Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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) ?


share|cite|improve this question
$K$ is not contained in the boundary of $B^n$; it does not even touch the boundary. –  Florian Nov 4 '11 at 9:38
@Florian : yes, the Dirichlet problem is solved in the domain $B^n \setminus K$. The solution is is harmonic on $B^n \setminus K$ and continuous on $\overline{B^n}$. –  vizietto Nov 4 '11 at 9:52

2 Answers 2

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.

share|cite|improve this answer
Marius Mitrea has a bunch of papers on the regularity of the Dirichlet problem on manifolds. –  Nilima Nigam Nov 25 '11 at 16:21

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.