Heat flow for \(p\)-harmonic maps with values in the circle
The authors investigate the initial-boundary value problem for the \(p\)-harmonic heat flow from the \(N\)-dimensional unit ball in \(\mathbb{R}^N\) into the unit circle \(S^1\subset \mathbb{R}^2\), \(p>1\). In explicit formulas, this problem asks for solutions \(u:[0,T]\times B^N \to S^1\) of the parabolic p. d. e. \[ \frac{\partial u}{\partial t}-\text{div}(|\nabla_x u|^{p-2}\nabla_x u)= |\nabla_x u|^p u \] with initial condition \(u(0, x) = u_0(x)\), \(x\in B^N\), and boundary condition \(u(t,x) =\Phi(x)\), \(t\geq 0\), \(x\in\partial B^N\). The authors prove the following result: given \(u_0\), \(\Phi\in W^{1,p}(B^N,S^1)\) with \(u_0 =\Phi\) on \(\partial B^N\), the above problem possesses a weak solution defined for all \(t > 0\) and such that \(\frac 1p\int|\nabla u(T,x)|^p dx+ \int^T_0 \int |\frac{\partial u}{\partial t}|^2dtdx\leq \frac 1p \int|\nabla u_0(x)|^p dx\). The authors show furthermore that, provided the initial datum \(u_0\) is weakly \(p\)-harmonic but not stationary with respect to variations of the independent variables, then the corresponding heat flow they construct is not constant in time. This shows that in this case the initial-boundary value problem has infinitely many solutions. The existence theorem is proved by a time-discretization scheme.
scientific article; zbMATH DE number 1512502