Potential space estimates for Green potentials in convex domains. (English) Zbl 0789.35047

Fromm, Stephen J.

Proc. Am. Math. Soc. 119, No. 1, 225-231 (1993).

The author proves some (1,1) weak type bounds for the operators \[ f\mapsto \int\nabla_ x \nabla_ x G(x,y)f(y)dy \qquad\text{and} \qquad f\mapsto\int \nabla_ x \nabla_ y G(x,y)f(y)dy, \] \(G\) being the Green operator associated to the Dirichlet problem for the Poisson equation in a bounded and convex domain in \(\mathbb{R}^ n\). The smoothing properties of the Green operator in the Bessel potential space is then studied, and an application to the restriction of Sobolev spaces is included.

Reviewer: I.Vrabie (Iaşi)

MSC:

35J25	Boundary value problems for second-order elliptic equations
35B65	Smoothness and regularity of solutions to PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

potential space estimate; Green operator; Dirichlet problem for the Poisson equation

References:

[1]	Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2]	Vilhelm Adolfsson, \?^{\?}-integrability of the second order derivatives of Green potentials in convex domains, Pacific J. Math. 159 (1993), no. 2, 201 – 225. · Zbl 0738.35017
[3]	Vilhelm Adolfsson, \?²-integrability of second-order derivatives for Poisson’s equation in nonsmooth domains, Math. Scand. 70 (1992), no. 1, 146 – 160. · Zbl 0761.35018
[4]	P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060
[5]	Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303 – 342. · Zbl 0485.35031
[6]	David Jerison and Carlos E. Kenig, The functional calculus for the Laplacian on Lipschitz domains, Journées ”Équations aux Dérivées Partielles” (Saint Jean de Monts, 1989) École Polytech., Palaiseau, 1989, pp. Exp. No. IV, 10. · Zbl 0725.35021
[7]	Peter Sjögren, Weak \?\(_{1}\) characterizations of Poisson integrals, Green potentials and \?^{\?} spaces, Trans. Amer. Math. Soc. 233 (1977), 179 – 196. · Zbl 0332.31003
[8]	Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[9]	H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978.