In this note we present the solution in W 1,2of the Robin boundary value problem for the Laplacian on a Lipschitz domain Ω ⊂ R nwith C`n- valued datum f ∈ L 2(bΩ) (see (R) below). This work originates from [6], where we considered the case of scalar-valued datum f ∈ L p(bΩ), 1 < p ≤ 2. In the present context of the Clifford algebra C`n, the direct relationship between the Clifford derivatives of the single layer potential and left Clifford-Cauchy integral operators allows for a more unified and direct approach to the solution of the problem. Because we are choosing the Robin coefficient b in the space L s(bΩ) with s greater than the critical exponent n − 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 affinity with the classical study of the Neu- mann problem for C 1-domains (see [3]). The treatment of the critical exponent case (namely, b ∈ L n−1(bΩ)) requires a different 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 Clifford 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 coefficient, and we state without proof the corresponding result in L p, with critical Robin coefficient.

The 𝒞𝓁n-valued Robin boundary value problem on Lipschitz domains in ℝn.

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 Ω ⊂ R nwith C`n- valued datum f ∈ L 2(bΩ) (see (R) below). This work originates from [6], where we considered the case of scalar-valued datum f ∈ L p(bΩ), 1 < p ≤ 2. In the present context of the Clifford algebra C`n, the direct relationship between the Clifford derivatives of the single layer potential and left Clifford-Cauchy integral operators allows for a more unified and direct approach to the solution of the problem. Because we are choosing the Robin coefficient b in the space L s(bΩ) with s greater than the critical exponent n − 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 affinity with the classical study of the Neu- mann problem for C 1-domains (see [3]). The treatment of the critical exponent case (namely, b ∈ L n−1(bΩ)) requires a different 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 Clifford 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 coefficient, and we state without proof the corresponding result in L p, with critical Robin coefficient.
2001
Clifford analysis and its applications
183
191
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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