We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we prove that the solution becomes immediately differentiable with Holder continuous derivatives. Therefore, the solutions to the equation are classical.
We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form \begin{equation*} \displaystyle\int_Ω \langle \mathcal{A}(x,Du) ,D(φ-u) \rangle dx \geq 0 \qquad \forall φ\in \mathcal{K}_ψ(Ω), \end{equation*} where $Ω$ is a bounded open subset of $\mathbb{R}^n$, $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω)= \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the…
In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces en…
We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \begin{gather*} \min \biggl\{ \int_Ω F(x,w,Dw) d x \ : \ w \in \mathcal{K}_ψ(Ω) \biggr\}, \end{gather*} with $F$ double phase functional of the form \begin{equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end{equation*} where $Ω$ is a bounded open subset of $\mathbb{R}^n$, $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω)= \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the class of admissible functions. Assuming that…
We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form \begin{gather*} \min \biggl\{ \displaystyle\int_Ω F(x,Dw)dx \ : \ w \in \mathcal{K}_ψ(Ω) \biggr\}, \end{gather*} where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \geq 2$, the function $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω) := \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the class of admissible functions. If the obstacle $ψ$ is locally bounded, we prove that the gradient of soluti…
We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{ö}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.
We consider semilinear stochastic evolution equations on Hilbert spaces with multiplicative Wiener noise and linear drift term of the type $A + \varepsilon G$, with $A$ and $G$ maximal monotone operators and $\varepsilon$ a "small" parameter, and study the differentiability of mild solutions with respect to $\varepsilon$. The operator $G$ can be a singular perturbation of $A$, in the sense that its domain can be strictly contained in the domain of $A$.
We establish Ambrosetti -Prodi type results for periodic solutions of one -dimensional nonlinear problems with drift term and drift -less whose principal operator is the fractional Laplacian of order $s\in(0,1)$. We establish conditions for the existence and nonexistence of solutions. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also establish a priori bounds in order to get multiplicity results. We also prove that the solutions are $C^{1,α}$ under some regularity assumptions in the nonlinearities, that is, th…
In this paper, we analyze the $s$-dependence of the solution $u_s$ to the fractional Poisson equation $(-Δ)^s_Ω = f$ in an open bounded set $Ω\subset \mathbb{R}^N$. Precisely, we show that the solution map $(0,1)\to L^2(Ω)$, $s\mapsto u_s$ is continuously differentiable. Moreover, when $f = λ_s u_s$, we also analyze the one-sided differentiability of the first nontrivial eigenvalue of $(-Δ)^s_Ω$ regarded as a function of $s \in (0,1)$.
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-Δ)^s +b\cdot\nabla$, in the subcritical regime $s>\frac{1}{2}$. Our main result states that once the free boundary is $C^1$ then it is $C^\infty$, whenever $s\not\in\mathbb{Q}$. In order to achieve this, we establish a fine boundary expansion for solutions to linear nonlocal equations with drift in terms of the powers of distance function. Quite interestingly, due to the drift term, the powers do not increase by natural numbers and the fact that $s$ is irrational plays …
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
We study the heat kernel of the supercritical fractional diffusion equation with the drift in the critical Hölder space. We show that such a drift can have point irregularities strong enough to make the heat kernel vanish at a point for all $t>0$.
We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf(x, Dv(x))\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions $v\in u_0+W^{1, p}_0(Ω)$ such that $v\geψ$ a. e. in $Ω$, where $u_0\in W^{1,p}(Ω)$ is a fixed boundary datum. Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the gra…
We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.
We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as follows \begin{equation}\label{PP} (-\triangle_{g})^αu+g(u)=K(x)f(x,u),\ \ \text{in}\ \mathbb{R}^{d},\tag{P} \end{equation} where $d\geq 3,\ (-\triangle_{g})^α$ is the fractional Orlicz g-Laplace operator, $f:\mathbb{R}^d\times\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $K$ is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term $f$, we prove that the …
We study an initial value problem with fractional Laplacian and a singular drift term, and obtain local and global existence theorems similar to the results in Jourdain et al.(2005).
In this work we investigate the well-posedness for difussion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the Hörmander classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators, and the corres…
In this paper we prove a fractional analogue of the classical Korn's first inequality. The inequality makes it possible to show the equivalence of a function space of vector field characterized by a Gagliardo-type seminorm with 'projected difference' with that of a corresponding fractional Sobolev space. As an application, we will use it to obtain a Caccioppoli-type inequality for a nonlinear system of nonlocal equations, which in turn is a key ingredient in applying known results to prove a higher fractional differentiability result for weak solutions of the nonlinear system of nonlocal equat…
