# Download An introduction to partial differential equations by Yehuda Pinchover and Jacob Rubinstein PDF

By Yehuda Pinchover and Jacob Rubinstein

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So how is it that we seem to have obtained a unique solution? The mystery is easily resolved by observing that in choosing the positive sign for the square root in the argument of ψ, we effectively reduced the solution to the ray {x > 0}. Indeed, in this region a characteristic intersects the projection of the initial curve only once. 7 Solve the equation u x + 3y 2/3 u y = 2 subject to the initial condition u(x, 1) = 1 + x. 30) x(0, s) = s, y(0, s) = 1, u(0, s) = 1 + s. 31) In this example we expect a unique solution in a neighborhood of the initial curve since the transversality condition holds: J= 1 3 = −3 = 0.

This is called a mixed boundary condition. Another possibility is to generalize the condition of the third kind and replace the normal derivative by a (smoothly dependent) directional derivative of u in any direction that is not tangent to the boundary. This is called an oblique boundary condition. Also, one can provide a nonlocal boundary condition. For example, one can provide a boundary condition relating the heat ﬂux at each point on the boundary to the integral of the temperature over the whole boundary.

It follows that one can expect at most a local existence theorem for a ﬁrst-order PDE, even if the PDE is linear. (2) The parametric representation of the integral surface might hide further difﬁculties. We shall demonstrate this in the sequel by obtaining naive-looking parametric representations of singular surfaces. The difﬁculty lies in the inversion of the transformation from the plane (t, s) to the plane (x, y). Recall that the implicit function theorem implies that such a transformation is invertible if the Jacobian J = ∂(x, y)/∂(t, s) = 0.