In this note we present the solution in W 1,2of the Robin boundary value problem for the Laplacian on a Lipschitz domain ā„¦ āŠ‚ R nwith C`n- valued datum f āˆˆ L 2(bā„¦) (see (R) below). This work originates from [6], where we considered the case of scalar-valued datum f āˆˆ L p(bā„¦), 1 < p ā‰¤ 2. In the present context of the Cliļ¬€ord algebra C`n, the direct relationship between the Cliļ¬€ord derivatives of the single layer potential and left Cliļ¬€ord-Cauchy integral operators allows for a more uniļ¬ed and direct approach to the solution of the problem. Because we are choosing the Robin coeļ¬ƒcient b in the space L s(bā„¦) with s greater than the critical exponent n āˆ’ 1, the solution operator for the Robin problem turns out to be a compact perturbation of the solution operator of the Neumann problem. In this respect, the situation we present here bears a close aļ¬ƒnity with the classical study of the Neu- mann problem for C 1-domains (see [3]). The treatment of the critical exponent case (namely, b āˆˆ L nāˆ’1(bā„¦)) requires a diļ¬€erent approach, which has been developed in [6]. The structure of this paper is as follows. In sections 2 and 3 we describe and summarize the features of the Cliļ¬€ord algebras, the function spaces and the singular integral operators that are involved in this work. In Section 4 we present a simple proof of the L 2-solution of the Robin problem with non-critical Robin coeļ¬ƒcient, and we state without proof the corresponding result in L p, with critical Robin coeļ¬ƒcient.

The š’žš“n-valued Robin boundary value problem on Lipschitz domains in ā„n / Loredana Lanzani. - STAMPA. - 25:(2001), pp. 183-191. [10.1007/978-94-010-0862-4_18]

The š’žš“n-valued Robin boundary value problem on Lipschitz domains in ā„n.

Loredana Lanzani
2001

Abstract

In this note we present the solution in W 1,2of the Robin boundary value problem for the Laplacian on a Lipschitz domain ā„¦ āŠ‚ R nwith C`n- valued datum f āˆˆ L 2(bā„¦) (see (R) below). This work originates from [6], where we considered the case of scalar-valued datum f āˆˆ L p(bā„¦), 1 < p ā‰¤ 2. In the present context of the Cliļ¬€ord algebra C`n, the direct relationship between the Cliļ¬€ord derivatives of the single layer potential and left Cliļ¬€ord-Cauchy integral operators allows for a more uniļ¬ed and direct approach to the solution of the problem. Because we are choosing the Robin coeļ¬ƒcient b in the space L s(bā„¦) with s greater than the critical exponent n āˆ’ 1, the solution operator for the Robin problem turns out to be a compact perturbation of the solution operator of the Neumann problem. In this respect, the situation we present here bears a close aļ¬ƒnity with the classical study of the Neu- mann problem for C 1-domains (see [3]). The treatment of the critical exponent case (namely, b āˆˆ L nāˆ’1(bā„¦)) requires a diļ¬€erent approach, which has been developed in [6]. The structure of this paper is as follows. In sections 2 and 3 we describe and summarize the features of the Cliļ¬€ord algebras, the function spaces and the singular integral operators that are involved in this work. In Section 4 we present a simple proof of the L 2-solution of the Robin problem with non-critical Robin coeļ¬ƒcient, and we state without proof the corresponding result in L p, with critical Robin coeļ¬ƒcient.
2001
Clifford analysis and its applications
183
191
The ����n-valued Robin boundary value problem on Lipschitz domains in ℝn / Loredana Lanzani. - STAMPA. - 25:(2001), pp. 183-191. [10.1007/978-94-010-0862-4_18]
Loredana Lanzani
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/919475
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