Analysis of Partial Differential Equations (APDE)
http://hdl.handle.net/20.500.11824/1
2019-05-18T22:08:34ZOn extension problem, trace hardy and Hardy’s inequalities for some fractional Laplacians
http://hdl.handle.net/20.500.11824/973
On extension problem, trace hardy and Hardy’s inequalities for some fractional Laplacians
Boggarapu P.; Roncal L.; Thangavelu S.
We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.
2019-09-01T00:00:00ZBloom type upper bounds in the product BMO setting
http://hdl.handle.net/20.500.11824/965
Bloom type upper bounds in the product BMO setting
Li K.; Martikainen H.; Vuorinen E.
We prove some Bloom type estimates in the product BMO setting. More specifically,
for a bounded singular integral $T_n$ in $\mathbb R^n$ and a bounded singular integral $T_m$ in $\mathbb R^m$ we prove that
$$
\| [T_n^1, [b, T_m^2]] \|_{L^p(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}} \|b\|_{{\rm{BMO}}_{\rm{prod}}(\nu)},
$$
where $p \in (1,\infty)$, $\mu, \lambda \in A_p$ and $\nu := \mu^{1/p}\lambda^{-1/p}$ is the Bloom weight. Here $T_n^1$ is $T_n$ acting on the first variable,
$T_m^2$ is $T_m$ acting on the second variable, $A_p$ stands for the bi-parameter weights of $\mathbb R^n \times \mathbb R^m$ and
${\rm{BMO}}_{\rm{prod}}(\nu)$ is a weighted product BMO space.
2019-04-08T00:00:00ZConvex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
http://hdl.handle.net/20.500.11824/964
Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
Rüland, A.; Taylor J. M.; Zillinger, C.
We study convex integration solutions in the context of the modelling of shape-memory
alloys. The purpose of the article is twofold, treating both rigidity and flexibility prop-
erties: Firstly, we relate the maximal regularity of convex integration solutions to the
presence of lower bounds in variational models with surface energy. Hence, variational
models with surface energy could be viewed as a selection mechanism allowing for
or excluding convex integration solutions. Secondly, we present the first numerical
implementations of convex integration schemes for the model problem of the geomet-
rically linearised two-dimensional hexagonal-to-rhombic phase transformation. We
discuss and compare the two algorithms from Rüland et al. (J Elast. 2019. https://doi.
org/10.1007/s10659-018-09719-3; SIAM J Math Anal 50(4):3791–3841, 2018) and
give a numerical estimate of the regularity attained.
2019-03-30T00:00:00ZTopological singular set of vector-valued maps, I: application to manifold-constrained Sobolev and BV spaces
http://hdl.handle.net/20.500.11824/962
Topological singular set of vector-valued maps, I: application to manifold-constrained Sobolev and BV spaces
Canevari G.; Orlandi G.
We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$.
In particular, this operator is defined on maps that take values in a closed submanifold $\mathcal{N}$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian
in case $\mathcal{N}$ is a sphere. More precisely, the range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $\mathcal{N}$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with $\mathcal{N}$-well potentials.
2019-03-30T00:00:00Z