# Are there existence results for the heat equation on unbounded Lipschitz domain?

I am looking for a reference/ ideas on the following problem. Let $\Omega\subset\Bbb R^2$ be a Lipschitz domain (if it helps, the domain can be piecewise smooth with only one "kink", for example the domain $\Omega=\{(x,y)\in\Bbb R^2:-\infty<x<y,y>\mu\}$ for some $\mu>0$), and consider for some $0<T<\infty$, $$\partial_tu-\Delta u=f\quad\text{in}\quad\Omega\times(0,T)$$ $$u|_{\partial\Omega\times(0,T)}=g$$ $$u|_{t=0}=h$$ where $f,g,h$ are sufficiently smooth etc.

I am also interested in the same problem with the second boundary condition $\beta(x,t)\cdot\nabla u+\alpha(x,t)u=g$ on $\partial\Omega\times(0,T)$, as well as the mixed problem, which has a Dirichlet boundary condition on $\{y=\mu\}$, and second boundary condition on the line $\{x=y\}$, the latter is for the case of the specific $\Omega$ given above.

I have seen existence uniqueness and regularity results for the non mixed problem (but for either type of boundary condition) for bounded Lipschitz domains, and unbounded sufficiently smooth domains, so really I am looking at a unison here i.e., unbounded and not sufficiently smooth.

Any help here would be greatly appreciated.

• Have you looked at the rectangular case at all? What have you done so far to address this problem? – DisintegratingByParts Jul 17 '15 at 16:46
• I've focused mainly on the specific domain I mentioned, I have tried altering the proof in ladyzhenskayas book, which assumes $C^2$ boundary, in the proof they construct an operator that is written (non trivially) as a sum of local coordinate constant coefficient operators. I wanted to try to construct a sequence of operators defined for a smoothed version of the domain, and hoped to take a limit to produce an operator, but so far no luck. – Ellya Jul 17 '15 at 17:08
• The aim being the at the operator $R$ that is given defines the inverse to the operator that defines the pde (and BC's), this operator ($R$) is bounded and thus we get existence and uniqueness – Ellya Jul 17 '15 at 17:12
• You could try conformally mapping this region to a rectangular one to see what you get. Might be interesting. – DisintegratingByParts Jul 17 '15 at 17:12
• An infinite rectangle or finite? – Ellya Jul 17 '15 at 17:13