Scattering problems

The stationary incoming wave u of frequency k is a solution to the perturbed Helmholtz equation (scattering by medium) (6.0.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGbbGaamyDaiabgkHiTiaadUgapaWaaWbaaSqabeaapeGaaGOm % aaaakiaadwhacqGH9aqpcaaIWaGaamyAaiaad6gatuuDJXwAK1uy0H % MmaeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab-1ris9aadaahaaWcbeqa % a8qacaaIZaaaaaaa!4B20! $$ Au - {k^2}u = 0in{\mathbb{R}^3} $$ (A is the elliptic operator − div(a ∇) + b · ∇ + c with ℜb, div b = 0, and ℑc ≤ 0, which coincides with the Laplace operator outside a ball B and which possesses the uniqueness of continuation property) or to the Helmholtz equation (scattering by an obstacle) (6.0.2) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqqHuoarcaWG1bGaey4kaSIaam4Aa8aadaahaaWcbeqaa8qacaaI % YaaaaOGaamyDaiabg2da9iaaicdapaGaaGjbV-qacaWGPbGaamOBa8 % aacaaMe8+dbiaadseapaWaaSbaaSqaa8qacaWGLbaapaqabaGcpeGa % eyypa0Zefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacq % WFDeIupaWaaWbaaSqabeaapeGaaG4maaaakiaacYfaceWGebGbaeba % aaa!5404! $$ \Delta u + {k^2}u = 0\;in\;{D_e} = {\mathbb{R}^3}\backslash \bar D $$ with the Dirichlet boundary data 6.0.3d % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2 % da9iaaicdacaaMe8Uaae4Baiaab6gacaaMe8UaeyOaIyRaamiraiaa % ysW7caGGOaGaae4Caiaab+gacaqGMbGaaeiDaiaaysW7caqGVbGaae % OyaiaabohacaqG0bGaaeyyaiaabogacaqGSbGaaeyzaiaaysW7caWG % ebGaaiykaiaac6caaaa!5282! $$ u = 0\;{\text{on}}\;\partial D\;({\text{soft}}\;{\text{obstacle}}\;D). $$ o the Neumann boundary data 6.0.3n % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaS % baaSqaaiaadAhaaeqaaOGaamyDaiabg2da9iaaicdacaaMe8Uaae4B % aiaab6gacaaMe8UaeyOaIyRaamiraiaaysW7caGGOaGaaeiAaiaabg % gacaqGYbGaaeizaiaaysW7caqGVbGaaeOyaiaabohacaqG0bGaaeyy % aiaabogacaqGSbGaaeyzaiaaysW7caWGebGaaiykaiaac6caaaa!54FC! $$ {\partial _v}u = 0\;{\text{on}}\;\partial D\;({\text{hard}}\;{\text{obstacle}}\;D). $$