In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain Ω with sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and exact observability for wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate with larger wave numbers interval.
Consider a solution f ∈ C 2 (Ω) of a prescribed mean curvature equationwhere Ω ⊂ IR 2 is a domain whose boundary has a corner at O = (0, 0) ∈ ∂Ω. If sup x∈Ω |f (x)| and sup x∈Ω |H(x, f (x))| are both finite and Ω has a reentrant corner at O, then the radial limits of f at O,f (r cos(θ), r sin(θ)), are shown to exist and to have a specific type of behavior, independent of the boundary behavior of f on ∂Ω \ {O}. If sup x∈Ω |f (x)| and sup x∈Ω |H(x, f (x))| are both finite and the trace of f on one side has a limit at O, then the radial limits of f at O exist and have a specific type of behavior.
Consider a solution f ∈ C 2 (Ω) of a prescribed mean curvature equationwhere Ω is a domain whose boundary has a corner at O = (0, 0) ∈ ∂Ω and the angular measure of this corner is 2α, for some α ∈ (0, π). Suppose sup x∈Ω |f (x)| and sup x∈Ω |H(x, f (x))| are both finite. If α > π 2, then the (nontangential) radial limits of f at O,were recently proven by the authors to exist, independent of the boundary behavior of f on ∂Ω, and to have a specific type of behavior.Suppose, the contact angle γ(·) that the graph of f makes with one side of ∂Ω has a limit (denoted γ2) at O and π − 2α < γ2 < 2α.We prove that the (nontangential) radial limits of f at O exist and the radial limits have a specific type of behavior, independent of the boundary behavior of f on the other side of ∂Ω. We also discuss the case α ∈ 0, π 2 .
In this paper, we investigate the interior inverse source problem for the Helmholtz equation with attenuation in the plane from boundary Cauchy data of multiple frequencies when the source term is assumed to be compactly supported in an arbitrary domain Ω with sufficiently smooth boundary. The main goal of this paper is to understand the dependence of increasing stability on the attenuation factor or constant damping. Using Fourier transform with respect to the wave numbers, explicit bounds for the analytic continuation and Hankel function and exact observability and Carleman estimates for the wave equation led us to our goal which is an increasing stability estimates with larger wave numbers interval.
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