Dirichlet Problem with L1(S) Boundary Values
Round 1
Reviewer 1 Report
REFEREE’S REPORT
on the paper Axioms-1801668
“Dirichlet problem with L1(S) boundary values”
by Alexander G. Ramm
The paper deals with the existence of solution of a Dirichlet problem for Laplace equation with L1(S) boundary values.
The paper is on five pages and has only one section Introduction. There are formulated two definitions, two lemmas, one theorem and one open problem. It can be divided on two or three sections with a proof of the main result. The abstract can be revised as well.
Under above considerations I can recommend the paper for publication in the journal “Axioms” after a major revision.
Author Response
Please see the attachment of details. Thank you.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The article investigates the Dirichlet problem with L 1 (S) boundary values without using the embedded theorems. In my view, the article presents a method of proof in interesting ways. However, the authors should revise the editing to improve paper readability. Here are some suggestions to enhance paper readability:
- The introduction section seems too short. The introduction section should be rewritten with a more detailed literature review.
- There are several spelling errors throughout the article. For instance, "the solution is unque"
- Without a conclusion, the article seems incomplete. The conclusion should be written in detail by stressing the importance of the proposed theory. Please show when the presented approach is applicable, what are the pros and cons of using it when it should be preferred over the other existing theory etc. A detailed illustration of why such a theory is a powerful tool in applications would increase the interest of the reader.
Author Response
Please see the attachment of details. Thank you.
Author Response File: Author Response.pdf
Reviewer 3 Report
In the paper the Dirichlet boundary value problem $\Delta u(x)=0, x\in D;\quad u|_{\partial D}=h$ is considered. The main result of the article is contained in Theorem 1. It is proved that the above problem has a solution for an arbitrary $h\in L^1(S)$ and this solution is unique. I think that the results obtained are interesting and deserve publication.
I have the following minor remarks.
1. The university website http://www.math.ksu.edu/ contains the author's e-mail address: [email protected], however, it differs from the address indicated in the article: [email protected]
2. Page 1. Probably the author made typos by writing "unque" in Abstract and "detrmines uniquely" in the first paragraph.
3. Page 2. It is not very clear why the word "Definition1" is in bold in the text.
4. Page 2. The factor $f''(0)$ is probably missing on the right side of the equality $J(0)=1/2$. In addition, the continuity of the kernel $A(s,t)$ for $n=2$ is a well-known result, but it will not be superfluous for readers to see its proof.
5. Page 3, Remark 2. It would be nice to describe in more detail why $u=((x-1)^2+y^2-1)/(x^2+y^2)\not\in L^1(S)$ .
6. Page 4. Why $D':=\Bbb R^3\setminus\bar D$.
7. Page 5. It is not very clear why in the second formula from above, after the first inequality $\le$, under the integral sign is $|g(x,s)-g(t,s)|$.
Author Response
Please see the attachment of details. Thank you.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
REFEREE’S REPORT
on the paper Axioms-1801668 R
“Dirichlet problem with L1(S) boundary values”
by Alexander G. Ramm
The paper deals with the existence of solution of a Dirichlet problem for Laplace equation with L1(S) boundary values.
The paper is extended, interesting and clearly written.
I recommend the paper for publication in the journal “Axioms”.
Reviewer 2 Report
The author updated the paper with the recommended changes. I recommend the paper for publication in the journal “Axioms.”
