Partial differential equation with a nowhere differentiable boundary
1
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. Now, what is the solution for a continuous but nowhere differentiable boundary such as a fractal curve?
pde boundary-value-problem fractals elliptic-equations
asked 6 hours ago
Hans
4,95021032
add a comment |
1
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. Now, what is the solution for a continuous but nowhere differentiable boundary such as a fractal curve?
pde boundary-value-problem fractals elliptic-equations
asked 6 hours ago
Hans
4,95021032
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.
– Xander Henderson
2 hours ago
@XanderHenderson: I am aware that this subject in general is broad, that is precisely why I would like to "consider the simple case of the Dirichlet boundary value problem of the Laplace's equation" "with a continuous but nowhere differentiable boundary such as a fractal surface". In any case, I have pared down the question to a completely specific one.
– Hans
1 hour ago
add a comment |
1
1
1
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. Now, what is the solution for a continuous but nowhere differentiable boundary such as a fractal curve?
pde boundary-value-problem fractals elliptic-equations
asked 6 hours ago
Hans
4,95021032
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. Now, what is the solution for a continuous but nowhere differentiable boundary such as a fractal curve?
pde boundary-value-problem fractals elliptic-equations
pde boundary-value-problem fractals elliptic-equations
asked 6 hours ago
Hans
4,95021032
asked 6 hours ago
Hans
4,95021032
edited 1 hour ago
asked 6 hours ago
Hans
4,95021032
asked 6 hours ago
Hans
4,95021032
asked 6 hours ago
Hans
4,95021032
4,95021032
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.
– Xander Henderson
2 hours ago
@XanderHenderson: I am aware that this subject in general is broad, that is precisely why I would like to "consider the simple case of the Dirichlet boundary value problem of the Laplace's equation" "with a continuous but nowhere differentiable boundary such as a fractal surface". In any case, I have pared down the question to a completely specific one.
– Hans
1 hour ago
add a comment |
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.
– Xander Henderson
2 hours ago
@XanderHenderson: I am aware that this subject in general is broad, that is precisely why I would like to "consider the simple case of the Dirichlet boundary value problem of the Laplace's equation" "with a continuous but nowhere differentiable boundary such as a fractal surface". In any case, I have pared down the question to a completely specific one.
– Hans
1 hour ago
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.
– Xander Henderson
2 hours ago
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.
– Xander Henderson
2 hours ago
@XanderHenderson: I am aware that this subject in general is broad, that is precisely why I would like to "consider the simple case of the Dirichlet boundary value problem of the Laplace's equation" "with a continuous but nowhere differentiable boundary such as a fractal surface". In any case, I have pared down the question to a completely specific one.
– Hans
1 hour ago
@XanderHenderson: I am aware that this subject in general is broad, that is precisely why I would like to "consider the simple case of the Dirichlet boundary value problem of the Laplace's equation" "with a continuous but nowhere differentiable boundary such as a fractal surface". In any case, I have pared down the question to a completely specific one.
– Hans
1 hour ago
add a comment |
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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.
– Xander Henderson
2 hours ago
@XanderHenderson: I am aware that this subject in general is broad, that is precisely why I would like to "consider the simple case of the Dirichlet boundary value problem of the Laplace's equation" "with a continuous but nowhere differentiable boundary such as a fractal surface". In any case, I have pared down the question to a completely specific one.
– Hans
1 hour ago