Finite time blow-up for the harmonic map heat flowSpringer Science and Business Media LLC - Tập 1 - Trang 231-236 - 1993
Joseph F. Grotowski
We consider the harmonic map heat flow from the three-dimensional ball to the two-sphere. We establish the existence of regular initial data leading to blow-up in finite time.
Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifoldsSpringer Science and Business Media LLC - Tập 35 - Trang 385-407 - 2008
Nicolas Saintier
We describe the asymptotic behaviour in Sobolev spaces of sequences of solutions of Paneitz-type equations [Eq. (E
α
) below] on a compact Riemannian manifold (M, g) which are invariant by a subgroup of the group of isometries of (M, g). We also prove pointwise estimates.
Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactnessSpringer Science and Business Media LLC - Tập 61 - Trang 1-29 - 2022
Michael Novack, Xiaodong Yan
We consider the 3D smectic energy
$$\begin{aligned} {\mathcal {E}}_{\epsilon }\left( u\right) =\frac{1}{2}\int _{\Omega }\frac{1}{\varepsilon } \left( \partial _{z}u-\frac{(\partial _{x}u)^{2}+(\partial _{y}u)^{2}}{2}\right) ^{2} +\varepsilon \left( \partial _{x}^{2}u+\partial _{y}^{2}u\right) ^{2}dx\,dy\,dz. \end{aligned}$$
...... hiện toàn bộ
Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functionsSpringer Science and Business Media LLC - Tập 63 - Trang 1-49 - 2024
Paz Hashash, Arkady Poliakovsky
In this paper we analyse functions in Besov spaces
$$B^{1/q}_{q,\infty }(\mathbb {R}^N,\mathbb {R}^d),q\in (1,\infty )$$
, and functions in fractional Sobolev spaces
$$W^{r,q}(\mathbb {R}^N,\mathbb {R}^d),r\in (0,1),q\in [1,\inft...... hiện toàn bộ
Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentialsSpringer Science and Business Media LLC - Tập 59 - Trang 1-38 - 2020
Lu Chen, Guozhen Lu, Maochun Zhu
In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in
$$H^{2}\left( {\mathbb {R}}^{4}\right) $$
which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solution...... hiện toàn bộ