Partial differential equation with a nowhere differentiable boundary
4
$begingroup$
Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. Consider the case when the boundary is continuous but nowhere differentiable such as a fractal curve. I would like to know the answers or see a rough survey of the answers, or be pointed to the references pertaining, to the following questions.
- What is an analog of the Green's function?
- What are some of the solution methods other than Green's function analog?
reference-request pde boundary-value-problem fractals elliptic-equations
asked Jan 4 at 2:19
HansHans
5,03421232
$endgroup$
|
show 10 more comments
4
$begingroup$
Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. Consider the case when the boundary is continuous but nowhere differentiable such as a fractal curve. I would like to know the answers or see a rough survey of the answers, or be pointed to the references pertaining, to the following questions.
- What is an analog of the Green's function?
- What are some of the solution methods other than Green's function analog?
reference-request pde boundary-value-problem fractals elliptic-equations
asked Jan 4 at 2:19
HansHans
5,03421232
$endgroup$
1
$begingroup$
While the question you ask is an interesting one, it is incredibly broad, and leads to a lot of interesting mathematics, some of which is still actively being researched. The actual shape of the domain matters (convex domains, star-convex domains, John domains, &c.), the boundary conditions matter (Dirichlet, von Neuman, Robin, &c.), and the differential operator under consideration matters (obviously?). Unfortunately, because this question is so broad, I don't think that it is a good fit for MSE, and I have voted to close it.
$endgroup$
– Xander Henderson
Jan 4 at 6:29
1
$begingroup$
Johan Thim wrote a fairly well known Masters thesis on nowhere differentiable continuous functions, and it appears that his research is now somewhat close to what you're asking --- My main area of research is analysis and partial differential equations, in particular on domains with minimal smoothness assumptions.
$endgroup$
– Dave L. Renfro
Jan 5 at 0:00
3
$begingroup$
As I said in another comment, I know almost nothing about this topic, but perhaps these two papers could help: The boundary-value problems for Laplace equation and domains with nonsmooth boundary by Dagmar Medková (1998) and Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture by Michel L. Lapidus (1991).
$endgroup$
– Dave L. Renfro
Jan 5 at 0:30
2
$begingroup$
OK, I didn't notice the ambiguity with "my". As I said, I don't know anything about this stuff, but I have seen references to Lapidus's paper many times over the years (due to my interest in nowhere differentiability, fractals, and the like). Possible search words might John domain, corkscrew domain, uniform domain, non'tangentially accessible domain, Lipschitz domain, plump domain, cigar domain, ball accessible domain, uniformly perfect domain, etc. An old but probably useful paper is Uniform domains by Jussi Väisälä (1988)
$endgroup$
– Dave L. Renfro
Jan 5 at 1:04
1
$begingroup$
@Hans: Given what I know about this, I'm not sure I would be able to ask anything more specific, other than I'm not sure whether I'd mention "Green's function" (since for all I know that's a specific technique that might be entirely inappropriate for the situation you're asking about), so I'll do this.
$endgroup$
– Dave L. Renfro
Jan 5 at 17:41
|
show 10 more comments
4
4
4
0
$begingroup$
Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. Consider the case when the boundary is continuous but nowhere differentiable such as a fractal curve. I would like to know the answers or see a rough survey of the answers, or be pointed to the references pertaining, to the following questions.
- What is an analog of the Green's function?
- What are some of the solution methods other than Green's function analog?
reference-request pde boundary-value-problem fractals elliptic-equations
asked Jan 4 at 2:19
HansHans
5,03421232
$endgroup$
Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. Consider the case when the boundary is continuous but nowhere differentiable such as a fractal curve. I would like to know the answers or see a rough survey of the answers, or be pointed to the references pertaining, to the following questions.
- What is an analog of the Green's function?
- What are some of the solution methods other than Green's function analog?
reference-request pde boundary-value-problem fractals elliptic-equations
reference-request pde boundary-value-problem fractals elliptic-equations
asked Jan 4 at 2:19
HansHans
5,03421232
asked Jan 4 at 2:19
HansHans
5,03421232
edited Jan 4 at 23:41
Hans
asked Jan 4 at 2:19
HansHans
5,03421232
asked Jan 4 at 2:19
HansHans
5,03421232
asked Jan 4 at 2:19
HansHans
5,03421232
5,03421232
1
$begingroup$
While the question you ask is an interesting one, it is incredibly broad, and leads to a lot of interesting mathematics, some of which is still actively being researched. The actual shape of the domain matters (convex domains, star-convex domains, John domains, &c.), the boundary conditions matter (Dirichlet, von Neuman, Robin, &c.), and the differential operator under consideration matters (obviously?). Unfortunately, because this question is so broad, I don't think that it is a good fit for MSE, and I have voted to close it.
$endgroup$
– Xander Henderson
Jan 4 at 6:29
1
$begingroup$
Johan Thim wrote a fairly well known Masters thesis on nowhere differentiable continuous functions, and it appears that his research is now somewhat close to what you're asking --- My main area of research is analysis and partial differential equations, in particular on domains with minimal smoothness assumptions.
