Denote by $B = \{(x,y) \in \mathbb{R}^2|x^2 + y^2 < 1\}$ the open unit ball in $\mathbb{R}^2$, and by $S$ it's boundary, i.e. the unit sphere.
For some $n > 0$ let $x_1,...,x_n \in B$, $y_1,...,y_n \in \mathbb{R}$ and $v_1,...,v_n \in \mathbb{R}^2$. Here one should think about $y_i$ as function values at $x_i$ and $v_i$ is the gradient of the function at said point.
I am looking to construct a function $u : B\cup S \to \mathbb{R}$ with the following restrictions:
- $u$ is continuous.
- $u$ is harmonic on $B$.
- For every $1 \leq i \leq n$, $u(x_i) = y_i$ and $\nabla u(x_i) = v_i$.
Generally, this should have many solutions (infinitely so even). One approach is to do a complex polynomial interpolation for the given points where we think about the $y_i$ as the real values of the polynomial.
I'm looking for solutions which are a bit more restricted. In particular, solutions $u$, such that for every $x \in S$, $|u(x)| \leq M$ for some pre-specified $M > 0$.
Obviously, those solutions will not always exist. For example if there exists an $1 \leq i \leq n$, such that $|y_i| > M$, then using the maximum principle, for any solution $u$ there has to be a point $x \in S$ such that $|u(x)| \geq |y_i|$.
Another bad example could be when $||v_i|| \geq M$, then using the interior gradient estimates for harmonic functions we will arrive at the same conclusion as in the previous example. This also means that we cannot have $2$ different points, such that $|y_i - y_j| \geq M||x_i - x_j||$.
My question is thus, under which conditions on $x_i,y_i,v_i$ can we expect to find a solution as described above? How will the solution look like?
More generally, I would be happy for any references which deal with similar problems.
Remark: giving the gradient values as data is a bit superfluous. Any comments which ignore those value are also much welcome.