# Tranforming boundary value problem (heat equation) to one with homogenous boundary condition

Let $\Omega \subseteq \mathbb R^n$ be a bounded region with smooth boundary, let $u$ be two times continously differentiable in $\Omega \times (0,\infty)$ and in $\overline \Omega \times [0,\infty)$ let it be a continous solution of the heat equation $$u_t - \Delta u = 0 \mbox{ in } \Omega \times (0,\infty) \tag{1}$$ with the boundary conditions $$u = g \mbox{ on } \Omega \times \{ 0 \} \mbox{ and } u = f \mbox{ on } \partial \Omega \times (0,\infty)$$ with given functions $g \in C(\overline \Omega)$ and $f \in C(\partial \Omega)$, where $f(x) = g(x)$ for all $x \in \partial \Omega$.

1) Show that (1) could be reduced, with the help of the solution of an appropriate Dirichlet-Problem, to a problem (1) with homogenous boundary conditions ($f = 0$).

I have no idea how to show that, any help?

• Let$v=u-f$, and you will find that $v$ satisfy a heat function with homogeneous boundary condition. – Lion Jun 5 '14 at 14:42
• Formatting remark: use $$... \tag{1}$$ to number formulas. – user147263 Jun 9 '14 at 3:15

## 1 Answer

with the help of the solution of an appropriate Dirichlet-Problem

The hint suggests: solve the Dirichlet problem for the Laplace operator with the boundary values $f$. That is, find the harmonic function $w$ on $\Omega$ that is continuous on the closure and satisfies $w=f$ on $\partial \Omega$. Note that $w$ does not involve time.

Introduce the function $v = u-w$ (which resembles Lion's suggestion). Observe that $v$ also satisfies the heat equation. The boundary condition for $v$ is homogeneous.