Mixed boundary value problems for elliptic and parabolic differential equations of second order
The author considers mixed boundary value problems either for elliptic or parabolic linear differential operators of second order. He is concerned with the existence, uniqueness and regularity of the solutions in fairly general circumstances. For instance, say that B is a first order oblique differential operator on a relatively open subset S of the boundary \(\partial \Omega\) of \(\Omega\) (a bounded connected open subset of \({\mathbb{R}}^ n)\) and consider the problem \[ Lu=f\quad in\quad \Omega;\quad Bu=g\quad on\quad S;\quad u=h\quad on\quad \partial \Omega \setminus S. \] The author's results include the case where \(\bar S\) and \(\partial \Omega \setminus S\) meet at arbitrary angles provided that \(\partial \Omega \setminus S\) is Lipschitz. The author proves the above result by using techniques of super and subsolutions (known as Perron's method) with appropriate adaptations due to the same author [Adv. Math. 55, 161-172 (1985; Zbl 0567.35037)].
first order oblique differential operator
scientific article; zbMATH DE number 3985741
scientific article