Solution of a weakly formulated pde involving p-Laplacian

[rajesh_d]

Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$

I want to solve this weakly formulated pde:

$$
0=\frac{A}{N^{d+1}} \sum_i \phi(x_i) (f(x_i)-d_i) |f(x_i)-d_i|^{d-1} + \int_\Omega \phi f |f|^{d-1} +Ad\int_\Omega \nabla \phi \cdot \nabla f |\nabla f|^{d-1}
$$
holds for all sufficiently smooth $$\phi$$.

In the space of continuous solutions, the solution $$f$$ exists, unique and is known to be Holder continuous.
PS :
I am looking for a fast algorithm to solve it numerically, even at a high dimensions, by leveraging the state of the art qualitative knowledge on this pde.
I already know that, among the space of continuous functions, the solution exists, is unique and is atleast Holder continuous with $$\alpha = \frac{1}{d+1}$$. I am not interested in solutions that have isolated discontinuities.

 

[jvicens]

Hello,

Thank you for sharing your problem with us. The PDE you have presented is quite interesting and challenging. It seems like you are trying to solve a non-linear elliptic equation with a combination of Dirichlet and Neumann boundary conditions. The fact that the solution exists in the space of continuous functions and is known to be Holder continuous is a great starting point.

To solve this problem numerically, you can use finite element methods or finite difference methods. Both of these methods have been extensively used to solve non-linear elliptic equations. For high dimensions, you can also consider using adaptive mesh refinement techniques to improve the accuracy and efficiency of your solution.

In terms of leveraging the state of the art qualitative knowledge on this PDE, you can start by looking at existing literature on similar problems. There might be some well-established numerical methods or algorithms that have been used to solve similar PDEs. Additionally, you can also consider using parallel computing techniques to speed up the numerical solution.

I hope this helps. Good luck with your research!