We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms.
As an application, we use such a Hopf-type lemma in combination with the method of moving planes to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $Ω\subset \mathbb R^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $Ω$; in turn, $Ω$ must be a ball.
Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only `local' assumptions are imposed on the solutions.
We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even symmetric perturbations, up to multiplicative constants). In proving this result, one also establishes an odd symmetric version of the maximum principle for nonlocal minimal graphs, according to which the odd symmetric minimizer is positive in the direction of the positive bump and negative in the direction of the negative bump.
Symmetry via antisymmetric maximum principles in nonlocal problems of variable order
考虑开有界集$Ω\subset\mathbb{R}^{N}$中的非线性问题\[(P) \;\; I u=f(x,u) \text{ in $Ω$,} \;\; u=0 \text{ on $\mathbb{R}^{N}\setminusΩ$ }\]，其中$I$是一个非局部算子，它可以是各向异性的，并且可以具有不同的阶.对$I$、$Ω$和非线性方程$f$关于固定方向（如$x_1$）作了适度的对称性和单调性假设，证明了$(P)$的任何非负弱解$u$在$x_1$中是对称的.此外，我们有以下选择：要么是$Ω$中的$u\equiv 0$，要么是$|x_1|$中的$u$严格递减。证明依赖于新的最大值原理反对称上解的相关类线性问题。
We prove that for antisymmetric vectorfield $Ω$ with small $L^2$-norm there exists a gauge $A \in L^\infty \cap \dot{W}^{1/2,2}(\mathbb{R}^1,GL(N))$ such that ${\rm div}_{\frac12} (AΩ- d_{\frac{1}{2}} A) = 0$. This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.
In this paper, we first establish Hopf's lemmas for parabolic fractional equations and parabolic fractional $p$-equations. Then we derive an asymptotic Hopf's lemma for antisymmetric solutions to parabolic fractional equations. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.
In this note we prove an estimate on the level sets of a function with $(p, q)$ growth that depends on the difference quotient of a bounded weak solution to a nonlocal double phase equation. This estimate is related to a self improving property of these solutions.
In this paper, we study the asymptotic behavior as $\varepsilon\to0^+$ of solutions $u\_\varepsilon$ to the nonlocal stationary Fisher-KPP type equation$$ \frac{1}{\varepsilon^m}\int\_{\mathbb{R}^N}J\_\varepsilon(x-y)(u\_\varepsilon(y)-u\_\varepsilon(x))\mathrm{d}y+u\_\varepsilon(x)(a(x)-u\_\varepsilon(x))=0\text{ in }\mathbb{R}^N, $$where $\varepsilon>0$ and $0\leq m<2$. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution $u\_\varepsilon$ and that $u\_\varepsilon\to a^+$ as $\varepsilon\to0^+$ where $a^+=\max\{0,a\}$. T…
In the present work we deal with the existence of solutions for optimal control problems associated to transport equations. The behaviour of a population of individuals will be influenced by the presence of a population of control agents whose role is to lead the dynamics of the individuals towards a specific goal. The dynamics of the population of individuals is described by a suitable nonlocal transport equation, while the role of the population of agents is designed by the optimal control problem. This model has been first studied in [12] for a class of continuous nonlocal potentials, while…
We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and propose some open questions in the subject.
In this paper we provide a rate of convergence for periodic homogenization of Hamilton-Jacobi-Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where the convexity plays a crucial role. Such regularity estimates are possible from the available literature once we provide a representation formula for the effective Hamiltonian, a result that has an independent interest.
In this paper, we consider the dual fractional parabolic problem in the right half space. We prove that the positive solutions are strictly increasing in $x_1$ direction without assuming the solutions be bounded. So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative $\partial_t^α$ and the fractional Laplacian $(-Δ)^s$. To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several …
