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Boundary value problems and Hardy spaces associated to the Helmholtz equation in Lipschitz domains. (English) Zbl 0858.35027
Summary: We introduce and discuss, in the Clifford algebra framework, certain Hardy-like spaces which are well suited for the study of the Helmholtz equation $\Delta u+k^2u=0$ in Lipschitz domains of $\bbfR^{n+1}$. In particular, in the second part of the paper, these results are used in connection with the classical boundary value problems for the Helmholtz equation in Lipschitz domains in arbitrary space dimensions. In this setting, existence, uniqueness, and optimal estimates are obtained by inverting the corresponding layer potential operators on $L^p$ for sharp ranges of $p$’s. Also, a detailed discussion of the Helmholtz eigenvalues of Lipschitz domains is presented.

35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
31B10Integral representations of harmonic functions (higher-dimensional)
42B30$H^p$-spaces (Fourier analysis)
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