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What are the conditions[General Criteria] for the existence or non existence of the solutions to a PDE[Elliptic type] subject to given boundary conditions?

A specific Example:

Let's consider the reduced wave equation: $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=-k^2u$$

In the simplest case if u vanishes on the square:$x=0,x=a,y=0,y=a$ the solution is: $$u=\sin\frac{m\pi x}{a}\sin\frac{n\pi\ y}{a}$$

where, $k^2=(m^2+n^2)\pi^2/a^2$.

Can we use this result to solve: $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=f(u)$$

One may to expand f(u) in terms of the eigenfunctions of the operator:

$$\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}$$ How should one go about the job if the boundary conditions on the same square are changed to a different continuous and differentiable functions of the type:

Example: For x=0 $u=f_1(x,y)$

For x=a $u=f_2(x,y)$

For y=0 $u=f_3(x,y)$

For x=a $u=f_4(x,y)$

  • 4
    Could you please try to make your question a bit more specific? What kind of PDEs? What kind of boundary conditions? As it is it is certainly overly broad and hardly answerable.2011-12-06
  • 1
    I still think this question is overly broad. There are entire books on this matter, like [Evans](http://books.google.com/books?id=Xnu0o_EJrCQC) or [Gilbarg-Trudinger](http://books.google.com/books?id=eoiGTf4cmhwC). See also the book recommendations in [this thread](http://math.stackexchange.com/questions/40115/how-to-prove-weyls-asymptotic-law-for-the-eigenvalues-of-the-dirichlet-laplacia), for example.2011-12-06
  • 0
    You may consider the reduced wave wave equation $\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}=-k^2u$. In the simplest case if u vanishes on the square x=0,x=a,y=0,y=a the solution is$u=\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{a}; m>0,n>0;k^2=\frac{(m^2+n^2)\pi^2}{a^2}$.Can we use this result to solve:$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=f(u)$?Suppose we expand f(u) in terms of the eigenfunctions of the operator $\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$--is it going to work?2011-12-06
  • 1
    You should add that info to your question. If you do that, I'll vote to re-open.2011-12-06
  • 0
    Thank you. I cast a vote to re-open.2011-12-06
  • 0
    Just a comment: the two books @t.b. linked to are still very relevant.2011-12-06

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