Algebraic investigation of a nonlinear and singular Dirichlet problem. (Analyse algébrique d’un problème de Dirichlet non linéaire et singulier.) (French) Zbl 0959.46028

Marti, J.-A.; Nuiro, S. P.

Topol. Methods Nonlinear Anal. 13, No. 2, 301-311 (1999).

Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^d\) with boundary of class \(C^\infty\) and suppose \(\theta\in C^\infty(\overline\Omega\times \mathbb{R})\) satisfies certain growth conditions. The authors first solve the regular Dirichlet problem \[ -\Delta u(x)+ \theta(x, u(x))= f(x), \] when \(x\in\Omega\), \(u(x)= g(x)\) when \(x\in\partial\Omega\) for a unique \(u\in C^\infty(\overline\Omega)\), where \(f\in C^\infty(\overline\Omega)\) and \(g\in C^\infty(\partial\Omega)\) are given. Their main result develops an idea of A. Delcroix and D. Scarpalezos [Integral Transforms, Spec. Funct. 6, No. 1-4, 181-190 (1998; Zbl 0919.46027)] in order to solve the Dirichlet problem when \(f\), \(g\), and \(u\) belong to a class of generalized functions derived from the algebraic structure of a quotient ring and the topological structure of a semi-normed algebra.

Reviewer: J.V.Whittaker (Vancouver)

MSC:

46F30	Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
46E25	Rings and algebras of continuous, differentiable or analytic functions
46E10	Topological linear spaces of continuous, differentiable or analytic functions
47F05	General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35D05	Existence of generalized solutions of PDE (MSC2000)

Keywords:

nonlinear and singular Dirichlet problem; Goursat algebra; Sobolev algebra; irregular data; regular Dirichlet problem; generalized functions; quotient ring; semi-normed algebra