Uniqueness Of Harmonic Functions Research Articles

Sets of uniqueness for harmonic and analytic functions and inverse problems for wave equations

Several examples of one-dimensional analytic sets of uniqueness for harmonic functions on the sphere in ℝ3 are given and some examples of analytic sets on the sphere in ℝn which cannot contain sets of uniqueness are presented. Analytic curves which are sets of uniqueness for real-analytic functions in ℝn, n ≥ 3, are constructed. The obtained results are used to justify the inhomogeneity sounding schemes when the inverse problem of acoustic scattering is solved under the conditions that the source and detector coordinates coincide.

Discrete harmonic functions on an orthant in $\mathbb{Z}^d$

We give a positive answer to a conjecture on the uniqueness of harmonic functions in the quarter plane stated by K. Raschel. More precisely we prove the existence and uniqueness of a positive discrete harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed at the boundary of an orthant in Zd. Our methodsallow on the other hand to generalize from the quarter plane to orthants in higher dimensions and to treat the spatially inhomogeneous walks.

On sets of uniqueness for harmonic functions in the unit circle

The resutls of this paper show that the structure of sets mentioned in the title is not trivial. For example, it is shown that there exist countalbe sets of uniqueness for logarithmic potential, i.e., closed countable subsets E of the unit circle\(\mathbb{T}\) such that $$f \in C(\mathbb{T}),f|_E = 0,U^f |_E = 0 \Rightarrow f \equiv 0.$$

On the Uniqueness of Harmonic Functions under Boundary Conditions

On the Uniqueness of Harmonic Functions under Boundary Conditions