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I've never studied PDE and I only studied a little about Sobolev Spaces by myself. I would like to understand why can I conclude that $u_{m,n}$ is a weak solution to the Dirichlet problem below. Is being used any theorem? Or is being used only a definition of a weak solution? or is it only a matter of developing calculations related to the previous inequality in 2.10? I also did not understand why there is an infimum before a maximum in (2.10), would be about any definition, or is being used any theorem?

P.S: I'm studying for my final project in my course however is very hard to understand many steps in the paper. Despite of this, I'm really enjoying everything I'm learning.

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Where (2.6), (2.7) and (2.8) are

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And, the functional $J_{m,n}$ being mentioned is

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    $\begingroup$ Try to calculate directional derivative first. Think of $\nabla$ as an ordinary linear operator. $\endgroup$
    – daw
    Feb 10, 2022 at 12:31
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    $\begingroup$ I wonder what is the derivative of $(v_+)^q$. $\endgroup$
    – Arctic Char
    Feb 10, 2022 at 16:11
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    $\begingroup$ @ArcticChar nothing else than $q(v^+)^{q-1}$ provided $q\ge1$ (assuming $v^+=\max(v,0)$). $\endgroup$
    – daw
    Feb 10, 2022 at 16:13
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    $\begingroup$ Um.... I guess so. To the OP, you've got your $L_n'(v)$ wrong. Look like you just add $\varphi$ to the product and did not differentiate. $\endgroup$
    – Arctic Char
    Feb 10, 2022 at 16:17
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    $\begingroup$ Much better than the first trial. The second term is missing a $\phi$ $\endgroup$
    – daw
    Feb 11, 2022 at 7:01

1 Answer 1

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You are using two theorems. The first theorem is the Mountain Pass Theorem , which tells you that if the functional $J_{m,n}$ satisfies the hypotheses (2.6) and (2.7), then it admits a critical point, that is , a point $u_{m,n}$ such that the derivative of the functional $J_{m,n}$ computed at $u_{m,n}$ is zero. Then you are using another theorem that says that critical points of your functional are weak solutions of a partial differential equations, called the Euler Lagrange equations of your functional. You can find this derivation in the book by Evans “Partial differential equations”.

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    $\begingroup$ Are you Giovanni Leoni? I can't believe you're responding to me by here. Lol I'm a fan of your book "A first course of Sobolev Spaces". This book is helping all the time in my final project. Thanks very much for your contributions! $\endgroup$
    – Silvinha
    Feb 18, 2022 at 16:24
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    $\begingroup$ Glad you like the book! $\endgroup$
    – Gio67
    Feb 18, 2022 at 21:38

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