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Francesca Da Lio

Publications and source records attributed to Francesca Da Lio.

31 records · Page 2Linked to original sources
arXiv2016

Horizontal $α$-Harmonic Maps

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $α$-harmonic maps}, $α\ge 1/2$, with respect to such a distribution. These are maps $u\in H^α({\mathbb R}^k,{\mathbb R}^m)$ satisfying $P_T\nabla u=\nabla u$ and $P_T(u)(-Δ)^αu=0$ in ${\mathcal D}'({\mathbb R}^k).$ If the distribution of planes is integrable then we recover the classical case of $α$-harmonic maps with values into a manifold. In this paper we shall focus our attention to the case $α=1/2$ in dimension $1$ and $α=2$ in dimension $2$ and we investigate the regularity of the {\em horizontal $α$-harmonic maps}. In both cases we show that such maps satisfy a Schrödinger type system with an antisymmetric potential, that permits us to apply the previous results obtained by the authors. Finally we study the regularity of {\em variational $α$-harmonic} maps which are critical points of $\|(-Δ)^{α/2} u\|^2_{L^2}$ under the constraint to be tangent (horizontal) to a given planes distribution. We produce a convexification of this variational problem which permits to write it's Euler Lagrange equations.

math.AP↗
arXiv2015

Blow-up analysis of a nonlocal Liouville-type equation

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\mathbb{R}$ \begin{equation} (-Δ)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.

math.DG↗
arXiv2012

Compactness and Bubbles Analysis for 1/2-harmonic Maps

In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps $u_k\colon\R\to {\cal{S}}^{m-1}$ such that $|u_k|_{\dot H^{1/2}(\R,{\cal{S}}^{m-1})}\le C.$ More precisely we show that there exist a weak 1/2-harmonic map $u_\infty\colon\R\to {\cal{S}}^{m-1}$, a possible empty set ${a_1,...,a_\ell}$ in $\R$ such that up to subsequences $$(|(-Δ)^{1/4}u_k|^2 \rightharpoonup |(-Δ)^{1/4}u_{\infty}|^2)dx+\sum_{i=1}^{\ell}λ_i δ_{a_i}, in Radon measure,$$ as $k\to +\infty$, with $λ_i\ge 0.$ The convergence of $u_k$ to $u_\infty$ is strong in $\dot W^{1/2,p}_{loc}(\R\setminus{a_1,...,a_\ell})$, for every $p\ge 1.$ We quantify the loss of energy in the weak convergence and we show that in the case of non-constant 1/2-harmonic maps with values in $ {\cal{S}}^2\,$ one has $λ_i=2 πn_i$, with $n_i$ a positive integer.

math.AP↗
arXiv2012

n/p-Harmonic maps: regularity for the sphere case

We introduce n/p-harmonic maps as critical points of E(v) the Lp-Norm of the alpha-laplacian of v, where pointwise v maps Rn into a sphere, and alpha = n/p. This energy combines the non-local behaviour of the fractional harmonic maps introduced by Riviere and the first author with the degenerate arguments of the n-laplacian. In this setting, we will prove Holder continuity.

math.AP↗
arXiv2010

Fractional Harmonic Maps into Manifolds in odd dimension n>1

In this paper we consider critical points of the following nonlocal energy {equation} {\cal{L}}_n(u)=\int_{\R^n}| ({-Δ})^{n/4} u(x)|^2 dx\,, {equation} where $u\colon H^{n/2}(\R^n)\to{\cal{N}}\,$ ${\cal{N}}\subset\R^m$ is a compact $k$ dimensional smooth manifold without boundary and $n>1$ is an odd integer. Such critical points are called $n/2$-harmonic maps into ${\cal{N}}$. We prove that $Δ^{n/2} u\in L^p_{loc}(\R^n)$ for every $p\ge 1$ and thus $u\in C^{0,α}_{loc}(\R^n)\,.$ The local Hölder continuity of $n/2$-harmonic maps is based on regularity results obtained in \cite{DL1} for nonlocal Schrödinger systems with an antisymmetric potential and on suitable {\it 3-terms commutators} estimates.

math.AP↗
arXiv2010

Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps

We consider nonlocal linear Schrödinger-type critical systems of the type \begin{equation}\label{eqabstr} Δ^{1/4} v=Ω\, v~~~\mbox{in $\R\,.$} \ \end{equation} where $Ω$ is antisymmetric potential in $L^2(\R,so(m))$, $v$ is a ${\R}^m$ valued map and $Ω\, v$ denotes the matrix multiplication. We show that every solution $v\in L^2(\R,\R^m)$ of \rec{eqabstr} is in fact in $L^p_{loc}(\R,\R^m)$, for every $2\le p<+\infty$, in other words, we prove that the system (\ref{eqabstr}) which is a-priori only critical in $L^2$ happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the $C^{0,α}_{loc}$ regularity of weak $1/2$-harmonic maps into $C^2$ compact manifold without boundary.

math.AP↗
arXiv2010

Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications

In this paper, we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.

math.AP↗
arXiv2009

Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions

We study periodic homogenization problems for second-order pde in half-space type domains with Neumann boundary conditions. In particular, we are interested in "singular problems" for which it is necessary to determine both the homogenized equation and boundary conditions. We provide new results for fully nonlinear equations and boundary conditions. Our results extend previous work of Tanaka in the linear, periodic setting in half-spaces parallel to the axes of the periodicity, and of Arisawa in a rather restrictive nonlinear periodic framework. The key step in our analysis is the study of associated ergodic problems in domains with similar structure.

math.AP↗
arXiv2009

Local $C^{0,α}$ Estimates for Viscosity Solutions of Neumann-type Boundary Value Problems

In this article, we prove the local $C^{0,α}$ regularity and provide $C^{0,α}$ estimates for viscosity solutions of fully nonlinear, possibly degenerate, elliptic equations associated to linear or nonlinear Neumann type boundary conditions. The interest of these results comes from the fact that they are indeed regularity results (and not only a priori estimates), from the generality of the equations and boundary conditions we are able to handle and the possible degeneracy of the equations we are able to take in account in the case of linear boundary conditions.

math.AP↗
arXiv2009

3-Commutators Estimates and the Regularity of 1/2 Harmonic Maps into Spheres

We prove the regularity of weak 1/2-harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2-harmonic map equation in the form of a non-local linear Schrödinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.

math.AP↗
arXiv2008

Uniqueness results for convex Hamilton-Jacobi equations under $p>1$ growth conditions on data

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like $o(1+|x|^p)$ at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like $O(1+|x|^p)$ at infinity. This latter case encompasses some equations related to backward stochastic differential equations.

math.AP↗