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A Solution of a weakly formulated pde involving p-Laplacian

  1. Aug 3, 2017 #1
    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. jcsd
  3. Aug 9, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
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