Return to Answer

First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is probably too complicated for a practical use. For unbounded domains, one needs an additional growth restriction to ensure uniqueness. For example, that the boundary data for a cylinder are bounded, but this condition can be substantially relaxed. Poisson formula for a slab is relatively simple and it is written in

MR4238605 Madych, W. R. Harmonic functions in slabs and half-spaces. Harmonic analysis and applications, 325–359, Springer Optim. Appl., 168, Springer, Cham, 2021.

For the circular cylinder you may look into

MR3595961 D. Khavinson, E. Lundberg, and H. Render, Dirichlet's problem with entire data posed on an ellipsoidal cylinder, Potential Anal. 46 (2017), no. 1, 55–62.

MR1094495 Yoshida, H. Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pacific J. Math. 148 (1991), no. 2, 369–395.

Qiao, Lei, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math. 144 (2018), 39–54.

MR2904138 Ancona, Alano, On positive harmonic functions in cones and cylinders, Rev. Mat. Iberoam. 28 (2012), no. 1, 201–230.

Look also at the reference list in these papers.

For the infinite cylinder with periodic $L^1$ boundary conditions, there is a unique periodic solution. It is also unique among Dirichlet solutions which grow slower than $x^\omega$, where $\omega$ is the smallest Dirichlet eigenvalue of $D$. Similar results are available for the slab.

First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is probably too complicated for a practical use. For unbounded domains, one needs an additional growth restriction to ensure uniqueness. For example, that the boundary data and the solution for a cylinder are bounded, but this condition can be substantially relaxed. Poisson formula for a slab is relatively simple and it is written in

MR4238605 Madych, W. R. Harmonic functions in slabs and half-spaces. Harmonic analysis and applications, 325–359, Springer Optim. Appl., 168, Springer, Cham, 2021.

For the circular cylinder you may look into

MR3595961 D. Khavinson, E. Lundberg, and H. Render, Dirichlet's problem with entire data posed on an ellipsoidal cylinder, Potential Anal. 46 (2017), no. 1, 55–62.

MR1094495 Yoshida, H. Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pacific J. Math. 148 (1991), no. 2, 369–395.

Qiao, Lei, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math. 144 (2018), 39–54.

MR2904138 Ancona, Alano, On positive harmonic functions in cones and cylinders, Rev. Mat. Iberoam. 28 (2012), no. 1, 201–230.

Look also at the reference list in these papers.

For the infinite cylinder with periodic $L^1$ boundary conditions, there is a unique periodic solution. It is also unique among Dirichlet solutions which grow slower than $x^\omega$, where $\omega$ is the smallest Dirichlet eigenvalue of $D$. Similar results are available for the slab.

First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is probably too complicated for a practical use. For unbounded domains, one needs an additional growth restriction to ensure uniqueness. For example, that the boundary data for a cylinder are bounded, but this condition can be substantially relaxed. Poisson formula for a slab is relatively simple and it is written in

MR4238605 Madych, W. R. Harmonic functions in slabs and half-spaces. Harmonic analysis and applications, 325–359, Springer Optim. Appl., 168, Springer, Cham, 2021.

For the circular cylinder you may look into

MR3595961 D. Khavinson, E. Lundberg, and H. Render, Dirichlet's problem with entire data posed on an ellipsoidal cylinder, Potential Anal. 46 (2017), no. 1, 55–62.

MR1094495 Yoshida, H. Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pacific J. Math. 148 (1991), no. 2, 369–395.

Qiao, Lei, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math. 144 (2018), 39–54.

Look also at the reference list in these papers.

For the infinite cylinder with periodic $L^1$ boundary conditions, there is a unique periodic solution. It is also unique among Dirichlet solutions which grow slower than $x^\omega$, where $\omega$ is the smallest Dirichlet eigenvalue of $D$. Similar results are available for the slab.

where $\omega$

First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is probably too complicated for a practical use. For unbounded domains, one needs an additional growth restriction to ensure uniqueness. For example, that the boundary data for a cylinder are bounded, but this condition can be substantially relaxed. Poisson formula for a slab is relatively simple and it is written in

MR4238605 Madych, W. R. Harmonic functions in slabs and half-spaces. Harmonic analysis and applications, 325–359, Springer Optim. Appl., 168, Springer, Cham, 2021.

For the circular cylinder you may look into

MR3595961 D. Khavinson, E. Lundberg, and H. Render, Dirichlet's problem with entire data posed on an ellipsoidal cylinder, Potential Anal. 46 (2017), no. 1, 55–62.

MR1094495 Yoshida, H. Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pacific J. Math. 148 (1991), no. 2, 369–395.

Qiao, Lei, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math. 144 (2018), 39–54.

MR2904138 Ancona, Alano, On positive harmonic functions in cones and cylinders, Rev. Mat. Iberoam. 28 (2012), no. 1, 201–230.

Look also at the reference list in these papers.

For the infinite cylinder with periodic $L^1$ boundary conditions, there is a unique periodic solution. It is also unique among Dirichlet solutions which grow slower than $x^\omega$, where $\omega$ is the smallest Dirichlet eigenvalue of $D$. Similar results are available for the slab.

First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is probably too complicated for a practical use. For unbounded domains, one needs an additional growth restriction to ensure uniqueness. For example, that the boundary data for a cylinder are bounded, but this condition can be substantially relaxed. Poisson formula for a slab is relatively simple and it is written in

MR4238605 Madych, W. R. Harmonic functions in slabs and half-spaces. Harmonic analysis and applications, 325–359, Springer Optim. Appl., 168, Springer, Cham, 2021.

For the circular cylinder you may look into

MR3595961 D. Khavinson, E. Lundberg, and H. Render, Dirichlet's problem with entire data posed on an ellipsoidal cylinder, Potential Anal. 46 (2017), no. 1, 55–62.

Look also at the reference list in these papers.

For the infinite cylinder with periodic $L^1$ boundary conditions, there is a unique periodic solution. It is also unique among Dirichlet solutions which grow slower than $x^\omega$, where $\omega$ is the smallest Dirichlet eigenvalue of $D$. Similar results are available for the slab.

where $\omega$

First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is probably too complicated for a practical use. For unbounded domains, one needs an additional growth restriction to ensure uniqueness. For example, that the boundary data for a cylinder are bounded, but this condition can be substantially relaxed. Poisson formula for a slab is relatively simple and it is written in

MR4238605 Madych, W. R. Harmonic functions in slabs and half-spaces. Harmonic analysis and applications, 325–359, Springer Optim. Appl., 168, Springer, Cham, 2021.

For the circular cylinder you may look into

MR3595961 D. Khavinson, E. Lundberg, and H. Render, Dirichlet's problem with entire data posed on an ellipsoidal cylinder, Potential Anal. 46 (2017), no. 1, 55–62.

MR1094495 Yoshida, H. Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pacific J. Math. 148 (1991), no. 2, 369–395.

Qiao, Lei, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math. 144 (2018), 39–54.

Look also at the reference list in these papers.

For the infinite cylinder with periodic $L^1$ boundary conditions, there is a unique periodic solution. It is also unique among Dirichlet solutions which grow slower than $x^\omega$, where $\omega$ is the smallest Dirichlet eigenvalue of $D$. Similar results are available for the slab.

where $\omega$