Question

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times {0}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator.

We wish to solve the Dirichlet problem (for harmonic functions) on $B^n \setminus K$ with boundary value $f = 0$ on $K$, and $f$ (let's say) is smooth on $\partial B^n$. Such a solution exists, is unique and we can find it within the class of Sobolev function.

My question is how this solution behaves near $K$. Is it Hölder continuous ? What is the best exponent of Hölder continuity one can expect ?

More generally, are there known criterias for the regularity at the boundary (not abstract continuity like with the Wiener criterion, but with a control of the modulus of continuity) ?

Thanks

Answer 1

Far from the rim of the inner disk one has a good regularity. At the rim though, I think you can get the solution behaviour from explicit constructions of potentials induced by a charged ellipsoid and collapsing one of axes, e.g., as given in Kellogg's book.

Answer 2

Grisvard's book has an extensive discussion of 2d elliptic problems with corners. Your problem is singular at the rim, and the singularity there is essentially the same one as a 2d problem with a 360 degree corner.