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In this work we discuss an exterior boundary value problem for the hyperbolic wave and Maxwell's equations where data is specified on a timelike tube and on a null hypersurface. Problems of this type arise when studying radiation, particularly in numerical applications. We first consider the linear wave equation. Although the Cauchy problem analysis is elementary, the null-timelike problem is very different and more complicated than even the exterior Cauchy problem. We prove existence for smooth data and then, using energy methods, we derive existence results in H¹. Energy methods do not suffice to produce estimates for all higher derivatives, but these can be instead derived using transport methods and estimates on the timelike boundary. However we find that this is only possible if much greater regularity is assumed for the initial data - there is a loss of half the derivatives. We also apply these results to Maxwell's equations in Minkowski space. We introduce a gauge adapted to the null-timelike problem, which allows us to eliminate all gauge freedoms and show that any solution of Maxwell's equations is uniquely gauge-equivalent to a solution in this gauge. We also show that Maxwell's equations can be solved uniquely in this gauge for appropriate data on the null-timelike boundary. This leads to a one-to-one correspondence between a class of boundary data ("free data") and the space of all gauge-equivalence classes of solutions of Maxwell's equations. |
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