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I have some questions concerning Dirichlet Problem and it would be very nice if somebody could give me some hints or some literature tips.

Actually, at the moment I am working on Dirichlet Problem and the quite similar Dirichlet Principle. My questions are mainly about the connections between the two of them. For my purposes the considered domain is always the unit ball $B$ in two dimensions. So first I'm wondering if the classical solution to the Dirichlet problem in $C(\overline{B})\cap C^2(B)$ is also the minimum to the following question:

find a function $u$ so that for a given function $v \in H^1_2(B)$ $$ \Delta u(x)=0 ,\quad x \in B$$ $$ u =v \text{ on}\ \partial B.$$ My problem is that I don't know if the classical solution has finite Dirichlet Integral, so I'm not sure whether it is in $ H^1_2(B).$ Is it generally possible to tackle the problem with a classical solution gained by Poisson's Integral for example? Or do you need Hilbert Space arguments already there?

My second question is the following: If I have a harmonic function (by Weyl's Lemma in my case) in $ H^1_2(B)$ which has continuous boundary values in the sense of the trace operator, then how do I know this function is continuous as a function on the closure of the ball $B$?

I read this in some books but I couldn't find a proof anywhere? Thanks in advance for every answer and I hope the questions are comprehensible.

Marcus
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2 Answers2

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A good starting point is Brezis' book on functional analysis. You can also read Zeidler's books on applied functional analysis. Anyway, a continuous function on a bounded set which is also twice differentiable in the interior of the domain must have a finite Dirichlet integral, since the gradient is a continuous function.

Under reasonable assumptions, the Dirichlet problem in uniquely solvable, so the solution can be found by any means. It would be different if the geometry of the domain or the irregularity of the boundary conditions could leave room for different kind of solutions.

Your last question is concerned with elliptic regularity theory. You can start with Evans' book on partial differential equations, for example.

Siminore
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  • Thank you for the book tips. About your argumentation that the gradient is a continous function I have a question. Are you sure that the gradient is continous on $\overline{B}$ if the function is twice differentiable only in the interior and continous on the closure? There are examoles of solutions to Dirichlet Problem for example in Courant's book about Dirichlet Principle whose Dirichlet Integral is not finite. Or am I wrong because there are different assumptions? – Marcus Aug 06 '15 at 10:49
  • Of course you must begin with a definition of solution. A classical solution is always a weak solution. Then the hard game is to prove that a weak solution is indeed classical, and elliptic regularity theory comes in. – Siminore Aug 06 '15 at 14:15
  • See also http://math.stackexchange.com/questions/292633/regularity-up-to-the-boundary – Siminore Aug 06 '15 at 16:40
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I think the interior regularity I need is covered by Weyl's Lemma completely. I am searching for an argument that a harmonic function whose boundary values exist in the sense of the trace operator is continous on the closure $\bar{B}.$ Is this obvious so that no one gives a proof or how can I tackle this problem. And my second problem is that I have no idea how to show that a solution to dirichlet's problem in $$C^2(B)\cap C(\bar{B})$$ has finite Dirichlet Integral if I only know that the boundary values it takes on are given by a function in H^1(B)? \par For both questions I already looked up a lot of literature but didn't quite exactly find anything. Or I did not see that what I read answered my questions. So you think I can find the answers in the books you recommended? Of course I did not read through all the chapters covering laplace equation in them yet, but I hoped to find anything I understand and so far I did not. I got the feeling the problems covered there have much weaker assumptions and therefore the results are weaker, too. My problem is sadly quite specific.

Marcus
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