$endgroup$
– Dave L. Renfro
Jan 5 at 0:00
3
$begingroup$
As I said in another comment, I know almost nothing about this topic, but perhaps these two papers could help: The boundary-value problems for Laplace equation and domains with nonsmooth boundary by Dagmar Medková (1998) and Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture by Michel L. Lapidus (1991).
$endgroup$
– Dave L. Renfro
Jan 5 at 0:30
2
$begingroup$
OK, I didn't notice the ambiguity with "my". As I said, I don't know anything about this stuff, but I have seen references to Lapidus's paper many times over the years (due to my interest in nowhere differentiability, fractals, and the like). Possible search words might John domain, corkscrew domain, uniform domain, non'tangentially accessible domain, Lipschitz domain, plump domain, cigar domain, ball accessible domain, uniformly perfect domain, etc. An old but probably useful paper is Uniform domains by Jussi Väisälä (1988)
$endgroup$
– Dave L. Renfro
Jan 5 at 1:04
1
$begingroup$
@Hans: Given what I know about this, I'm not sure I would be able to ask anything more specific, other than I'm not sure whether I'd mention "Green's function" (since for all I know that's a specific technique that might be entirely inappropriate for the situation you're asking about), so I'll do this.
$endgroup$
– Dave L. Renfro
Jan 5 at 17:41
|
show 10 more comments
1
$begingroup$
While the question you ask is an interesting one, it is incredibly broad, and leads to a lot of interesting mathematics, some of which is still actively being researched. The actual shape of the domain matters (convex domains, star-convex domains, John domains, &c.), the boundary conditions matter (Dirichlet, von Neuman, Robin, &c.), and the differential operator under consideration matters (obviously?). Unfortunately, because this question is so broad, I don't think that it is a good fit for MSE, and I have voted to close it.
$endgroup$
– Xander Henderson
Jan 4 at 6:29
1
$begingroup$
Johan Thim wrote a fairly well known Masters thesis on nowhere differentiable continuous functions, and it appears that his research is now somewhat close to what you're asking --- My main area of research is analysis and partial differential equations, in particular on domains with minimal smoothness assumptions.
$endgroup$
– Dave L. Renfro
Jan 5 at 0:00
3
$begingroup$
As I said in another comment, I know almost nothing about this topic, but perhaps these two papers could help: The boundary-value problems for Laplace equation and domains with nonsmooth boundary by Dagmar Medková (1998) and Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture by Michel L. Lapidus (1991).
$endgroup$
– Dave L. Renfro
Jan 5 at 0:30
2
$begingroup$
OK, I didn't notice the ambiguity with "my". As I said, I don't know anything about this stuff, but I have seen references to Lapidus's paper many times over the years (due to my interest in nowhere differentiability, fractals, and the like). Possible search words might John domain, corkscrew domain, uniform domain, non'tangentially accessible domain, Lipschitz domain, plump domain, cigar domain, ball accessible domain, uniformly perfect domain, etc. An old but probably useful paper is Uniform domains by Jussi Väisälä (1988)
$endgroup$
– Dave L. Renfro
Jan 5 at 1:04
1
$begingroup$
@Hans: Given what I know about this, I'm not sure I would be able to ask anything more specific, other than I'm not sure whether I'd mention "Green's function" (since for all I know that's a specific technique that might be entirely inappropriate for the situation you're asking about), so I'll do this.
$endgroup$
– Dave L. Renfro
Jan 5 at 17:41
1
1
$begingroup$
While the question you ask is an interesting one, it is incredibly broad, and leads to a lot of interesting mathematics, some of which is still actively being researched. The actual shape of the domain matters (convex domains, star-convex domains, John domains, &c.), the boundary conditions matter (Dirichlet, von Neuman, Robin, &c.), and the differential operator under consideration matters (obviously?). Unfortunately, because this question is so broad, I don't think that it is a good fit for MSE, and I have voted to close it.
$endgroup$
– Xander Henderson
Jan 4 at 6:29
$begingroup$
While the question you ask is an interesting one, it is incredibly broad, and leads to a lot of interesting mathematics, some of which is still actively being researched. The actual shape of the domain matters (convex domains, star-convex domains, John domains, &c.), the boundary conditions matter (Dirichlet, von Neuman, Robin, &c.), and the differential operator under consideration matters (obviously?). Unfortunately, because this question is so broad, I don't think that it is a good fit for MSE, and I have voted to close it.
$endgroup$
– Xander Henderson
Jan 4 at 6:29
1
1
$begingroup$
Johan Thim wrote a fairly well known Masters thesis on nowhere differentiable continuous functions, and it appears that his research is now somewhat close to what you're asking --- My main area of research is analysis and partial differential equations, in particular on domains with minimal smoothness assumptions.
