# Solvability of the Dirichlet problem for a general second-order elliptic equation

## Abstract

The paper is concerned with studying the solvability of the Dirichlet problem for the second-order elliptic equation ; in a bounded domain Q subset of R{sub n}, n{>=}2, with C{sup 1}-smooth boundary and boundary condition u{sub 0} element of L{sub 2}({partial deriv}Q). Conditions for the existence of an (n-1)-dimensionally continuous solution are obtained, the resulting solvability condition is shown to be similar in form to the solvability condition in the conventional generalized setting (in W{sub 2}{sup 1}(Q)). In particular, the problem is shown to have an (n-1)-dimensionally continuous solution for all u{sub 0} element of L{sub 2}({partial deriv}Q) and all f and F from the appropriate function spaces, provided that the homogeneous problem (with zero boundary conditions and zero right-hand side) has no nonzero solutions in W{sub 2}{sup 1}(Q). Bibliography: 14 titles.

- Authors:

- Yerevan State University, Yerevan (Armenia)

- Publication Date:

- OSTI Identifier:
- 21592532

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 202; Journal Issue: 7; Other Information: DOI: 10.1070/SM2011v202n07ABEH004174; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; BOUNDARY CONDITIONS; DIRICHLET PROBLEM; EQUATIONS; FUNCTIONS; MATHEMATICAL SOLUTIONS; MATHEMATICAL SPACE; ONE-DIMENSIONAL CALCULATIONS; SMOOTH MANIFOLDS; BOUNDARY-VALUE PROBLEMS; MATHEMATICAL MANIFOLDS; SPACE

### Citation Formats

```
Dumanyan, Vagram Zh.
```*Solvability of the Dirichlet problem for a general second-order elliptic equation*. United States: N. p., 2011.
Web. doi:10.1070/SM2011V202N07ABEH004174; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Dumanyan, Vagram Zh.
```*Solvability of the Dirichlet problem for a general second-order elliptic equation*. United States. doi:10.1070/SM2011V202N07ABEH004174; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Dumanyan, Vagram Zh. Sun .
"Solvability of the Dirichlet problem for a general second-order elliptic equation". United States. doi:10.1070/SM2011V202N07ABEH004174; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21592532,
```

title = {Solvability of the Dirichlet problem for a general second-order elliptic equation},

author = {Dumanyan, Vagram Zh},

abstractNote = {The paper is concerned with studying the solvability of the Dirichlet problem for the second-order elliptic equation ; in a bounded domain Q subset of R{sub n}, n{>=}2, with C{sup 1}-smooth boundary and boundary condition u{sub 0} element of L{sub 2}({partial deriv}Q). Conditions for the existence of an (n-1)-dimensionally continuous solution are obtained, the resulting solvability condition is shown to be similar in form to the solvability condition in the conventional generalized setting (in W{sub 2}{sup 1}(Q)). In particular, the problem is shown to have an (n-1)-dimensionally continuous solution for all u{sub 0} element of L{sub 2}({partial deriv}Q) and all f and F from the appropriate function spaces, provided that the homogeneous problem (with zero boundary conditions and zero right-hand side) has no nonzero solutions in W{sub 2}{sup 1}(Q). Bibliography: 14 titles.},

doi = {10.1070/SM2011V202N07ABEH004174; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 7,

volume = 202,

place = {United States},

year = {2011},

month = {7}

}