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Finite time blow-up for the harmonic map heat flow
Springer 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.
Parabolic De Giorgi classes of order p and the Harnack inequality
Springer Science and Business Media LLC - Tập 26 - Trang 379-399 - 2006
Ugo Gianazza, Vincenzo Vespri
We define suitable homogeneous parabolic De Giorgi classes of order p and show that in this context a Harnack inequality holds with a time gap of order ρ p .
Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifolds
Springer 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.
Solutions of Aronsson equation near isolated points
Springer Science and Business Media LLC - - 2010
Vesa Julin
Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactness
Springer 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 functions
Springer 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 $$W^{r,q}(\mathbb {R}^N,\mathbb {R}^d),r\in (0,1),q\in [1,\inft...... hiện toàn bộ
$$C^{2,\alpha }$$ C 2 , α estimates for nonlinear elliptic equations of twisted type
Springer Science and Business Media LLC - Tập 55 Số 1 - 2016
Tristan C. Collins
$$L^\infty $$-estimates in optimal transport for non quadratic costs
Springer Science and Business Media LLC - - 2022
Cristian E. Gutiérrez, Annamaria Montanari
Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials
Springer Science and Business Media LLC - Tập 59 - Trang 1-38 - 2020
Lu Chen, Guozhen Lu, Maochun Zhu
In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in $$H^{2}\left( {\mathbb {R}}^{4}\right) $$ which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solution...... hiện toàn bộ
Least energy solutions to a cooperative system of Schrödinger equations with prescribed $$L^2$$ -bounds: at least $$L^2$$ -critical growth
Springer Science and Business Media LLC - Tập 61 - Trang 1-31 - 2021
Jarosław Mederski, Jacopo Schino
We look for least energy solutions to the cooperative systems of coupled Schrödinger equations $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u)\quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3,\\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \end{array...... hiện toàn bộ
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