 1952 (1952) |

[11] | Hörmander, Comm Pure Appl Math 14 pp 371– (1961) |

[12] | Linear partial differential operators, 4th printing. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 116. Springer, Berlin–New York, 1976. |

[13] | John, Amer J Math 63 pp 141– (1941) |

[14] | Lax, Comm Pure Appl Math 8 pp 615– (1955) |

[15] | Lax, Comm Pure Appl Math 13 pp 427– (1960) |

[16] | Lupo, Commun Contemp Math 2 pp 535– (2000) |

[17] | Lupo, Comm Pure Appl Math 56 pp 403– (2003) |

[18] | Morawetz, Proc Roy Soc London Ser A 236 pp 141– (1956) |

[19] | Morawetz, Comm Pure Appl Math 91011 pp 45– (1956) |

[20] | Morawetz, Comm Pure Appl Math 11 pp 315– (1958) |

[21] | Morawetz, Comm Pure Appl Math 23 pp 587– (1970) |

[22] | Morawetz, J Hyperbolic Differ Equ 1 pp 1– (2004) |

[23] | Payne, Houston J Math 23 pp 709– (1997) |

[24] | Picone, Rend Circ Mat Palermo 30 pp 349– (1910) |

[25] | Pilant, J Math Anal Appl 106 pp 321– (1985) |

[26] | Protter, J Rational Mech Anal 4 pp 721– (1955) |

[27] | Sarason, Comm Pure Appl Math 15 pp 237– (1962) |

[28] | Equations of mixed type. Translations of Mathematical Monographs, 51. American Mathematical Society, Providence, R.I., 1978. |

[29] | Locally convex spaces and linear partial differential equations. Die Grundlehren der mathematischen Wissenschaften, 146. Springer, New York, 1967. · doi:10.1007/978-3-642-87371-3 |

[30] | Tricomi, Atti Acad Naz Lincei Mem Cl Fis Mat Nat (5) 14 pp 134– (1923) |

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