# A Solution of a weakly formulated pde involving p-Laplacian

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1. Aug 3, 2017

### 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.

2. Aug 9, 2017

### PF_Help_Bot

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