Oblique derivative boundary condition ($a{\partial }_{n}q+b{\partial }_{\tau }q=g$)

An oblique derivative boundary condition prescribes a linear combination of the outward normal derivative and the tangential derivative on a boundary. Its strictly oblique case has a nonzero normal coefficient; a zero normal coefficient is a limiting tangential condition. Both trace directions must be tracked when constructing a global relation. On unbounded domains, regularity at corners and a growth condition at infinity are separate parts of an admissible boundary value problem.

For a solution with a continuous gradient at a quadrant corner, the prescribed derivative directions are $(-\sin {\beta }_1,\cos {\beta }_1)$ and $(\cos {\beta }_2,-\sin {\beta }_2)$. When ${\beta }_1+{\beta }_2=\pi /2$, these vectors are negatives of one another. Their corner data must therefore be opposite. Imposing $h_1(0)=h_2(0)$ as well forces both to vanish. Smooth decay of separate boundary functions alone does not ensure this local compatibility condition for an overdetermined linear system.

// Target: analysis.bigb

For a complex-valued harmonic function in the first quadrant, write $\Phi =q_z$ and $\Psi (\bar{z})=q_{\bar{z}}$ as independent holomorphic functions. Prescribe $i(e^{i{\beta }_1}\Phi -e^{-i{\beta }_1}\Psi )=g_1$ on the imaginary axis and $e^{-i{\beta }_2}\Phi +e^{i{\beta }_2}\Psi =g_2$ on the real axis. If ${\beta }_1+{\beta }_2=n\pi /2$, set $ϵ=(-1{)}^n$. Reflection into four quadrants produces a scalar Riemann-Hilbert problem with parity $H(-z)=-ϵH(z)$ and known jumps. Provided its corner and large-arc contributions vanish, the Cauchy integral formula givesComplex boundary data do not justify replacing $q_{\bar{z}}$ by the conjugate of $q_z$.

For real harmonic $q$, prescribe $-\sin {\beta }_1{q}_x+\cos {\beta }_1{q}_y=h_1$ on the vertical axis and $\cos {\beta }_2{q}_x-\sin {\beta }_2{q}_y=h_2$ on the horizontal axis. Their orthogonal derivatives $j_1,j_2$ determine the missing gradient components. If $H_1,J_1$ use the kernel $e^{ky}$ and $H_2,J_2$ use $e^{-ikx}$, the clockwise spectra ${\hat{q}}_1={\int }_0^{i\infty }{e}^{-ikz}{q}_{z}dz$, ${\hat{q}}_2=-{\int }_0^{\infty }{e}^{-ikz}{q}_{z}dz$ have the displayed expressions. Their global relation is ${\hat{q}}_1+{\hat{q}}_2=0$ in the third spectral quadrant.

// Target: analysis.bigb

Reality of the boundary derivatives gives $H_1(\bar{k})=\overline{H_1(k)}$ and $H_2(-\bar{k})=\overline{H_2(k)}$, and the same identities for $J_1,J_2$. The global relation and its conjugate express both quarter-plane oblique boundary spectral functions in terms of the known $H_1,H_2$ and the single bounded holomorphic function $\Phi$ in the upper half-plane. For $\gamma ={\beta }_1+{\beta }_2\in \{0,\pi /2,\pi \}$, its coefficients on the positive real and positive imaginary reconstruction rays are opposite, because $e^{4i\gamma }=1$. Closing the first-quadrant contour removes $\Phi$ by the Cauchy integral theorem.

// Target: analysis.bigb

For tangential data with ${\beta }_1={\beta }_2=0$, the harmonic function $q=xy$ has zero prescribed data on both axes but a growing nonzero derivative. For normal data with ${\beta }_1={\beta }_2=\pi /2$, $q=\log |z|$ also has zero prescribed data, and $q_z=1/(2z)$ decays at infinity but is singular at the corner. Thus decay of the data, and even decay of the solution gradient, does not alone determine the quadrant representation. Requiring the reflected gradient to decay at infinity and satisfy $|z|H(z)\to 0$ at the corner removes both examples. Without such conditions, add homogeneous holomorphic terms obeying the reflection parity.