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We consider time-harmonic acoustic scattering by planar sound-soft(Dirichlet) and sound-hard (Neumann) screens embedded in $\mathbb{R}^n$ for $n= 2$ or $3$. In contrast to previous studies in which the screen is assumed tobe a bounded Lipschitz (or smoother) relatively open subset of the plane, weconsider screens occupying an arbitrary bounded subset of the plane. Thus ourstudy includes cases where the screen is a relatively open set with a boundarythat is fractal, or indeed has positive surface measure, and cases where thescreen has empty interior and is fractal, or indeed has positive surfacemeasure. We elucidate for which screen geometries the classical formulations ofscreen scattering are well-posed, showing that the classical formulation forsound-hard scattering is not well-posed if the screen boundary has Hausdorffdimension greater than $n-2$. We also propose novel well-posed boundaryintegral equation (BIE) and boundary value problem (BVP) formulations, validfor arbitrary bounded screens. In fact, we show that for sufficiently irregularscreens there exist whole families of well-posed formulations, with infinitelymany distinct solutions, the distinct formulations distinguished by the sensein which the boundary conditions are understood. To select the physicallycorrect solution we propose limiting geometry principles, taking the limit ofsolutions for a convergent sequence of more regular screens, this a naturalprocedure for those fractal screens for which there exists a standard sequenceof prefractal approximations. We present examples exhibiting interestingphysical behaviours, including penetration of waves through screens with"holes" in them, where the "holes" have no interior points, so that the screenand its closure scatter differently. Our results depend on subtle andinteresting properties of fractional Sobolev spaces on non-Lipschitz sets.

Well-posed PDE and integral equation formulations for scattering by fractal screens