Green's function for wave equations in R² or R³ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:31:26Z http://mathoverflow.net/feeds/question/38811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38811/greens-function-for-wave-equations-in-r-or-r Green's function for wave equations in R² or R³ Anand 2010-09-15T11:51:46Z 2010-09-15T13:05:52Z <p>Hello,</p> <p>For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we can get the Green's function. But this requirement is too strong. </p> <p>What we would like to have is that: the wave is confined in a convex, sufficiently smooth domain. On the boundary, either Dirichlet or Neumann's conditions can be put. To impose these conditions is just to avoid the diffraction problem, which can be too much complicated for us.</p> <p>During this searching, I encountered books by Prof. Melrose, Prof. Michael E. Taylor and also the formidable three volumns by Prof. Hormander. I still feel hopeless in finding that.</p> <p>Thanks in advance for any comments!</p> <p>Best!</p> http://mathoverflow.net/questions/38811/greens-function-for-wave-equations-in-r-or-r/38817#38817 Answer by Denis Serre for Green's function for wave equations in R² or R³ Denis Serre 2010-09-15T13:05:52Z 2010-09-15T13:05:52Z <p>What do you mean by <em>getting the Green's function</em> ? If you mean in closed form, then this is hopeless for most domains.</p> <p>Otherwise, the proper way to express the solution of $$u_{tt}=\Delta u,\qquad u(0)=u_0,\qquad u_t(0)=u_1$$ with homogeneous boundary conditions BC (say Dirichlet or Neumann) is to use the Laplace transform $\hat u$ of $u$ as an auxiliary function: $$\hat u(z):=\int_0^{+\infty}\exp(-sz)u(s)ds.$$ For each $z$ of positive real part, $\hat u(z)$ solves the elliptic problem $$(-\Delta+z^2)w=u_1+zu_0.$$ The above problem, with BC, is well-posed, for every $z$ away from the imaginary axis, and the map $z\mapsto w$ is holomorphic. One recovers $u$ through a Cauchy integral along an appropriate contour in the complex plane. This amounts to express the Green's function of the wave equation as a Cauchy integral in terms of the Green's functions of the elliptic problems parametrized by $z$. This expression may be used to analyze the singularities of the Green's function.</p>