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Register a new userLevel-set inequalities on fractional maximal distribution functions and applications to regularity theory
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The aim of this paper is to establish an abstract theory based on the so-called fractional-maximal distribution functions (FMDs). From the rough ideas introduced in~cite{AM2007}, we develop and prove some abstract results related to the level-set inequalities and norm-comparisons by using the language of such FMDs. Particularly interesting is the applicability of our approach that has been shown in regularity and Calderon-Zygmund type estimates. In this paper, due to our research experience, we will establish global regularity estimates for two types of general quasilinear problems (problems with divergence form and double obstacles), via fractional-maximal operators and FMDs. The range of applications of these abstract results is large. Apart from these two examples of the regularity theory for elliptic equations discussed, it is also promising to indicate further possible applications of our approach for other special topics.
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We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.
This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phase field theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow.
Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, $minbigl{(-Delta)^su,,u-varphibigr}=0$ in $mathbb R^n$, for general obstacles $varphi$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in cite{GP} to all $sin(0,1)$.
Let $(mathbb M, d,mu)$ be a metric measure space with upper and lower densities: $$ begin{cases} |||mu|||_{beta}:=sup_{(x,r)in mathbb Mtimes(0,infty)} mu(B(x,r))r^{-beta}<infty; |||mu|||_{beta^{star}}:=inf_{(x,r)in mathbb Mtimes(0,infty)} mu(B(x,r))r^{-beta^{star}}>0, end{cases} $$ where $beta, beta^{star}$ are two positive constants which are less than or equal to the Hausdorff dimension of $mathbb M$. Assume that $p_t(cdot,cdot)$ is a heat kernel on $mathbb M$ satisfying Gaussian upper estimates and $mathcal L$ is the generator of the semigroup associated with $p_t(cdot,cdot)$. In this paper, via a method independent of Fourier transform, we establish the decay estimates for the kernels of the fractional heat semigroup ${e^{-t mathcal{L}^{alpha}}}_{t>0}$ and the operators ${{mathcal{L}}^{theta/2} e^{-t mathcal{L}^{alpha}}}_{t>0}$, respectively. By these estimates, we obtain the regularity for the Cauchy problem of the fractional dissipative equation associated with $mathcal L$ on $(mathbb M, d,mu)$. Moreover, based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in $mathbb{M}times(0,infty)$, we also characterize a nonnegative Randon measure $ u$ on $mathbb Mtimes(0,infty)$ such that $R_alpha L^p(mathbb M)subseteq L^q(mathbb Mtimes(0,infty), u)$ under $(alpha,p,q)in (0,1)times(1,infty)times(1,infty)$, where $u=R_alpha f$ is the weak solution of the fractional diffusion equation $(partial_t+ mathcal{L}^alpha)u(t,x)=0$ in $mathbb Mtimes(0,infty)$ subject to $u(0,x)=f(x)$ in $mathbb M$.
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