Abstract
The purpose of these lectures is to present a rather elementary and selfcontained approach to the weak first and second boundary value problem for the Laplacian in Lq where 1 < q < ∞. These problems are basic for a lot of applications in mathematical physics, like as e.g. Stokes’ problem. From the viewpoint of applications it is necessary to consider as well bounded as exterior domains. Our approach rests on two variational inequalities in Lq and a type of regularity argument. The results presented here are part of a joint work with H. Sohr (Paderborn/FRG) [9].
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References
Alt, H.W.: Lineare Funktionalanalysis. Berlin, Heidelberg, New York: Springer 1985.
Fujiwara, D. and Morimoto, H.: An L,.-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo, Sec. I 24, 685–700 (1977).
Galdi, G.P. and Simader, C.G.: Existence, uniqueness and LQ-estimates for the Stokes problem in an exterior domain. To appear in: Arch. Ratl. Mech. Anal.
Gilbarg, D. and Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1977.
Kufner, A., John, O. and Fucik, S.: Function spaces. Prague: Academia 1977.
Necas, J.: Les méthodes directes en théorie des équations elliptiques. Paris-Prague: Masson, Academia 1967.
Simader, C.G.: On Dirichlet’s boundary value problem. Lecture Notes in Math. 268, Springer-Verlag 1972.
Simader, C.G. and Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in L4-spaces for bounded and exterior domains. To appear.
Simader, C.G. and Sohr, H.: The weak Dirichlet and Neumann problem in LQ for second order elliptic equations in bounded and exterior problems. To appear as a book.
Stein, E.M.: Singular integrals and differentialbility properties of functions. Princeton: Princeton University Press 1970.
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Simader, C.G. (1990). The weak Dirichlet and Neumann problem for the Laplacian in Lq for bounded and exterior domains. Applications.. In: Krbec, M., Kufner, A., Opic, B., Rákosník, J. (eds) Nonlinear Analysis, Function Spaces and Applications Vol. 4. TEUBNER-TEXTE zur Mathematik. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-01272-6_7
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DOI: https://doi.org/10.1007/978-3-663-01272-6_7
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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