Existence of solution to Cauchy boundary value problem in Lipschitz class of functions

Question

For a research question I have run into the following problem that seems intuitively true but I do not know how to prove it and am not sure in which generality.

Let $\Omega\subset \mathbb{R}^2$ be a simply connected domain with piecewise smooth boundary (In fact I know that it has a finite number of cusps and is smooth otw).

I have a function $g$ defined outside $\Omega$ for which both the values and gradients extend continuously as I approach $\partial\Omega$. It is known that as I walk around $\partial \Omega$ the gradients $\nabla g$ take one turn around the boundary of a convex region $\mathcal{H} \subset \mathbb{R}^2$.

Now the question is whether $g$ can be extended to the inside of $\Omega$ such that all of its gradients inside $\Omega$ lie in $\bar{\mathcal{H}}$.

I'm not quite sure how to approach this. Clearly something like a mean value property for a solution to the corresponding Cauchy boundary value problem would be enough but I don't know if there is a PDE that has a guaranteed solution to this kind of boundary data and the property of gradients being in the convex hull of the boundary gradients. At the same time it feel like one could possibly construct a solution by hand but again I'm not sure.

Any help, references or information would be appreciated.

Answer 1

Just for posterity, the concept I was looking for is that of Lipschitz extensions.

It turns out that a $K$-Lipschitz function outside any $\Omega$ can be extended to $\Omega$ with the same Lipschitz constant. This is quite easy to do. You construct a conus with inclines $K$ around every point of $\partial\Omega$ and define $g(x)$ to be the infimum of all coni over $x$.

The extension to not $K$-Lipschitz but gradients in a set $\mathcal{H}$ is then trivial. Just construct the coni based on $\mathcal{H}$.

Also see e.g. here for a reference.

Sorry that this ended up having nothing to do with PDEs as I assumed at first.