Existence of Variational Solutions in Noncylindrical Domains
Verena Bögelein, Frank Duzaar, Christoph Scheven, Thomas Singer · SIAM Journal on Mathematical Analysis · 2018
We study gradient flows of integral functionals in noncylindrical bounded domains $E\subset\mathbb{R}^n\times[0,T)$. The systems of differential equations take the form $\partial_t u -{div} D_\xi f(x,u,Du) = -D_u f(x,u,Du)\ {on $E$}$, for an integrand $f(x,u,Du)$ that is convex and coercive with respect to the $W^{1,p}$-norm for $p>1$. We prove the existence of variational solutions on noncylindrical domains under the only assumption that $\mathcal{L}^{n+1}(\partial E)=0$, even for functionals that do not admit a growth condition from above. For nondecreasing domains, the solutions are unique and admit a time-derivative in $L^2(E)$. For domains that decrease the most with bounded speed and integrands that satisfy a $p$-growth condition, we prove that the constructed solutions are continuous in time with respect to the $L^2$-norm and solve the above system of differential equations in the weak sense. Under the additional assumption that the domain also increases the most at finite speed, we establish the uniqueness of solutions.