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Let $n\geq 2$ be an integer and $\Omega\in R^n$ be a bounded domain with boundary $\partial\Omega$ . Consider the following IBVP:

$u_t=\Delta u$, for $x\in \Omega$, $t>0$;

$u(x, 0)=f(x), x\in\Omega$ (intial condition )

$u= g(x,t), x\in\partial\Omega, t>0$ (boundary condition).

When $n=1$ we can easily homogenize the boundary condition and then use separation of variables or any other standard method to show the existence of the solution. When $n\geq 2$ and $g(x, t)=0$ (means zero boundary condition) we can use Galerkin's method to prove the existence of the solution. I try to understand how to prove the existence of above problem if $g(x,t)\neq 0$ and if $n\geq 2$.

Of course the above problem is just an example to explain my point. The equation could be any nonlinear partial differential differential equation. In my previous post I asked if there is a method to handle for such problems (problems with nonhomogeneous boundary condition in higher dimensions as given above). Any help is appreciated.

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    $\begingroup$ A simple trick is to subtract a reference function which satisfies the boundary conditions and then look for the difference as a solution of a problem with homogeneous conditions. If this is what you are looking for, this is not an appropriate question for this forum. The problem with this simple approach is that it does not typically yield optimal regularity results. If those are what you are looking for, you need to make the question much more specific. $\endgroup$ Jun 2, 2016 at 20:06
  • $\begingroup$ @Michael Renardy: For example, for $\Omega\subset R^n$ we consider the linear homogeneous diffusion equation $u_t=\Delta u$ defined in the region $\Omega\times (0, T)$ with the initial condition $u(x, 0)=f(x)$ and boundary condition $u|_{\partial\Omega}=g(x, t)$, $t>0$. $\endgroup$
    – daulomb
    Jun 2, 2016 at 21:42
  • $\begingroup$ I dont know whether contraction mapping theorem or any other fixed point theorems can be used.This is what I need. $\endgroup$
    – daulomb
    Jun 2, 2016 at 21:52

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