arXiv subjects

Michael Struwe

Publications and source records attributed to Michael Struwe.

8 recordsLinked to original sources

arXiv2026

Flowing to free boundary minimal surfaces

We introduce a flow that is designed to flow maps $u:\Sigma\to \mathbb{R}^n$ which map the boundary of a general domain surface $\Sigma$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free boundary (branched) minimal immersion supported by $N$. In the case when $\Sigma$ is the unit disc $D$, this task can be achieved by means of the Plateau-flow introduced in the work [15] of the second author. When $\Sigma\neq D$, however, also the conformal type of the domain metric plays a role and it no longer suffices to deform the trace of the given map into a half-harmonic map as in [15]. In order to overcome this issue, here we combine ideas of the Plateau-flow from [15] with ideas of the Teichm\"uller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0\colon\partial \Sigma\to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial \Sigma$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion.

math.AP↗

arXiv2024

The prescribed curvature flow on the disc

For given functions $f$ and $j$ on the disc $B$ and its boundary $\partial B=S^1$, we study the existence of conformal metrics $g=e^{2u}g_0$ with prescribed Gauss curvature $K_g=f$ and boundary geodesic curvature $k_g=j$. Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz (2018), we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work Struwe (2005) on the prescribed curvature problem on the sphere, we are able to exhibit a $2$-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz (2021) by degree-theory.

math.AP↗

arXiv2022

Plateau flow or the heat flow for half-harmonic maps

Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target manifold $N$, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author's 1985 results for the harmonic map heat flow of surfaces and in similar generality. When $N$ is a smoothly embedded, oriented closed curve $\Gamma$ the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces.

math.DG↗

arXiv2015

Well-posedness of the supercritical Lane-Emden heat flow in Morrey spaces

For any smoothly bounded domain $Ω\subset\mathbb R^n$, $n\geq 3$, and any exponent $p>2^*=2n/(n-2)$ we study the Lane-Emden heat flow $u_t-Δu = |u|^{p-2}u$ on $Ω\times]0,\infty[$ and establish local and global well-posedness results for the initial value problem with suitably small initial data $u\big|_{t=0}=u_0$ in the Morrey space $L^{2,λ}(Ω)$, where $λ=4/(p-2)$. We contrast our results with results on instantaneous complete blow-up of the flow for certain large data in this space, similar to ill-posedness results of Galaktionov-Vazquez for the Lane-Emden flow on $\mathbb R^n$.

math.AP↗

arXiv2010

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $Ω\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(Ω)$ to the equation $(-Δ)^m u_k=λ_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<λ_k\to 0$. Assuming that $$Λ:=\lim_{k\to\infty}\int_Ωu_k(-Δ)^m u_k dx<\infty,$$ we prove that $Λ$ is an integer multiple of $Λ_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.

math.FA↗

arXiv2009

The heat flow with a critical exponential nonlinearity

We analyze the possible concentration behavior of heat flows related to the Moser-Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser-Trudinger energy in a supercritical regime.

math.AP↗

arXiv2006

Partial regularity for harmonic maps, and related problems

Via Gauge theory, we give a new proof of partial regularity for harmonic maps in dimension m>2 into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of "minimal" C^2 regularity. The proof we present moreover extends to a large class of elliptic systems of quadratic growth.

math.AP↗

arXiv1992

Semilinear wave equations

We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the $u^5$-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.

math.AP↗