$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with nonlocal Dirichlet boundary conditions. The typical problem I might write down is for example
$$
\begin{cases}
\partial_t u=\Delta u & \mbox{in }(0,T)\times\Omega\\
u\rvert_{\partial\Omega}=\int_\Omega u\, \mathrm d x & \mbox{on }(0,T)\times\partial\Omega\\
u\rvert_{t=0}=u_0 & \mbox{in }\Omega.
\end{cases}
$$
Here $\Omega\subset \R^d$ is as usual your favourite smooth, bounded, open set.
This looks like a standard Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ except that $g=\int_\Omega u$ depends nonlocally on the values of $u$ inside the domain at any given time $t>0$.
Has anyone seen anything vaguely related to this kind of models, and if so can anyone provide references? Any help is much appreciated!
I am sorry I cannot really provide more details or a more focussed problem.
This popped up in my research as a very secondary question, and as usual I would rather avoid reinventing the wheel if possible. And a quick bibliographical search did not provide much useful information (except for a few exotic papers in dimension $d=1$ only).
Best Answer
I keep the reduction of @Andrè Schlicting, assume $|\Omega|=1$ and consider the operator $(I-P)\Delta$ in $L^2_0=\{u \in L^2(\Omega),\ \int_\Omega u=0\}$, where $Pu=\int_{\Omega} u$.
Assuming regularity for $\Omega$, then $\Delta$ has domain $H^2 \cap H^1_0$ so that the domain of $(I-P)\Delta$ in $L^2_0$ is $H^2\cap H^1_0 \cap L^2_0$.
Let me also consider the usual inner product $a(u,v)=\int_{\Omega} \nabla u \cdot \nabla v$ which is equivalent to the inner product of $H^1_0$, by Poincare inequality.
It is immediate to see that, if $u \in H^2\cap H^1_0 \cap L^2_0$ and $(I-P)u=-f\in L^2_0$, then $a(u,v)=\int_\Omega f v$ for every $v \in H^1_0 \cap L^2_0$.
Conversely, let $u \in H^1_0 \cap L^2_0$ satisfy $a(u,v)=\int_\Omega f v$ for every $v \in H^1_0 \cap L^2_0$, with $f \in L^2_0$. Let me show that $u \in H^2$ and $(I-P) \Delta u=0$. Let $w \in H^1_0$ and $\phi$ smooth with a compact support and mass 1. Then $v=w-c\phi \in H^1_0 \cap L^2_0\ $, $c=\int_\Omega w$. Therefore $$ \int_\Omega \nabla u \cdot (\nabla w-c\nabla\phi)= \int_\Omega f(w-c\phi) $$ or $$ \int_\Omega \nabla u \cdot \nabla w= \int_\Omega w(f-a) $$ with $a=\int_\Omega (\nabla u \nabla \phi-f \phi)$. By elliptic regularity, since $w \in H^1_0$ is arbitrary, $u \in H^2$. Integrating by parts we then obtain $$ -\int_\Omega \Delta u ( w-c\phi)= \int_\Omega f(w-c\phi) $$ Take now a sequence $\phi_k \to 1$ each with mass 1 (and all dominated by a fixed constant). Letting $k \to \infty$ above we get, since $f$ has zero mean,
$$ -\int_\Omega \Delta u (w-c)= \int_\Omega f(w-c)=\int_\Omega fw $$ which says that $(I-P)\Delta u=-f$.
Therefore the operator $(I-P)\Delta$ is associated to the form $a$ in $L^2_0$ and then is self-adjoint in $L^2_0$ and the solvability of the parabolic problem follows.