# A mixed boundary value problem for $u_{xy}=f(x,y,u,u_x,u_y)$

## TL;DR

This paper investigates a hyperbolic PDE boundary value problem, revealing conditions under which uniqueness of solutions fails and establishing local existence using Picard iteration with additional data.

## Contribution

It demonstrates non-uniqueness in certain boundary configurations and provides a constructive method for local existence of solutions.

## Key findings

- Uniqueness fails when boundary curves are positioned with $M$ below $N$.
- Existence of local solutions is established under a Lipschitz condition on $f$.
- A Picard iteration scheme with additional data is used for construction.

## Abstract

Consider a single hyperbolic PDE $u_{xy}=f(x,y,u,u_x,u_y)$, with locally prescribed data: $u$ along a non-characteristic curve $M$ and $u_x$ along a non-characteristic curve $N$. We assume that $M$ and $N$ are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the $(x,y)$-plane. It is known that if $M$ is located above $N$, then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when $M$ lies below $N$, the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function $f$). The construction, via Picard iteration, makes use of a careful choice of additional $u$-data which are updated in each iteration step.