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 $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$(adsbygoogle = window.adsbygoogle || []).push({});

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.

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

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