Solve a partial differential equation in two variables by specifing value of the solution in a region

Is it possible, when using DSolve to solve a partial differential equation, to specify the value of the solution in a finite domain? For example, to find the function $\phi(x_1,x_2)$, (with $x_1,x_2 \in \mathbb{R}$) that satisfies the equation

$-i(\partial_{x_1} + \partial_{x_2}) \phi(x_1,x_2) = A \phi(x_1,x_2)$,

with the condition that $\phi(x_1,x_2) = f(x_1,x_2)$ in the third quadrant, i.e. for $x_1,x_2<0$.

$f(x_1,x_2)$ is a known function and $A$ is just a parameter.

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In all but the third quadrant, ϕ satisfies

s = DSolveValue[-I (D[ϕ[x1, x2], x1] + D[ϕ[x1, x2], x2]) ==
a ϕ[x1, x2], ϕ[x1, x2], {x1, x2}]
(* E^(I a x1) C[-x1 + x2] *)

where C is an arbitrary function of x2 - x1. Boundary conditions are applied at the edges of the third quadrant,

(s /. x1 -> 0) == f[0, x2] (* x2 < 0 *)
(* C[x2] == f[0, x2] *)

(s /. x2 -> 0) == f[x1, 0] (* x1 < 0 *)
(* E^(I a x1) C[-x1] == f[x1, 0] *)

It follows, therefore, that C is given by

C[x2-x1] -> Piecewise[{{Exp[I a (x2 - x1)] f[x1 - x2, 0], x2 - x1 > 0},
{f[0, x2 - x1], x2 - x1 < 0}}]

and ϕ is given by

Piecewise[{{E^(I*a*x2)*f[x1 - x2, 0], x1 < x2}, {E^(I*a*x1)*f[0, -x1 + x2], x1 > x2}}, 0]

The validity of this expression can be verified by inserting it into the PDE and the two boundary conditions given above.

To illustrate the solution, consider

f[x_, y_] := x + y

designate the solution given above by ss, substitute a -> 1, and set the solution to f[x1, x2] in the third quadrant. Then,

Plot3D[Evaluate[ReIm[Piecewise[{{f[x1, x2], x1 < 0 && x2 < 0}}, ss /. a -> 1]]],
{x1, -1, 1}, {x2, -1, 1}, AxesLabel -> {x1, x2, ϕ}, Exclusions -> None,
PlotPoints -> 50, LabelStyle -> Directive[Bold, Black, Medium], ImageSize -> Large] where the real part of the solution is in orange, and the imaginary part in blue. Visibly the solution is continuous everywhere. Its first derivative need not be continuous, however, because the PDE is only first-order.

• Thanks for the answer! It's very clear and helpful! – Michele Cotrufo Nov 15 '17 at 21:10