Abstract
In this paper we study the problem of constructing reflector surfaces from the near field data. The light is transmitted as a collinear beam and the reflected rays illuminate a given domain on the fixed receiver surface. We consider two types of weak solutions and prove their equivalence under some convexity assumptions on the target domain. The regularity of weak solutions is a very delicate problem and the positive answer depends on a number of conditions characterizing the geometric positioning of the reflector and receiver. In fact, we show that there is a domain \({\mathcal{D}}\) in the ambient space such that the weak solution is smooth if and only if its graph lies in \({\mathcal{D}}\).
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Abbreviations
- C, C 0, C n , . . .:
-
Generic constants,
- \({\overline{\mathcal{U}}}\) :
-
Closure of a set \({\mathcal{U}}\),
- \({\partial \mathcal{U}}\) :
-
Boundary of a set \({\mathcal{U}}\) ,
- \({\widehat {X}}\) :
-
\({\widehat X=(x^1,\ldots,x^{n},0)}\) projection of \({X=(x^1,\ldots ,x^{n+1}) \in \mathbb{R}^{n+1}}\) ,
- ⟨·, ·⟩:
-
Inner product in \({\mathbb{R}^{n+1}}\) ,
- B r (x):
-
\({\{y \in \mathbb{R}^n: |y - x| < \}}\) , Open ball centered at y,
- B r :
-
B r (0),
- Γ u :
-
Graph of function u,
- ∂ i u, D i u, Du :
-
\({\partial_iu=D_iu=\frac{\partial{u}}{\partial{x^i}}}\) and Du = (D 1 u, . . . , D n u),
- ψ :
-
Defining function of receiver \({\Sigma=\{Z \in \mathbb{R}^{n+1} : \psi(Z)=0\}}\) ,
- ∇ψ :
-
(n + 1)-Dimensional gradient of receiver \({\psi: \mathbb{R}^{n+1} \longrightarrow \mathbb{R}}\),
- \({\widehat{\nabla}\psi}\) :
-
(ψ 1, . . . ψ n ,0), Projection of ∇ψ,
- Π :
-
Hyperplane \({\{X \in \mathbb{R}^{n+1} : x^{n+1}=0\}}\) ,
- \({\mathbb{S}^{n+1}}\) :
-
Units sphere in \({\mathbb{R}^{n+1}}\) ,
- |E|:
-
n-Dimensional Lebesgue measure of E ⊂ Π,
- \({\mathcal{H}_{\Sigma}^{n}}\) :
-
n-Dimensional Hausdorff measure restricted on Σ
- \({\mathbb{P}_L(\mathcal{U, V})}\) :
-
See (5.5),
- \({\mathbb{W}^+(\mathcal{U}, \mathcal{V}), \mathbb{W}_0^+(\mathcal{U}, \mathcal{V})}\) :
-
See Definitions 5.1 and 5.2,
- \({\mathcal{AS}^+(\mathcal{U},\Sigma)}\) :
-
See Definition 11.1
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Karakhanyan, A.L. Existence and Regularity of the Reflector Surfaces in \({\mathbb{R}^{n+1}}\) . Arch Rational Mech Anal 213, 833–885 (2014). https://doi.org/10.1007/s00205-014-0743-z
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DOI: https://doi.org/10.1007/s00205-014-0743-z