Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBoundary regularity for minimizing biharmonic maps
72
0
0.0
(
0
)
Authors
Katarzyna Mazowiecka
Ask ChatGPT about the research
No Arabic abstract
We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Schevens interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong--Lamm--Wang.
rate research
Read More
We prove an $epsilon$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets. Instead, we follow the spirit of the last-named authors recent work on free boundary harmonic maps and choose a good frame directly at the free boundary. This leads to growth estimates, which, in the critical regime $p=n$, imply Holder regularity of solutions. In the supercritical regime, $p < n$, we combine the growth estimate with the geometric reflection argument: the reflected equation is super-critical, but, under the assumption of growth estimates, solutions are regular. In the case $p<n$, for stationary $p$-harmonic maps with free boundary, as a consequence of a monotonicity formula we obtain partial regularity up to the boundary away from a set of $(n-p)$-dimensional Hausdorff measure.
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $sin(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $sin(0,1)$ and the stationarity/minimality assumption.
In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $mgeq 3$, we show that minimizing $1/2$-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For $m=2$, we prove that, up to an orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous minimizing $1/2$-harmonic map from the plane into the circle $mathbb{S}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d domain into $mathbb{S}^1$ has a topological charge equal to $pm1$.
In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary.
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $pge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Holder regularity up to the boundary, that is, $u/d^sin C^alpha(overlineOmega)$ for some $alphain(0,1)$, $d$ being the distance from the boundary.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