$endgroup$
– Dave L. Renfro
Jan 5 at 0:00
$begingroup$
Johan Thim wrote a fairly well known Masters thesis on nowhere differentiable continuous functions, and it appears that his research is now somewhat close to what you're asking --- My main area of research is analysis and partial differential equations, in particular on domains with minimal smoothness assumptions.
$endgroup$
– Dave L. Renfro
Jan 5 at 0:00
3
3
$begingroup$
As I said in another comment, I know almost nothing about this topic, but perhaps these two papers could help: The boundary-value problems for Laplace equation and domains with nonsmooth boundary by Dagmar Medková (1998) and Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture by Michel L. Lapidus (1991).
$endgroup$
– Dave L. Renfro
Jan 5 at 0:30
$begingroup$
As I said in another comment, I know almost nothing about this topic, but perhaps these two papers could help: The boundary-value problems for Laplace equation and domains with nonsmooth boundary by Dagmar Medková (1998) and Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture by Michel L. Lapidus (1991).
$endgroup$
– Dave L. Renfro
Jan 5 at 0:30
2
2
$begingroup$
OK, I didn't notice the ambiguity with "my". As I said, I don't know anything about this stuff, but I have seen references to Lapidus's paper many times over the years (due to my interest in nowhere differentiability, fractals, and the like). Possible search words might John domain, corkscrew domain, uniform domain, non'tangentially accessible domain, Lipschitz domain, plump domain, cigar domain, ball accessible domain, uniformly perfect domain, etc. An old but probably useful paper is Uniform domains by Jussi Väisälä (1988)
$endgroup$
– Dave L. Renfro
Jan 5 at 1:04
$begingroup$
OK, I didn't notice the ambiguity with "my". As I said, I don't know anything about this stuff, but I have seen references to Lapidus's paper many times over the years (due to my interest in nowhere differentiability, fractals, and the like). Possible search words might John domain, corkscrew domain, uniform domain, non'tangentially accessible domain, Lipschitz domain, plump domain, cigar domain, ball accessible domain, uniformly perfect domain, etc. An old but probably useful paper is Uniform domains by Jussi Väisälä (1988)
$endgroup$
– Dave L. Renfro
Jan 5 at 1:04
1
1
$begingroup$
@Hans: Given what I know about this, I'm not sure I would be able to ask anything more specific, other than I'm not sure whether I'd mention "Green's function" (since for all I know that's a specific technique that might be entirely inappropriate for the situation you're asking about), so I'll do this.
$endgroup$
– Dave L. Renfro
Jan 5 at 17:41
$begingroup$
@Hans: Given what I know about this, I'm not sure I would be able to ask anything more specific, other than I'm not sure whether I'd mention "Green's function" (since for all I know that's a specific technique that might be entirely inappropriate for the situation you're asking about), so I'll do this.
$endgroup$
– Dave L. Renfro
Jan 5 at 17:41
|
show 10 more comments
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1
$begingroup$
While the question you ask is an interesting one, it is incredibly broad, and leads to a lot of interesting mathematics, some of which is still actively being researched. The actual shape of the domain matters (convex domains, star-convex domains, John domains, &c.), the boundary conditions matter (Dirichlet, von Neuman, Robin, &c.), and the differential operator under consideration matters (obviously?). Unfortunately, because this question is so broad, I don't think that it is a good fit for MSE, and I have voted to close it.
$endgroup$
– Xander Henderson
Jan 4 at 6:29
1
$begingroup$
Johan Thim wrote a fairly well known Masters thesis on nowhere differentiable continuous functions, and it appears that his research is now somewhat close to what you're asking --- My main area of research is analysis and partial differential equations, in particular on domains with minimal smoothness assumptions.
$endgroup$
– Dave L. Renfro
Jan 5 at 0:00
3
$begingroup$
As I said in another comment, I know almost nothing about this topic, but perhaps these two papers could help: The boundary-value problems for Laplace equation and domains with nonsmooth boundary by Dagmar Medková (1998) and Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture by Michel L. Lapidus (1991).
$endgroup$
– Dave L. Renfro
Jan 5 at 0:30
2
$begingroup$
OK, I didn't notice the ambiguity with "my". As I said, I don't know anything about this stuff, but I have seen references to Lapidus's paper many times over the years (due to my interest in nowhere differentiability, fractals, and the like). Possible search words might John domain, corkscrew domain, uniform domain, non'tangentially accessible domain, Lipschitz domain, plump domain, cigar domain, ball accessible domain, uniformly perfect domain, etc. An old but probably useful paper is Uniform domains by Jussi Väisälä (1988)
$endgroup$
– Dave L. Renfro
Jan 5 at 1:04
1
$begingroup$
@Hans: Given what I know about this, I'm not sure I would be able to ask anything more specific, other than I'm not sure whether I'd mention "Green's function" (since for all I know that's a specific technique that might be entirely inappropriate for the situation you're asking about), so I'll do this.
$endgroup$
– Dave L. Renfro
Jan 5 at 17:41