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$$-\Delta u = \lambda u \space, \space\space\space\ x∈Ω$$ $$u(x) = 0\space, \space \space\space\space\space x∈\partial Ω$$

What functional would form a good correspondence, and whose minimization could show that the above function, $u$, has a weak solution in $H^1_0(\Omega)$? Is this analogous solving the case of:

$$-\Delta u = f(x) \space, \space\space\space\ x∈Ω$$ $$u(x) = 0\space, \space \space\space\space\space x∈\partial Ω$$ taking $f(x)=\lambda u$, or is the there a better way to solve this specific case?

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  • $\begingroup$ It's more complicated than that. Without an $f$ on the right hand side it becomes an eigenvalue problem and you would need two functionals. See mathreview.uwaterloo.ca/archive/voli/2/nica.pdf $\endgroup$
    – toypajme
    May 23, 2013 at 2:12
  • $\begingroup$ In this case, we are more interested in what $\lambda$'s values are, rather than $u$ itself. $\endgroup$
    – Shuhao Cao
    May 23, 2013 at 4:25
  • $\begingroup$ Just to be clear, I'm only working right now on showing the existence of a weak solution in ${H^1_0}$, not on identifying it (yet, at least). $\endgroup$
    – BBB
    May 23, 2013 at 5:09

1 Answer 1

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If $\lambda\leq 0$ and is known, then the equation is nothing but a regular Poisson equation with an $L^2$-term.


If $\lambda >0$ is is unknown, then this is an eigenvalue problem, the link toypajme's comment is pretty nice and readable. Here I just add a bit more about motivation. Normally for eigenvalue problem, we are minimizing the Rayleigh quotient: $$ F(u) = \frac{\| \nabla u\|_{L^2(\Omega)}^2}{\| u\|_{L^2(\Omega)}^2},\tag{1} $$ whereas $$\int_\Omega u w_i = 0\quad \text{for eigenfunction } w_i, i=1,\ldots,n-1,$$ and the minimizer $u = w_n$ is another eigenfunction while $F(w_n)$ is the $n$-th eigenvalue. Notice problem (1) can be written as the same constraint minimization problem in the link in the toypajme's comment.


If $\lambda = \omega^2$ is known and is not one of the eigenvalues for the operator $-\Delta$, also we have a right hand side $f$, then this is a Helmholtz equation, and it weak formulation $$ \int_{\Omega}\nabla u\cdot \nabla v - \omega^2 \int_{\Omega}uv = \int_\Omega fv $$ has a unique solution in $H^1_0$ by Fredholm alternative.

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