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The L P Approach to the Dirichlet Problem

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Il principio di minimo e sue applicazioni alle equazioni funzionali

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 17))

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Abstract

In this paper we present a L p approach to the Dirichlet problem and to related regularity problems for higher order elliptic equations. Although this approach is not as simple as the well known Hilbert space approach developed by Yishik [32] Garding [14], Browder [6 ; 7], Friedrichs [12], Mor-rey [22], Nirenberg [23], Lions [18] and others, it has the advantage of a greater generality. Thus, for example, we shall be able to treat the non-homogeneous Dirichlet problem in a much more general situation not restricted to solutions having a finite Dirichlet integral (in this connection see Ma-genes-Stampacchia [19, § 9] and the recent paper of Miranda [20]). The method is also applicable to elliptic operators which are not necessarily strongly elliptic. We remark further that the same method could be used to solve a general class of boundary value problems. This will be done in a subsequent paper where we shall also derive L p integral inequalities for a system of differential operators acting on functions satisfying general boundary conditions, simular to the « coercive » L 2 inequalities derived by Aronszajn [4] Agmon [2] and Schechter [25].

Recently Schechter [26 ; 27] presented a Hilbert space approach to general boundary value problems including the Dirichlet problem for non-stron-gly elliptic equations. His method is based on the L 2 estimates of Agmon-Douglis-Nirenberg [3] (see also [2 ; 25]) and on known L 2 regularity theorems. Our L p method which utilizes new regularity theorems is quite different and the results we obtain are stronger in various respects. Other existence results utilizing the continuity method were given by Agmon-Douglis-Ni-renberg [3].

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Authors
  1. Shmuel Agmon
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S. Faedo

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Agmon, S. (2011). The L P Approach to the Dirichlet Problem. In: Faedo, S. (eds) Il principio di minimo e sue applicazioni alle equazioni funzionali. C.I.M.E. Summer Schools, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10926-3_2

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