# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${\Delta }u+{k}^{2}u=0$ in Lipschitz domains of ${ℝ}^{n+1}$. 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.
##### MSC:
 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 31B10 Integral representations of harmonic functions (higher-dimensional) 42B30 ${H}^{p}$-spaces (Fourier analysis)