Let $p(x)$ be a positive measurable function on $(-1,1)$. Consider the Prandtl equation $$ u(x)-\frac{p(x)}\pi \int_{-1}^1 \frac{u'(t)}{t-x}dt=p(x)h_0(x),\quad u(1)=u(-1)=0.\quad\quad(\star) $$ What is state of the art of the existsence and regularity theory for $(\star)$, what is a standard reference? Recall that $(\star)$ may be rewritten as $u+p(x)\sqrt{-d^2/dx^2}u=p(x)h_0(x)$, where $-d^2/dx^2$ is Dirichlet Laplacian on $[-1,1]$, that relates this to the regularity theory for fractional Laplacians.
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2$\begingroup$ For the fractional Laplacian formulation: for the existence, one can refer, for example, to the 20-year-old paper by K. Bogdan and T. Byczkowski Potential theory of Schrödinger operator based on fractional Laplacian. Regularity might be treated in a much more recent paper The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach by J.I. Díaz, D. Gómez-Castro and J.L. Vázquez. The latter is purely analytic and gives a lot of references. $\endgroup$– Mateusz KwaśnickiJul 11, 2020 at 15:05
1 Answer
With $q=1/p$, let me write your equation as $$ \vert D\vert u+ q u= h_0, \quad u(\pm 1)=0. $$ Multiplying the equation by $u$, we get $$\Vert{u}\Vert_{H^{1/2}_0}^2\le \Vert{u}\Vert_{H^{1/2}_0}^2+\underbrace{\langle qu, u\rangle_{L^2}}_{\ge 0}=\langle h_0, u\rangle_{L^2}\le \Vert{u}\Vert_{H^{1/2}_0}\Vert{h_0}\Vert_{H^{-1/2}}, $$ entailing $ \Vert{u}\Vert_{H^{1/2}_0}\le \Vert{h_0}\Vert_{H^{-1/2}}. $ Assuming $h_0\in H^{-1/2}$ will give $u$ in $H^{1/2}_0$ (completion of smooth compactly functions supported in $(-1,1)$ for the $H^{1/2}$ norm; here $H^{-1/2}$ stands for the dual space of $H^{1/2}_0$). With no more information on the regularity of $1/p$, it seems questionable to improve the regularity $H^{1/2}$.